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Introduction-to-Data-Mining-2nd-Edition-by-Pang-Ning-Tan.pdf1.pdf
INTRODUCTION TO DATA MINING
INTRODUCTION TO DATA MINING SECOND EDITION
PANG-NING TAN
Michigan State Universit
MICHAEL STEINBACH
University of Minnesota
ANUJ KARPATNE
University of Minnesota
VIPIN KUMAR
University of Minnesota
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Library of Congress Cataloging-in-Publication Data on File
Names: Tan, Pang-Ning, author. | Steinbach, Michael, author. | Karpatne, Anuj, author. | Kumar, Vipin, 1956- author.
Title: Introduction to Data Mining / Pang-Ning Tan, Michigan State University, Michael Steinbach, University of Minnesota, Anuj Karpatne, University of Minnesota, Vipin Kumar, University of Minnesota.
Description: Second edition. | New York, NY : Pearson Education, [2019] | Includes bibliographical references and index.
Identifiers: LCCN 2017048641 | ISBN 9780133128901 | ISBN 0133128903
Subjects: LCSH: Data mining.
Classification: LCC QA76.9.D343 T35 2019 | DDC 006.3/12–dc23 LC record available at https://lccn.loc.gov/2017048641
1 18
ISBN-10: 0133128903
ISBN-13: 9780133128901
To our families …
Preface to the Second Edition Since the first edition, roughly 12 years ago, much has changed in the field of data analysis. The volume and variety of data being collected continues to increase, as has the rate (velocity) at which it is being collected and used to make decisions. Indeed, the term, Big Data, has been used to refer to the massive and diverse data sets now available. In addition, the term data science has been coined to describe an emerging area that applies tools and techniques from various fields, such as data mining, machine learning, statistics, and many others, to extract actionable insights from data, often big data.
The growth in data has created numerous opportunities for all areas of data analysis. The most dramatic developments have been in the area of predictive modeling, across a wide range of application domains. For instance, recent advances in neural networks, known as deep learning, have shown impressive results in a number of challenging areas, such as image classification, speech recognition, as well as text categorization and understanding. While not as dramatic, other areas, e.g., clustering, association analysis, and anomaly detection have also continued to advance. This new edition is in response to those advances.
Overview As with the first edition, the second edition of the book provides a comprehensive introduction to data mining and is designed to be accessible and useful to students, instructors, researchers, and professionals. Areas covered include data preprocessing, predictive modeling, association analysis, cluster analysis, anomaly detection, and avoiding false discoveries. The goal is to present fundamental concepts and algorithms for each topic, thus providing the reader with the necessary background for the application of data mining to real problems. As before, classification, association analysis and cluster analysis, are each covered in a pair of chapters. The introductory
chapter covers basic concepts, representative algorithms, and evaluation techniques, while the more following chapter discusses advanced concepts and algorithms. As before, our objective is to provide the reader with a sound understanding of the foundations of data mining, while still covering many important advanced topics. Because of this approach, the book is useful both as a learning tool and as a reference.
To help readers better understand the concepts that have been presented, we provide an extensive set of examples, figures, and exercises. The solutions to the original exercises, which are already circulating on the web, will be made public. The exercises are mostly unchanged from the last edition, with the exception of new exercises in the chapter on avoiding false discoveries. New exercises for the other chapters and their solutions will be available to instructors via the web. Bibliographic notes are included at the end of each chapter for readers who are interested in more advanced topics, historically important papers, and recent trends. These have also been significantly updated. The book also contains a comprehensive subject and author index.
What is New in the Second Edition? Some of the most significant improvements in the text have been in the two chapters on classification. The introductory chapter uses the decision tree classifier for illustration, but the discussion on many topics—those that apply across all classification approaches—has been greatly expanded and clarified, including topics such as overfitting, underfitting, the impact of training size, model complexity, model selection, and common pitfalls in model evaluation. Almost every section of the advanced classification chapter has been significantly updated. The material on Bayesian networks, support vector machines, and artificial neural networks has been significantly expanded. We have added a separate section on deep networks to address the current developments in this area. The discussion of evaluation, which occurs in the section on imbalanced classes, has also been updated and improved.
The changes in association analysis are more localized. We have completely reworked the section on the evaluation of association patterns (introductory chapter), as well as the sections on sequence and graph mining (advanced
chapter). Changes to cluster analysis are also localized. The introductory chapter added the K-means initialization technique and an updated the discussion of cluster evaluation. The advanced clustering chapter adds a new section on spectral graph clustering. Anomaly detection has been greatly revised and expanded. Existing approaches—statistical, nearest neighbor/density-based, and clustering based—have been retained and updated, while new approaches have been added: reconstruction-based, one- class classification, and information-theoretic. The reconstruction-based approach is illustrated using autoencoder networks that are part of the deep learning paradigm. The data chapter has been updated to include discussions of mutual information and kernel-based techniques.
The last chapter, which discusses how to avoid false discoveries and produce valid results, is completely new, and is novel among other contemporary textbooks on data mining. It supplements the discussions in the other chapters with a discussion of the statistical concepts (statistical significance, p-values, false discovery rate, permutation testing, etc.) relevant to avoiding spurious results, and then illustrates these concepts in the context of data mining techniques. This chapter addresses the increasing concern over the validity and reproducibility of results obtained from data analysis. The addition of this last chapter is a recognition of the importance of this topic and an acknowledgment that a deeper understanding of this area is needed for those analyzing data.
The data exploration chapter has been deleted, as have the appendices, from the print edition of the book, but will remain available on the web. A new appendix provides a brief discussion of scalability in the context of big data.
To the Instructor As a textbook, this book is suitable for a wide range of students at the advanced undergraduate or graduate level. Since students come to this subject with diverse backgrounds that may not include extensive knowledge of statistics or databases, our book requires minimal prerequisites. No database knowledge is needed, and we assume only a modest background in statistics or mathematics, although such a background will make for easier going in
some sections. As before, the book, and more specifically, the chapters covering major data mining topics, are designed to be as self-contained as possible. Thus, the order in which topics can be covered is quite flexible. The core material is covered in chapters 2 (data), 3 (classification), 5 (association analysis), 7 (clustering), and 9 (anomaly detection). We recommend at least a cursory coverage of Chapter 10 (Avoiding False Discoveries) to instill in students some caution when interpreting the results of their data analysis. Although the introductory data chapter (2) should be covered first, the basic classification (3), association analysis (5), and clustering chapters (7), can be covered in any order. Because of the relationship of anomaly detection (9) to classification (3) and clustering (7), these chapters should precede Chapter 9. Various topics can be selected from the advanced classification, association analysis, and clustering chapters (4, 6, and 8, respectively) to fit the schedule and interests of the instructor and students. We also advise that the lectures be augmented by projects or practical exercises in data mining. Although they are time consuming, such hands-on assignments greatly enhance the value of the course.
Support Materials Support materials available to all readers of this book are available at http://www-users.cs.umn.edu/~kumar/dmbook.
PowerPoint lecture slides
Suggestions for student projects
Data mining resources, such as algorithms and data sets
Online tutorials that give step-by-step examples for selected data mining techniques described in the book using actual data sets and data analysis software
Additional support materials, including solutions to exercises, are available only to instructors adopting this textbook for classroom use. The book’s resources will be mirrored at www.pearsonhighered.com/cs-resources.
Comments and suggestions, as well as reports of errors, can be sent to the authors through [email protected].
Acknowledgments Many people contributed to the first and second editions of the book. We begin by acknowledging our families to whom this book is dedicated. Without their patience and support, this project would have been impossible.
We would like to thank the current and former students of our data mining groups at the University of Minnesota and Michigan State for their contributions. Eui-Hong (Sam) Han and Mahesh Joshi helped with the initial data mining classes. Some of the exercises and presentation slides that they created can be found in the book and its accompanying slides. Students in our data mining groups who provided comments on drafts of the book or who contributed in other ways include Shyam Boriah, Haibin Cheng, Varun Chandola, Eric Eilertson, Levent Ertöz, Jing Gao, Rohit Gupta, Sridhar Iyer, Jung-Eun Lee, Benjamin Mayer, Aysel Ozgur, Uygar Oztekin, Gaurav Pandey, Kashif Riaz, Jerry Scripps, Gyorgy Simon, Hui Xiong, Jieping Ye, and Pusheng Zhang. We would also like to thank the students of our data mining classes at the University of Minnesota and Michigan State University who worked with early drafts of the book and provided invaluable feedback. We specifically note the helpful suggestions of Bernardo Craemer, Arifin Ruslim, Jamshid Vayghan, and Yu Wei.
Joydeep Ghosh (University of Texas) and Sanjay Ranka (University of Florida) class tested early versions of the book. We also received many useful suggestions directly from the following UT students: Pankaj Adhikari, Rajiv Bhatia, Frederic Bosche, Arindam Chakraborty, Meghana Deodhar, Chris Everson, David Gardner, Saad Godil, Todd Hay, Clint Jones, Ajay Joshi, Joonsoo Lee, Yue Luo, Anuj Nanavati, Tyler Olsen, Sunyoung Park, Aashish Phansalkar, Geoff Prewett, Michael Ryoo, Daryl Shannon, and Mei Yang.
Ronald Kostoff (ONR) read an early version of the clustering chapter and offered numerous suggestions. George Karypis provided invaluable LATEX assistance in creating an author index. Irene Moulitsas also provided
assistance with LATEX and reviewed some of the appendices. Musetta Steinbach was very helpful in finding errors in the figures.
We would like to acknowledge our colleagues at the University of Minnesota and Michigan State who have helped create a positive environment for data mining research. They include Arindam Banerjee, Dan Boley, Joyce Chai, Anil Jain, Ravi Janardan, Rong Jin, George Karypis, Claudia Neuhauser, Haesun Park, William F. Punch, György Simon, Shashi Shekhar, and Jaideep Srivastava. The collaborators on our many data mining projects, who also have our gratitude, include Ramesh Agrawal, Maneesh Bhargava, Steve Cannon, Alok Choudhary, Imme Ebert-Uphoff, Auroop Ganguly, Piet C. de Groen, Fran Hill, Yongdae Kim, Steve Klooster, Kerry Long, Nihar Mahapatra, Rama Nemani, Nikunj Oza, Chris Potter, Lisiane Pruinelli, Nagiza Samatova, Jonathan Shapiro, Kevin Silverstein, Brian Van Ness, Bonnie Westra, Nevin Young, and Zhi-Li Zhang.
The departments of Computer Science and Engineering at the University of Minnesota and Michigan State University provided computing resources and a supportive environment for this project. ARDA, ARL, ARO, DOE, NASA, NOAA, and NSF provided research support for Pang-Ning Tan, Michael Stein-bach, Anuj Karpatne, and Vipin Kumar. In particular, Kamal Abdali, Mitra Basu, Dick Brackney, Jagdish Chandra, Joe Coughlan, Michael Coyle, Stephen Davis, Frederica Darema, Richard Hirsch, Chandrika Kamath, Tsengdar Lee, Raju Namburu, N. Radhakrishnan, James Sidoran, Sylvia Spengler, Bhavani Thuraisingham, Walt Tiernin, Maria Zemankova, Aidong Zhang, and Xiaodong Zhang have been supportive of our research in data mining and high-performance computing.
It was a pleasure working with the helpful staff at Pearson Education. In particular, we would like to thank Matt Goldstein, Kathy Smith, Carole Snyder, and Joyce Wells. We would also like to thank George Nichols, who helped with the art work and Paul Anagnostopoulos, who provided LATEX support.
We are grateful to the following Pearson reviewers: Leman Akoglu (Carnegie Mellon University), Chien-Chung Chan (University of Akron), Zhengxin Chen (University of Nebraska at Omaha), Chris Clifton (Purdue University), Joy-deep Ghosh (University of Texas, Austin), Nazli Goharian (Illinois
Institute of Technology), J. Michael Hardin (University of Alabama), Jingrui He (Arizona State University), James Hearne (Western Washington University), Hillol Kargupta (University of Maryland, Baltimore County and Agnik, LLC), Eamonn Keogh (University of California-Riverside), Bing Liu (University of Illinois at Chicago), Mariofanna Milanova (University of Arkansas at Little Rock), Srinivasan Parthasarathy (Ohio State University), Zbigniew W. Ras (University of North Carolina at Charlotte), Xintao Wu (University of North Carolina at Charlotte), and Mohammed J. Zaki (Rensselaer Polytechnic Institute).
Over the years since the first edition, we have also received numerous comments from readers and students who have pointed out typos and various other issues. We are unable to mention these individuals by name, but their input is much appreciated and has been taken into account for the second edition.
Contents 1. Preface to the Second Edition v
1. 1 Introduction 1
1. 1.1 What Is Data Mining? 4
2. 1.2 Motivating Challenges 5
3. 1.3 The Origins of Data Mining 7
4. 1.4 Data Mining Tasks 9
5. 1.5 Scope and Organization of the Book 13
6. 1.6 Bibliographic Notes 15
1. 1.7 Exercises 21
2. 2 Data 23
1. 2.1 Types of Data 26
1. 2.1.1 Attributes and Measurement 27
2. 2.1.2 Types of Data Sets 34
2. 2.2 Data Quality 42
1. 2.2.1 Measurement and Data Collection Issues 42
2. 2.2.2 Issues Related to Applications 49
3. 2.3 Data Preprocessing 50
1. 2.3.1 Aggregation 51
2. 2.3.2 Sampling 52
3. 2.3.3 Dimensionality Reduction 56
4. 2.3.4 Feature Subset Selection 58
5. 2.3.5 Feature Creation 61
6. 2.3.6 Discretization and Binarization 63
7. 2.3.7 Variable Transformation 69
4. 2.4 Measures of Similarity and Dissimilarity 71
1. 2.4.1 Basics 72
2. 2.4.2 Similarity and Dissimilarity between Simple Attributes 74
3. 2.4.3 Dissimilarities between Data Objects 76
4. 2.4.4 Similarities between Data Objects 78
5. 2.4.5 Examples of Proximity Measures 79
6. 2.4.6 Mutual Information 88
7. 2.4.7 Kernel Functions* 90
8. 2.4.8 Bregman Divergence* 94
9. 2.4.9 Issues in Proximity Calculation 96
10. 2.4.10 Selecting the Right Proximity Measure 98
5. 2.5 Bibliographic Notes 100
1. 2.6 Exercises 105
3. 3 Classification: Basic Concepts and Techniques 113
1. 3.1 Basic Concepts 114
2. 3.2 General Framework for Classification 117
3. 3.3 Decision Tree Classifier 119
1. 3.3.1 A Basic Algorithm to Build a Decision Tree 121
2. 3.3.2 Methods for Expressing Attribute Test Conditions 124
3. 3.3.3 Measures for Selecting an Attribute Test Condition 127
4. 3.3.4 Algorithm for Decision Tree Induction 136
5. 3.3.5 Example Application: Web Robot Detection 138
6. 3.3.6 Characteristics of Decision Tree Classifiers 140
4. 3.4 Model Overfitting 147
1. 3.4.1 Reasons for Model Overfitting 149
5. 3.5 Model Selection 156
1. 3.5.1 Using a Validation Set 156
2. 3.5.2 Incorporating Model Complexity 157
3. 3.5.3 Estimating Statistical Bounds 162
4. 3.5.4 Model Selection for Decision Trees 162
6. 3.6 Model Evaluation 164
1. 3.6.1 Holdout Method 165
2. 3.6.2 Cross-Validation 165
7. 3.7 Presence of Hyper-parameters 168
1. 3.7.1 Hyper-parameter Selection 168
2. 3.7.2 Nested Cross-Validation 170
8. 3.8 Pitfalls of Model Selection and Evaluation 172
1. 3.8.1 Overlap between Training and Test Sets 172
2. 3.8.2 Use of Validation Error as Generalization Error 172
9. 3.9 Model Comparison* 173
1. 3.9.1 Estimating the Confidence Interval for Accuracy 174
2. 3.9.2 Comparing the Performance of Two Models 175
10. 3.10 Bibliographic Notes 176
1. 3.11 Exercises 185
4. 4 Classification: Alternative Techniques 193
1. 4.1 Types of Classifiers 193
2. 4.2 Rule-Based Classifier 195
1. 4.2.1 How a Rule-Based Classifier Works 197
2. 4.2.2 Properties of a Rule Set 198
3. 4.2.3 Direct Methods for Rule Extraction 199
4. 4.2.4 Indirect Methods for Rule Extraction 204
5. 4.2.5 Characteristics of Rule-Based Classifiers 206
3. 4.3 Nearest Neighbor Classifiers 208
1. 4.3.1 Algorithm 209
2. 4.3.2 Characteristics of Nearest Neighbor Classifiers 210
4. 4.4 Naïve Bayes Classifier 212
1. 4.4.1 Basics of Probability Theory 213
2. 4.4.2 Naïve Bayes Assumption 218
5. 4.5 Bayesian Networks 227
1. 4.5.1 Graphical Representation 227
2. 4.5.2 Inference and Learning 233
3. 4.5.3 Characteristics of Bayesian Networks 242
6. 4.6 Logistic Regression 243
1. 4.6.1 Logistic Regression as a Generalized Linear Model 244
2. 4.6.2 Learning Model Parameters 245
3. 4.6.3 Characteristics of Logistic Regression 248
7. 4.7 Artificial Neural Network (ANN) 249
1. 4.7.1 Perceptron 250
2. 4.7.2 Multi-layer Neural Network 254
3. 4.7.3 Characteristics of ANN 261
8. 4.8 Deep Learning 262
1. 4.8.1 Using Synergistic Loss Functions 263
2. 4.8.2 Using Responsive Activation Functions 266
3. 4.8.3 Regularization 268
4. 4.8.4 Initialization of Model Parameters 271
5. 4.8.5 Characteristics of Deep Learning 275
9. 4.9 Support Vector Machine (SVM) 276
1. 4.9.1 Margin of a Separating Hyperplane 276
2. 4.9.2 Linear SVM 278
3. 4.9.3 Soft-margin SVM 284
4. 4.9.4 Nonlinear SVM 290
5. 4.9.5 Characteristics of SVM 294
10. 4.10 Ensemble Methods 296
1. 4.10.1 Rationale for Ensemble Method 297
2. 4.10.2 Methods for Constructing an Ensemble Classifier 297
3. 4.10.3 Bias-Variance Decomposition 300
4. 4.10.4 Bagging 302
5. 4.10.5 Boosting 305
6. 4.10.6 Random Forests 310
7. 4.10.7 Empirical Comparison among Ensemble Methods 312
11. 4.11 Class Imbalance Problem 313
1. 4.11.1 Building Classifiers with Class Imbalance 314
2. 4.11.2 Evaluating Performance with Class Imbalance 318
3. 4.11.3 Finding an Optimal Score Threshold 322
4. 4.11.4 Aggregate Evaluation of Performance 323
12. 4.12 Multiclass Problem 330
13. 4.13 Bibliographic Notes 333
1. 4.14 Exercises 345
5. 5 Association Analysis: Basic Concepts and Algorithms 357
1. 5.1 Preliminaries 358
2. 5.2 Frequent Itemset Generation 362
1. 5.2.1 The Apriori Principle 363
2. 5.2.2 Frequent Itemset Generation in the Apriori Algorithm 364
3. 5.2.3 Candidate Generation and Pruning 368
4. 5.2.4 Support Counting 373
5. 5.2.5 Computational Complexity 377
3. 5.3 Rule Generation 380
1. 5.3.1 Confidence-Based Pruning 380
2. 5.3.2 Rule Generation in Apriori Algorithm 381
3. 5.3.3 An Example: Congressional Voting Records 382
4. 5.4 Compact Representation of Frequent Itemsets 384
1. 5.4.1 Maximal Frequent Itemsets 384
2. 5.4.2 Closed Itemsets 386
5. 5.5 Alternative Methods for Generating Frequent Itemsets* 389
6. 5.6 FP-Growth Algorithm* 393
1. 5.6.1 FP-Tree Representation 394
2. 5.6.2 Frequent Itemset Generation in FP-Growth Algorithm 397
7. 5.7 Evaluation of Association Patterns 401
1. 5.7.1 Objective Measures of Interestingness 402
2. 5.7.2 Measures beyond Pairs of Binary Variables 414
3. 5.7.3 Simpson’s Paradox 416
8. 5.8 Effect of Skewed Support Distribution 418
9. 5.9 Bibliographic Notes 424
1. 5.10 Exercises 438
6. 6 Association Analysis: Advanced Concepts 451
1. 6.1 Handling Categorical Attributes 451
2. 6.2 Handling Continuous Attributes 454
1. 6.2.1 Discretization-Based Methods 454
2. 6.2.2 Statistics-Based Methods 458
3. 6.2.3 Non-discretization Methods 460
3. 6.3 Handling a Concept Hierarchy 462
4. 6.4 Sequential Patterns 464
1. 6.4.1 Preliminaries 465
2. 6.4.2 Sequential Pattern Discovery 468
3. 6.4.3 Timing Constraints∗ 473
4. 6.4.4 Alternative Counting Schemes∗ 477
5. 6.5 Subgraph Patterns 479
1. 6.5.1 Preliminaries 480
2. 6.5.2 Frequent Subgraph Mining 483
3. 6.5.3 Candidate Generation 487
4. 6.5.4 Candidate Pruning 493
5. 6.5.5 Support Counting 493
6. 6.6 Infrequent Patterns∗ 493
1. 6.6.1 Negative Patterns 494
2. 6.6.2 Negatively Correlated Patterns 495
3. 6.6.3 Comparisons among Infrequent Patterns, Negative Patterns, and Negatively Correlated Patterns 496
4. 6.6.4 Techniques for Mining Interesting Infrequent Patterns 498
5. 6.6.5 Techniques Based on Mining Negative Patterns 499
6. 6.6.6 Techniques Based on Support Expectation 501
7. 6.7 Bibliographic Notes 505
1. 6.8 Exercises 510
7. 7 Cluster Analysis: Basic Concepts and Algorithms 525
1. 7.1 Overview 528
1. 7.1.1 What Is Cluster Analysis? 528
2. 7.1.2 Different Types of Clusterings 529
3. 7.1.3 Different Types of Clusters 531
2. 7.2 K-means 534
1. 7.2.1 The Basic K-means Algorithm 535
2. 7.2.2 K-means: Additional Issues 544
3. 7.2.3 Bisecting K-means 547
4. 7.2.4 K-means and Different Types of Clusters 548
5. 7.2.5 Strengths and Weaknesses 549
6. 7.2.6 K-means as an Optimization Problem 549
3. 7.3 Agglomerative Hierarchical Clustering 554
1. 7.3.1 Basic Agglomerative Hierarchical Clustering Algorithm 555
2. 7.3.2 Specific Techniques 557
3. 7.3.3 The Lance-Williams Formula for Cluster Proximity 562
4. 7.3.4 Key Issues in Hierarchical Clustering 563
5. 7.3.5 Outliers 564
6. 7.3.6 Strengths and Weaknesses 565
4. 7.4 DBSCAN 565
1. 7.4.1 Traditional Density: Center-Based Approach 565
2. 7.4.2 The DBSCAN Algorithm 567
3. 7.4.3 Strengths and Weaknesses 569
5. 7.5 Cluster Evaluation 571
1. 7.5.1 Overview 571
2. 7.5.2 Unsupervised Cluster Evaluation Using Cohesion and Separation 574
3. 7.5.3 Unsupervised Cluster Evaluation Using the Proximity Matrix 582
4. 7.5.4 Unsupervised Evaluation of Hierarchical Clustering 585
5. 7.5.5 Determining the Correct Number of Clusters 587
6. 7.5.6 Clustering Tendency 588
7. 7.5.7 Supervised Measures of Cluster Validity 589
8. 7.5.8 Assessing the Significance of Cluster Validity Measures 594
9. 7.5.9 Choosing a Cluster Validity Measure 596
6. 7.6 Bibliographic Notes 597
1. 7.7 Exercises 603
8. 8 Cluster Analysis: Additional Issues and Algorithms 613
1. 8.1 Characteristics of Data, Clusters, and Clustering Algorithms 614
1. 8.1.1 Example: Comparing K-means and DBSCAN 614
2. 8.1.2 Data Characteristics 615
3. 8.1.3 Cluster Characteristics 617
4. 8.1.4 General Characteristics of Clustering Algorithms 619
2. 8.2 Prototype-Based Clustering 621
1. 8.2.1 Fuzzy Clustering 621
2. 8.2.2 Clustering Using Mixture Models 627
3. 8.2.3 Self-Organizing Maps (SOM) 637
3. 8.3 Density-Based Clustering 644
1. 8.3.1 Grid-Based Clustering 644
2. 8.3.2 Subspace Clustering 648
3. 8.3.3 DENCLUE: A Kernel-Based Scheme for Density-Based Clustering 652
4. 8.4 Graph-Based Clustering 656
1. 8.4.1 Sparsification 657
2. 8.4.2 Minimum Spanning Tree (MST) Clustering 658
3. 8.4.3 OPOSSUM: Optimal Partitioning of Sparse Similarities Using METIS 659
4. 8.4.4 Chameleon: Hierarchical Clustering with Dynamic Modeling 660
5. 8.4.5 Spectral Clustering 666
6. 8.4.6 Shared Nearest Neighbor Similarity 673
7. 8.4.7 The Jarvis-Patrick Clustering Algorithm 676
8. 8.4.8 SNN Density 678
9. 8.4.9 SNN Density-Based Clustering 679
5. 8.5 Scalable Clustering Algorithms 681
1. 8.5.1 Scalability: General Issues and Approaches 681
2. 8.5.2 BIRCH 684
3. 8.5.3 CURE 686
6. 8.6 Which Clustering Algorithm? 690
7. 8.7 Bibliographic Notes 693
1. 8.8 Exercises 699
9. 9 Anomaly Detection 703
1. 9.1 Characteristics of Anomaly Detection Problems 705
1. 9.1.1 A Definition of an Anomaly 705
2. 9.1.2 Nature of Data 706
3. 9.1.3 How Anomaly Detection is Used 707
2. 9.2 Characteristics of Anomaly Detection Methods 708
3. 9.3 Statistical Approaches 710
1. 9.3.1 Using Parametric Models 710
2. 9.3.2 Using Non-parametric Models 714
3. 9.3.3 Modeling Normal and Anomalous Classes 715
4. 9.3.4 Assessing Statistical Significance 717
5. 9.3.5 Strengths and Weaknesses 718
4. 9.4 Proximity-based Approaches 719
1. 9.4.1 Distance-based Anomaly Score 719
2. 9.4.2 Density-based Anomaly Score 720
3. 9.4.3 Relative Density-based Anomaly Score 722
4. 9.4.4 Strengths and Weaknesses 723
5. 9.5 Clustering-based Approaches 724
1. 9.5.1 Finding Anomalous Clusters 724
2. 9.5.2 Finding Anomalous Instances 725
3. 9.5.3 Strengths and Weaknesses 728
6. 9.6 Reconstruction-based Approaches 728
1. 9.6.1 Strengths and Weaknesses 731
7. 9.7 One-class Classification 732
1. 9.7.1 Use of Kernels 733
2. 9.7.2 The Origin Trick 734
3. 9.7.3 Strengths and Weaknesses 738
8. 9.8 Information Theoretic Approaches 738
1. 9.8.1 Strengths and Weaknesses 740
9. 9.9 Evaluation of Anomaly Detection 740
10. 9.10 Bibliographic Notes 742
1. 9.11 Exercises 749
10. 10 Avoiding False Discoveries 755
1. 10.1 Preliminaries: Statistical Testing 756
1. 10.1.1 Significance Testing 756
2. 10.1.2 Hypothesis Testing 761
3. 10.1.3 Multiple Hypothesis Testing 767
4. 10.1.4 Pitfalls in Statistical Testing 776
2. 10.2 Modeling Null and Alternative Distributions 778
1. 10.2.1 Generating Synthetic Data Sets 781
2. 10.2.2 Randomizing Class Labels 782
3. 10.2.3 Resampling Instances 782
4. 10.2.4 Modeling the Distribution of the Test Statistic 783
3. 10.3 Statistical Testing for Classification 783
1. 10.3.1 Evaluating Classification Performance 783
2. 10.3.2 Binary Classification as Multiple Hypothesis Testing 785
3. 10.3.3 Multiple Hypothesis Testing in Model Selection 786
4. 10.4 Statistical Testing for Association Analysis 787
1. 10.4.1 Using Statistical Models 788
2. 10.4.2 Using Randomization Methods 794
5. 10.5 Statistical Testing for Cluster Analysis 795
1. 10.5.1 Generating a Null Distribution for Internal Indices 796
2. 10.5.2 Generating a Null Distribution for External Indices 798
3. 10.5.3 Enrichment 798
6. 10.6 Statistical Testing for Anomaly Detection 800
7. 10.7 Bibliographic Notes 803
1. 10.8 Exercises 808
1. Author Index 816
2. Subject Index 829
3. Copyright Permissions 839
1 Introduction Rapid advances in data collection and storage technology, coupled with the ease with which data can be generated and disseminated, have triggered the explosive growth of data, leading to the current age of big data. Deriving actionable insights from these large data sets is increasingly important in decision making across almost all areas of society, including business and industry; science and engineering; medicine and biotechnology; and government and individuals. However, the amount of data (volume), its complexity (variety), and the rate at which it is being collected and processed (velocity) have simply become too great for humans to analyze unaided. Thus, there is a great need for automated tools for extracting useful information from the big data despite the challenges posed by its enormity and diversity.
Data mining blends traditional data analysis methods with sophisticated algorithms for processing this abundance of data. In this introductory chapter, we present an overview of data mining and outline the key topics to be covered in this book. We start with a description of some applications that require more advanced techniques for data analysis.
Business and Industry Point-of-sale data collection (bar code scanners, radio frequency identification (RFID), and smart card technology) have allowed retailers to collect up-to-the-minute data about customer purchases at the checkout counters of their stores. Retailers can utilize this information, along with other business-critical data, such as web server logs from e-commerce websites and customer service records from call centers, to help them better understand the needs of their customers and make more informed business decisions.
Data mining techniques can be used to support a wide range of business intelligence applications, such as customer profiling, targeted marketing, workflow management, store layout, fraud detection, and automated buying and selling. An example of the last application is high-speed stock trading, where decisions on buying and selling have to be made in less than a second
using data about financial transactions. Data mining can also help retailers answer important business questions, such as “Who are the most profitable customers?” “What products can be cross-sold or up-sold?” and “What is the revenue outlook of the company for next year?” These questions have inspired the development of such data mining techniques as association analysis (Chapters 5 and 6).
As the Internet continues to revolutionize the way we interact and make decisions in our everyday lives, we are generating massive amounts of data about our online experiences, e.g., web browsing, messaging, and posting on social networking websites. This has opened several opportunities for business applications that use web data. For example, in the e-commerce sector, data about our online viewing or shopping preferences can be used to provide personalized recommendations of products. Data mining also plays a prominent role in supporting several other Internet-based services, such as filtering spam messages, answering search queries, and suggesting social updates and connections. The large corpus of text, images, and videos available on the Internet has enabled a number of advancements in data mining methods, including deep learning, which is discussed in Chapter 4. These developments have led to great advances in a number of applications, such as object recognition, natural language translation, and autonomous driving.
Another domain that has undergone a rapid big data transformation is the use of mobile sensors and devices, such as smart phones and wearable computing devices. With better sensor technologies, it has become possible to collect a variety of information about our physical world using low-cost sensors embedded on everyday objects that are connected to each other, termed the Internet of Things (IOT). This deep integration of physical sensors in digital systems is beginning to generate large amounts of diverse and distributed data about our environment, which can be used for designing convenient, safe, and energy-efficient home systems, as well as for urban planning of smart cities.
Medicine, Science, and Engineering Researchers in medicine, science, and engineering are rapidly accumulating data that is key to significant new discoveries. For example, as an important step toward improving our
understanding of the Earth’s climate system, NASA has deployed a series of Earth-orbiting satellites that continuously generate global observations of the land surface, oceans, and atmosphere. However, because of the size and spatio-temporal nature of the data, traditional methods are often not suitable for analyzing these data sets. Techniques developed in data mining can aid Earth scientists in answering questions such as the following: “What is the relationship between the frequency and intensity of ecosystem disturbances such as droughts and hurricanes to global warming?” “How is land surface precipitation and temperature affected by ocean surface temperature?” and “How well can we predict the beginning and end of the growing season for a region?”
As another example, researchers in molecular biology hope to use the large amounts of genomic data to better understand the structure and function of genes. In the past, traditional methods in molecular biology allowed scientists to study only a few genes at a time in a given experiment. Recent breakthroughs in microarray technology have enabled scientists to compare the behavior of thousands of genes under various situations. Such comparisons can help determine the function of each gene, and perhaps isolate the genes responsible for certain diseases. However, the noisy, high- dimensional nature of data requires new data analysis methods. In addition to analyzing gene expression data, data mining can also be used to address other important biological challenges such as protein structure prediction, multiple sequence alignment, the modeling of biochemical pathways, and phylogenetics.
Another example is the use of data mining techniques to analyze electronic health record (EHR) data, which has become increasingly available. Not very long ago, studies of patients required manually examining the physical records of individual patients and extracting very specific pieces of information pertinent to the particular question being investigated. EHRs allow for a faster and broader exploration of such data. However, there are significant challenges since the observations on any one patient typically occur during their visits to a doctor or hospital and only a small number of details about the health of the patient are measured during any particular visit.
Currently, EHR analysis focuses on simple types of data, e.g., a patient’s
blood pressure or the diagnosis code of a disease. However, large amounts of more complex types of medical data are also being collected, such as electrocardiograms (ECGs) and neuroimages from magnetic resonance imaging (MRI) or functional Magnetic Resonance Imaging (fMRI). Although challenging to analyze, this data also provides vital information about patients. Integrating and analyzing such data, with traditional EHR and genomic data is one of the capabilities needed to enable precision medicine, which aims to provide more personalized patient care.
1.1 What Is Data Mining? Data mining is the process of automatically discovering useful information in large data repositories. Data mining techniques are deployed to scour large data sets in order to find novel and useful patterns that might otherwise remain unknown. They also provide the capability to predict the outcome of a future observation, such as the amount a customer will spend at an online or a brick-and-mortar store.
Not all information discovery tasks are considered to be data mining. Examples include queries, e.g., looking up individual records in a database or finding web pages that contain a particular set of keywords. This is because such tasks can be accomplished through simple interactions with a database management system or an information retrieval system. These systems rely on traditional computer science techniques, which include sophisticated indexing structures and query processing algorithms, for efficiently organizing and retrieving information from large data repositories. Nonetheless, data mining techniques have been used to enhance the performance of such systems by improving the quality of the search results based on their relevance to the input queries.
Data Mining and Knowledge Discovery in Databases Data mining is an integral part of knowledge discovery in databases (KDD), which is the overall process of converting raw data into useful information, as shown in Figure 1.1. This process consists of a series of steps, from data preprocessing to postprocessing of data mining results.
Figure 1.1. The process of knowledge discovery in databases (KDD).
Figure 1.1. Full Alternative Text
The input data can be stored in a variety of formats (flat files, spreadsheets, or relational tables) and may reside in a centralized data repository or be distributed across multiple sites. The purpose of preprocessing is to transform the raw input data into an appropriate format for subsequent analysis. The steps involved in data preprocessing include fusing data from multiple sources, cleaning data to remove noise and duplicate observations, and selecting records and features that are relevant to the data mining task at hand. Because of the many ways data can be collected and stored, data preprocessing is perhaps the most laborious and time-consuming step in the overall knowledge discovery process.
“Closing the loop” is a phrase often used to refer to the process of integrating data mining results into decision support systems. For example, in business applications, the insights offered by data mining results can be integrated with campaign management tools so that effective marketing promotions can be conducted and tested. Such integration requires a postprocessing step to ensure that only valid and useful results are incorporated into the decision support system. An example of postprocessing is visualization, which allows analysts to explore the data and the data mining results from a variety of viewpoints. Hypothesis testing methods can also be applied during
postprocessing to eliminate spurious data mining results. (See Chapter 10.)
1.2 Motivating Challenges As mentioned earlier, traditional data analysis techniques have often encountered practical difficulties in meeting the challenges posed by big data applications. The following are some of the specific challenges that motivated the development of data mining.
Scalability Because of advances in data generation and collection, data sets with sizes of terabytes, petabytes, or even exabytes are becoming common. If data mining algorithms are to handle these massive data sets, they must be scalable. Many data mining algorithms employ special search strategies to handle exponential search problems. Scalability may also require the implementation of novel data structures to access individual records in an efficient manner. For instance, out-of-core algorithms may be necessary when processing data sets that cannot fit into main memory. Scalability can also be improved by using sampling or developing parallel and distributed algorithms. A general overview of techniques for scaling up data mining algorithms is given in Appendix F.
High Dimensionality It is now common to encounter data sets with hundreds or thousands of attributes instead of the handful common a few decades ago. In bioinformatics, progress in microarray technology has produced gene expression data involving thousands of features. Data sets with temporal or spatial components also tend to have high dimensionality. For example, consider a data set that contains measurements of temperature at various locations. If the temperature measurements are taken repeatedly for an extended period, the number of dimensions (features) increases in proportion to the number of measurements taken. Traditional data analysis techniques
that were developed for low-dimensional data often do not work well for such high-dimensional data due to issues such as curse of dimensionality (to be discussed in Chapter 2). Also, for some data analysis algorithms, the computational complexity increases rapidly as the dimensionality (the number of features) increases.
Heterogeneous and Complex Data Traditional data analysis methods often deal with data sets containing attributes of the same type, either continuous or categorical. As the role of data mining in business, science, medicine, and other fields has grown, so has the need for techniques that can handle heterogeneous attributes. Recent years have also seen the emergence of more complex data objects. Examples of such non-traditional types of data include web and social media data containing text, hyperlinks, images, audio, and videos; DNA data with sequential and three-dimensional structure; and climate data that consists of measurements (temperature, pressure, etc.) at various times and locations on the Earth’s surface. Techniques developed for mining such complex objects should take into consideration relationships in the data, such as temporal and spatial autocorrelation, graph connectivity, and parent-child relationships between the elements in semi-structured text and XML documents.
Data Ownership and Distribution Sometimes, the data needed for an analysis is not stored in one location or owned by one organization. Instead, the data is geographically distributed among resources belonging to multiple entities. This requires the development of distributed data mining techniques. The key challenges faced by distributed data mining algorithms include the following: (1) how to reduce the amount of communication needed to perform the distributed computation, (2) how to effectively consolidate the data mining results obtained from multiple sources, and (3) how to address data security and privacy issues.
Non-traditional Analysis The traditional statistical approach is based on a hypothesize-and-test paradigm. In other words, a hypothesis is proposed, an experiment is designed to gather the data, and then the data is analyzed with respect to the hypothesis. Unfortunately, this process is extremely labor-intensive. Current data analysis tasks often require the generation and evaluation of thousands of hypotheses, and consequently, the development of some data mining techniques has been motivated by the desire to automate the process of hypothesis generation and evaluation. Furthermore, the data sets analyzed in data mining are typically not the result of a carefully designed experiment and often represent opportunistic samples of the data, rather than random samples.
1.3 The Origins of Data Mining While data mining has traditionally been viewed as an intermediate process within the KDD framework, as shown in Figure 1.1, it has emerged over the years as an academic field within computer science, focusing on all aspects of KDD, including data preprocessing, mining, and postprocessing. Its origin can be traced back to the late 1980s, following a series of workshops organized on the topic of knowledge discovery in databases. The workshops brought together researchers from different disciplines to discuss the challenges and opportunities in applying computational techniques to extract actionable knowledge from large databases. The workshops quickly grew into hugely popular conferences that were attended by researchers and practitioners from both the academia and industry. The success of these conferences, along with the interest shown by businesses and industry in recruiting new hires with data mining background, have fueled the tremendous growth of this field.
The field was initially built upon the methodology and algorithms that researchers had previously used. In particular, data mining researchers draw upon ideas, such as (1) sampling, estimation, and hypothesis testing from statistics and (2) search algorithms, modeling techniques, and learning theories from artificial intelligence, pattern recognition, and machine learning. Data mining has also been quick to adopt ideas from other areas, including optimization, evolutionary computing, information theory, signal processing, visualization, and information retrieval, and extending them to solve the challenges of mining big data.
A number of other areas also play key supporting roles. In particular, database systems are needed to provide support for efficient storage, indexing, and query processing. Techniques from high performance (parallel) computing are often important in addressing the massive size of some data sets. Distributed techniques can also help address the issue of size and are essential when the data cannot be gathered in one location. Figure 1.2 shows the relationship of data mining to other areas.
Figure 1.2. Data mining as a confluence of many disciplines.
Data Science and Data-Driven Discovery Data science is an interdisciplinary field that studies and applies tools and techniques for deriving useful insights from data. Although data science is regarded as an emerging field with a distinct identity of its own, the tools and techniques often come from many different areas of data analysis, such as data mining, statistics, AI, machine learning, pattern recognition, database technology, and distributed and parallel computing. (See Figure 1.2.)
The emergence of data science as a new field is a recognition that, often, none of the existing areas of data analysis provides a complete set of tools for the data analysis tasks that are often encountered in emerging applications. Instead, a broad range of computational, mathematical, and statistical skills is often required. To illustrate the challenges that arise in analyzing such data,
consider the following example. Social media and the Web present new opportunities for social scientists to observe and quantitatively measure human behavior on a large scale. To conduct such a study, social scientists work with analysts who possess skills in areas such as web mining, natural language processing (NLP), network analysis, data mining, and statistics. Compared to more traditional research in social science, which is often based on surveys, this analysis requires a broader range of skills and tools, and involves far larger amounts of data. Thus, data science is, by necessity, a highly interdisciplinary field that builds on the continuing work of many fields.
The data-driven approach of data science emphasizes the direct discovery of patterns and relationships from data, especially in large quantities of data, often without the need for extensive domain knowledge. A notable example of the success of this approach is represented by advances in neural networks, i.e., deep learning, which have been particularly successful in areas which have long proved challenging, e.g., recognizing objects in photos or videos and words in speech, as well as in other application areas. However, note that this is just one example of the success of data-driven approaches, and dramatic improvements have also occurred in many other areas of data analysis. Many of these developments are topics described later in this book.
Some cautions on potential limitations of a purely data-driven approach are given in the Bibliographic Notes.
1.4 Data Mining Tasks Data mining tasks are generally divided into two major categories:
Predictive tasks The objective of these tasks is to predict the value of a particular attribute based on the values of other attributes. The attribute to be predicted is commonly known as the target or dependent variable, while the attributes used for making the prediction are known as the explanatory or independent variables.
Descriptive tasks Here, the objective is to derive patterns (correlations, trends, clusters, trajectories, and anomalies) that summarize the underlying relationships in data. Descriptive data mining tasks are often exploratory in nature and frequently require postprocessing techniques to validate and explain the results.
Figure 1.3 illustrates four of the core data mining tasks that are described in the remainder of this book.
Figure 1.3. Four of the core data mining tasks.
Figure 1.3. Full Alternative Text
Predictive modeling refers to the task of building a model for the target variable as a function of the explanatory variables. There are two types of predictive modeling tasks: classification, which is used for discrete target variables, and regression, which is used for continuous target variables. For example, predicting whether a web user will make a purchase at an online bookstore is a classification task because the target variable is binary-valued. On the other hand, forecasting the future price of a stock is a regression task because price is a continuous-valued attribute. The goal of both tasks is to
learn a model that minimizes the error between the predicted and true values of the target variable. Predictive modeling can be used to identify customers who will respond to a marketing campaign, predict disturbances in the Earth’s ecosystem, or judge whether a patient has a particular disease based on the results of medical tests.
Example 1.1 (Predicting the Type of a Flower). Consider the task of predicting a species of flower based on the characteristics of the flower. In particular, consider classifying an Iris flower as one of the following three Iris species: Setosa, Versicolour, or Virginica. To perform this task, we need a data set containing the characteristics of various flowers of these three species. A data set with this type of information is the well-known Iris data set from the UCI Machine Learning Repository at http://www.ics.uci.edu/~mlearn. In addition to the species of a flower, this data set contains four other attributes: sepal width, sepal length, petal length, and petal width. Figure 1.4 shows a plot of petal width versus petal length for the 150 flowers in the Iris data set. Petal width is broken into the categories low, medium, and high, which correspond to the intervals [0, 0.75), [0.75, 1.75), [1.75, ∞), respectively. Also, petal length is broken into categories low, medium,and high, which correspond to the intervals [0, 2.5), [2.5, 5), [5, ∞), respectively. Based on these categories of petal width and length, the following rules can be derived:
Petal width low and petal length low implies Setosa.
Petal width medium and petal length medium implies Versicolour.
Petal width high and petal length high implies Virginica.
While these rules do not classify all the flowers, they do a good (but not perfect) job of classifying most of the flowers. Note that flowers from the Setosa species are well separated from the Versicolour and Virginica species with respect to petal width and length, but the latter two species overlap
somewhat with respect to these attributes.
Figure 1.4. Petal width versus petal length for 150 Iris flowers.
Figure 1.4. Full Alternative Text
Association analysis is used to discover patterns that describe strongly associated features in the data. The discovered patterns are typically represented in the form of implication rules or feature subsets. Because of the exponential size of its search space, the goal of association analysis is to extract the most interesting patterns in an efficient manner. Useful applications of association analysis include finding groups of genes that have related functionality, identifying web pages that are accessed together, or
understanding the relationships between different elements of Earth’s climate system.
Example 1.2 (Market Basket Analysis). The transactions shown in Table 1.1 illustrate point-of-sale data collected at the checkout counters of a grocery store. Association analysis can be applied to find items that are frequently bought together by customers. For example, we may discover the rule {Diapers}→{Milk}, which suggests that customers who buy diapers also tend to buy milk. This type of rule can be used to identify potential cross-selling opportunities among related items.
Table 1.1. Market basket data. Transaction ID Items
1 {Bread, Butter, Diapers, Milk} 2 {Coffee, Sugar, Cookies, Salmon} 3 {Bread, Butter, Coffee, Diapers, Milk, Eggs} 4 {Bread, Butter, Salmon, Chicken} 5 {Eggs, Bread, Butter} 6 {Salmon, Diapers, Milk} 7 {Bread, Tea, Sugar, Eggs} 8 {Coffee, Sugar, Chicken, Eggs} 9 {Bread, Diapers, Milk, Salt} 10 {Tea, Eggs, Cookies, Diapers, Milk}
Cluster analysis seeks to find groups of closely related observations so that observations that belong to the same cluster are more similar to each other than observations that belong to other clusters. Clustering has been used to group sets of related customers, find areas of the ocean that have a significant
impact on the Earth’s climate, and compress data.
Example 1.3 (Document Clustering). The collection of news articles shown in Table 1.2 can be grouped based on their respective topics. Each article is represented as a set of word-frequency pairs (w : c), where w is a word and c is the number of times the word appears in the article. There are two natural clusters in the data set. The first cluster consists of the first four articles, which correspond to news about the economy, while the second cluster contains the last four articles, which correspond to news about health care. A good clustering algorithm should be able to identify these two clusters based on the similarity between words that appear in the articles.
Table 1.2. Collection of news articles. Article Word-frequency pairs
1 dollar: 1, industry: 4, country: 2, loan: 3, deal: 2,government: 2
2 machinery: 2, labor: 3, market: 4, industry: 2, work: 3,country: 1
3 job: 5, inflation: 3, rise: 2, jobless: 2, market: 3, country: 2,index: 3 4 domestic: 3, forecast: 2, gain: 1, market: 2, sale: 3, price: 2
5 patient: 4, symptom: 2, drug: 3, health: 2, clinic: 2, doctor:2 6 pharmaceutical: 2, company: 3, drug: 2, vaccine: 1, flu: 3 7 death: 2, cancer: 4, drug: 3, public: 4, health: 3, director: 2 8 medical: 2, cost: 3, increase: 2, patient: 2, health: 3, care: 1
Anomaly detection is the task of identifying observations whose characteristics are significantly different from the rest of the data. Such observations are known as anomalies or outliers. The goal of an anomaly detection algorithm is to discover the real anomalies and avoid falsely labeling normal objects as anomalous. In other words, a good anomaly detector must have a high detection rate and a low false alarm rate. Applications of anomaly detection include the detection of fraud, network intrusions, unusual patterns of disease, and ecosystem disturbances, such as droughts, floods, fires, hurricanes, etc.
Example 1.4 (Credit Card Fraud Detection). A credit card company records the transactions made by every credit card holder, along with personal information such as credit limit, age, annual income, and address. Since the number of fraudulent cases is relatively small compared to the number of legitimate transactions, anomaly detection techniques can be applied to build a profile of legitimate transactions for the users. When a new transaction arrives, it is compared against the profile of the user. If the characteristics of the transaction are very different from the previously created profile, then the transaction is flagged as potentially fraudulent.
1.5 Scope and Organization of the Book This book introduces the major principles and techniques used in data mining from an algorithmic perspective. A study of these principles and techniques is essential for developing a better understanding of how data mining technology can be applied to various kinds of data. This book also serves as a starting point for readers who are interested in doing research in this field.
We begin the technical discussion of this book with a chapter on data (Chapter 2), which discusses the basic types of data, data quality, preprocessing techniques, and measures of similarity and dissimilarity. Although this material can be covered quickly, it provides an essential foundation for data analysis. Chapters 3 and 4 cover classification. Chapter 3 provides a foundation by discussing decision tree classifiers and several issues that are important to all classification: overfitting, underfitting, model selection, and performance evaluation. Using this foundation, Chapter 4 describes a number of other important classification techniques: rule-based systems, nearest neighbor classifiers, Bayesian classifiers, artificial neural networks, including deep learning, support vector machines, and ensemble classifiers, which are collections of classifiers. The multiclass and imbalanced class problems are also discussed. These topics can be covered independently.
Association analysis is explored in Chapters 5 and 6. Chapter 5 describes the basics of association analysis: frequent itemsets, association rules, and some of the algorithms used to generate them. Specific types of frequent itemsets— maximal, closed, and hyperclique—that are important for data mining are also discussed, and the chapter concludes with a discussion of evaluation measures for association analysis. Chapter 6 considers a variety of more advanced topics, including how association analysis can be applied to categorical and continuous data or to data that has a concept hierarchy. (A concept hierarchy is a hierarchical categorization of objects, e.g., store itemsstore items→clothing→shoes→sneakers.) This chapter also describes
how association analysis can be extended to find sequential patterns (patterns involving order), patterns in graphs, and negative relationships (if one item is present, then the other is not).
Cluster analysis is discussed in Chapters 7 and 8. Chapter 7 first describes the different types of clusters, and then presents three specific clustering techniques: K-means, agglomerative hierarchical clustering, and DBSCAN. This is followed by a discussion of techniques for validating the results of a clustering algorithm. Additional clustering concepts and techniques are explored in Chapter 8, including fuzzy and probabilistic clustering, Self- Organizing Maps (SOM), graph-based clustering, spectral clustering, and density-based clustering. There is also a discussion of scalability issues and factors to consider when selecting a clustering algorithm.
Chapter 9, is on anomaly detection. After some basic definitions, several different types of anomaly detection are considered: statistical, distance- based, density-based, clustering-based, reconstruction-based, one-class classification, and information theoretic. The last chapter, Chapter 10, supplements the discussions in the other Chapters with a discussion of the statistical concepts important for avoiding spurious results, and then discusses those concepts in the context of data mining techniques studied in the previous chapters. These techniques include statistical hypothesis testing, p-values, the false discovery rate, and permutation testing. Appendices A through F give a brief review of important topics that are used in portions of the book: linear algebra, dimensionality reduction, statistics, regression, optimization, and scaling up data mining techniques for big data.
The subject of data mining, while relatively young compared to statistics or machine learning, is already too large to cover in a single book. Selected references to topics that are only briefly covered, such as data quality, are provided in the Bibliographic Notes section of the appropriate chapter. References to topics not covered in this book, such as mining streaming data and privacy-preserving data mining are provided in the Bibliographic Notes of this chapter.
1.6 Bibliographic Notes The topic of data mining has inspired many textbooks. Introductory textbooks include those by Dunham [16], Han et al. [29], Hand et al. [31], Roiger and Geatz [50], Zaki and Meira [61], and Aggarwal [2]. Data mining books with a stronger emphasis on business applications include the works by Berry and Linoff [5], Pyle [47], and Parr Rud [45]. Books with an emphasis on statistical learning include those by Cherkassky and Mulier [11], and Hastie et al. [32]. Similar books with an emphasis on machine learning or pattern recognition are those by Duda et al. [15], Kantardzic [34], Mitchell [43], Webb [57], and Witten and Frank [58]. There are also some more specialized books: Chakrabarti [9] (web mining), Fayyad et al. [20] (collection of early articles on data mining), Fayyad et al. [18] (visualization), Grossman et al. [25] (science and engineering), Kargupta and Chan [35] (distributed data mining), Wang et al. [56] (bioinformatics), and Zaki and Ho [60] (parallel data mining).
There are several conferences related to data mining. Some of the main conferences dedicated to this field include the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), the IEEE International Conference on Data Mining (ICDM), the SIAM International Conference on Data Mining (SDM), the European Conference on Principles and Practice of Knowledge Discovery in Databases (PKDD), and the Pacific- Asia Conference on Knowledge Discovery and Data Mining (PAKDD). Data mining papers can also be found in other major conferences such as the Conference and Workshop on Neural Information Processing Systems (NIPS),the International Conference on Machine Learning (ICML), the ACM SIGMOD/PODS conference, the International Conference on Very Large Data Bases (VLDB), the Conference on Information and Knowledge Management (CIKM), the International Conference on Data Engineering (ICDE), the National Conference on Artificial Intelligence (AAAI), the IEEE International Conference on Big Data, and the IEEE International Conference on Data Science and Advanced Analytics (DSAA).
Journal publications on data mining include IEEE Transactions on
Knowledge and Data Engineering, Data Mining and Knowledge Discovery, Knowledge and Information Systems, ACM Transactions on Knowledge Discovery from Data, Statistical Analysis and Data Mining, and Information Systems. There are various open-source data mining software available, including Weka [27] and Scikit-learn [46]. More recently, data mining software such as Apache Mahout and Apache Spark have been developed for large-scale problems on the distributed computing platform.
There have been a number of general articles on data mining that define the field or its relationship to other fields, particularly statistics. Fayyad et al. [19] describe data mining and how it fits into the total knowledge discovery process. Chen et al. [10] give a database perspective on data mining. Ramakrishnan and Grama [48] provide a general discussion of data mining and present several viewpoints. Hand [30] describes how data mining differs from statistics, as does Friedman [21]. Lambert [40] explores the use of statistics for large data sets and provides some comments on the respective roles of data mining and statistics. Glymour et al. [23] consider the lessons that statistics may have for data mining. Smyth et al. [53] describe how the evolution of data mining is being driven by new types of data and applications, such as those involving streams, graphs, and text. Han et al. [28] consider emerging applications in data mining and Smyth [52] describes some research challenges in data mining. Wu et al. [59] discuss how developments in data mining research can be turned into practical tools. Data mining standards are the subject of a paper by Grossman et al. [24]. Bradley [7] discusses how data mining algorithms can be scaled to large data sets.
The emergence of new data mining applications has produced new challenges that need to be addressed. For instance, concerns about privacy breaches as a result of data mining have escalated in recent years, particularly in application domains such as web commerce and health care. As a result, there is growing interest in developing data mining algorithms that maintain user privacy. Developing techniques for mining encrypted or randomized data is known as privacy-preserving data mining. Some general references in this area include papers by Agrawal and Srikant [3], Clifton et al. [12] and Kargupta et al. [36]. Vassilios et al. [55] provide a survey. Another area of concern is the bias in predictive models that may be used for some applications, e.g., screening job applicants or deciding prison parole [39].
Assessing whether such applications are producing biased results is made more difficult by the fact that the predictive models used for such applications are often black box models, i.e., models that are not interpretable in any straightforward way.
Data science, its constituent fields, and more generally, the new paradigm of knowledge discovery they represent [33], have great potential, some of which has been realized. However, it is important to emphasize that data science works mostly with observational data, i.e., data that was collected by various organizations as part of their normal operation. The consequence of this is that sampling biases are common and the determination of causal factors becomes more problematic. For this and a number of other reasons, it is often hard to interpret the predictive models built from this data [42, 49]. Thus, theory, experimentation and computational simulations will continue to be the methods of choice in many areas, especially those related to science.
More importantly, a purely data-driven approach often ignores the existing knowledge in a particular field. Such models may perform poorly, for example, predicting impossible outcomes or failing to generalize to new situations. However, if the model does work well, e.g., has high predictive accuracy, then this approach may be sufficient for practical purposes in some fields. But in many areas, such as medicine and science, gaining insight into the underlying domain is often the goal. Some recent work attempts to address these issues in order to create theory-guided data science, which takes pre-existing domain knowledge into account [17, 37].
Recent years have witnessed a growing number of applications that rapidly generate continuous streams of data. Examples of stream data include network traffic, multimedia streams, and stock prices. Several issues must be considered when mining data streams, such as the limited amount of memory available, the need for online analysis, and the change of the data over time. Data mining for stream data has become an important area in data mining. Some selected publications are Domingos and Hulten [14] (classification), Giannella et al. [22] (association analysis), Guha et al. [26] (clustering), Kifer et al. [38] (change detection), Papadimitriou et al. [44] (time series), and Law et al. [41] (dimensionality reduction).
Another area of interest is recommender and collaborative filtering systems
[1, 6, 8, 13, 54], which suggest movies, television shows, books, products, etc. that a person might like. In many cases, this problem, or at least a component of it, is treated as a prediction problem and thus, data mining techniques can be applied [4, 51].
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1.7 Exercises 1. 1. Discuss whether or not each of the following activities is a data
mining task.
1. Dividing the customers of a company according to their gender.
2. Dividing the customers of a company according to their profitability.
3. Computing the total sales of a company.
4. Sorting a student database based on student identification numbers.
5. Predicting the outcomes of tossing a (fair) pair of dice.
6. Predicting the future stock price of a company using historical records.
7. Monitoring the heart rate of a patient for abnormalities.
8. Monitoring seismic waves for earthquake activities.
9. Extracting the frequencies of a sound wave.
2. 2. Suppose that you are employed as a data mining consultant for an Internet search engine company. Describe how data mining can help the company by giving specific examples of how techniques, such as clustering, classification, association rule mining, and anomaly detection can be applied.
3. 3. For each of the following data sets, explain whether or not data privacy is an important issue.
1. Census data collected from 1900–1950.
2. IP addresses and visit times of web users who visit your website.
3. Images from Earth-orbiting satellites.
4. Names and addresses of people from the telephone book.
5. Names and email addresses collected from the Web.
2 Data This chapter discusses several data-related issues that are important for successful data mining:
The Type of Data Data sets differ in a number of ways. For example, the attributes used to describe data objects can be of different types—quantitative or qualitative—and data sets often have special characteristics; e.g., some data sets contain time series or objects with explicit relationships to one another. Not surprisingly, the type of data determines which tools and techniques can be used to analyze the data. Indeed, new research in data mining is often driven by the need to accommodate new application areas and their new types of data.
The Quality of the Data Data is often far from perfect. While most data mining techniques can tolerate some level of imperfection in the data, a focus on understanding and improving data quality typically improves the quality of the resulting analysis. Data quality issues that often need to be addressed include the presence of noise and outliers; missing, inconsistent, or duplicate data; and data that is biased or, in some other way, unrepresentative of the phenomenon or population that the data is supposed to describe.
Preprocessing Steps to Make the Data More Suitable for Data Mining Often, the raw data must be processed in order to make it suitable for analysis. While one objective may be to improve data quality, other goals focus on modifying the data so that it better fits a specified data mining technique or tool. For example, a continuous attribute, e.g., length, sometimes needs to be transformed into an attribute with discrete categories, e.g., short, medium, or long, in order to apply a particular technique. As another example, the number of attributes in a data set is often reduced because many techniques are more effective when the data has a relatively small number of attributes.
Analyzing Data in Terms of Its Relationships One approach to data analysis is to find relationships among the data objects and then perform the remaining analysis using these relationships rather than the data objects
themselves. For instance, we can compute the similarity or distance between pairs of objects and then perform the analysis—clustering, classification, or anomaly detection—based on these similarities or distances. There are many such similarity or distance measures, and the proper choice depends on the type of data and the particular application.
Example 2.1 (An Illustration of Data-Related Issues). To further illustrate the importance of these issues, consider the following hypothetical situation. You receive an email from a medical researcher concerning a project that you are eager to work on.
Hi,
I’ve attached the data file that I mentioned in my previous email. Each line contains the information for a single patient and consists of five fields. We want to predict the last field using the other fields. I don’t have time to provide any more information about the data since I’m going out of town for a couple of days, but hopefully that won’t slow you down too much. And if you don’t mind, could we meet when I get back to discuss your preliminary results? I might invite a few other members of my team.
Thanks and see you in a couple of days.
Despite some misgivings, you proceed to analyze the data. The first few rows of the file are as follows:
012 232 33.5 0 10.7 020 121 16.9 2 210.1 027 165 24.0 0 427.6 ⋮
A brief look at the data reveals nothing strange. You put your doubts aside
and start the analysis. There are only 1000 lines, a smaller data file than you had hoped for, but two days later, you feel that you have made some progress. You arrive for the meeting, and while waiting for others to arrive, you strike up a conversation with a statistician who is working on the project. When she learns that you have also been analyzing the data from the project, she asks if you would mind giving her a brief overview of your results.
Statistician: So, you got the data for all the patients? Data Miner: Yes. I haven’t had much time for analysis, but I do have a few interesting results. Statistician: Amazing. There were so many data issues with this set of patients that I couldn’t do much. Data Miner: Oh? I didn’t hear about any possible problems. Statistician: Well, first there is field 5, the variable we want to predict. It’s common knowledge among people who analyze this type of data that results are better if you work with the log of the values, but I didn’t discover this until later. Was it mentioned to you? Data Miner: No. Statistician: But surely you heard about what happened to field 4? It’s supposed to be measured on a scale from 1 to 10, with 0 indicating a missing value, but because of a data entry error, all 10’s were changed into 0’s. Unfortunately, since some of the patients have missing values for this field, it’s impossible to say whether a 0 in this field is a real 0 or a 10. Quite a few of the records have that problem. Data Miner: Interesting. Were there any other problems? Statistician: Yes, fields 2 and 3 are basically the same, but I assume that you probably noticed that. Data Miner: Yes, but these fields were only weak predictors of field 5. Statistician: Anyway, given all those problems, I’m surprised you were able to accomplish anything. Data Miner: True, but my results are really quite good. Field 1 is a very strong predictor of field 5. I’m surprised that this wasn’t noticed before. Statistician: What? Field 1 is just an identification number. Data Miner: Nonetheless, my results speak for themselves. Statistician: Oh, no! I just remembered. We assigned ID numbers after we sorted the records based on field 5. There is a strong connection, but it’s meaningless. Sorry.
Although this scenario represents an extreme situation, it emphasizes the importance of “knowing your data.” To that end, this chapter will address each of the four issues mentioned above, outlining some of the basic challenges and standard approaches.
2.1 Types of Data A data set can often be viewed as a collection of data objects. Other names for a data object are record, point, vector, pattern, event, case, sample, instance, observation, or entity. In turn, data objects are described by a number of attributes that capture the characteristics of an object, such as the mass of a physical object or the time at which an event occurred. Other names for an attribute are variable, characteristic, field, feature, or dimension.
Example 2.2 (Student Information). Often, a data set is a file, in which the objects are records (or rows) in the file and each field (or column) corresponds to an attribute. For example, Table 2.1 shows a data set that consists of student information. Each row corresponds to a student and each column is an attribute that describes some aspect of a student, such as grade point average (GPA) or identification number (ID).
Table 2.1. A sample data set containing student information. Student ID Year Grade Point Average (GPA) …
⋮
1034262 Senior 3.24 … 1052663 Freshman 3.51 … 1082246 Sophomore 3.62 …
Although record-based data sets are common, either in flat files or relational database systems, there are other important types of data sets and systems for
storing data. In Section 2.1.2, we will discuss some of the types of data sets that are commonly encountered in data mining. However, we first consider attributes.
2.1.1 Attributes and Measurement In this section, we consider the types of attributes used to describe data objects. We first define an attribute, then consider what we mean by the type of an attribute, and finally describe the types of attributes that are commonly encountered.
What Is an Attribute? We start with a more detailed definition of an attribute.
Definition 2.1. An attribute is a property or characteristic of an object that can vary, either from one object to another or from one time to another.
For example, eye color varies from person to person, while the temperature of an object varies over time. Note that eye color is a symbolic attribute with a small number of possible values {brown, black, blue, green, hazel, etc.} , while temperature is a numerical attribute with a potentially unlimited number of values.
At the most basic level, attributes are not about numbers or symbols. However, to discuss and more precisely analyze the characteristics of objects, we assign numbers or symbols to them. To do this in a well-defined way, we need a measurement scale.
Definition 2.2.
A measurement scale is a rule (function) that associates a numerical or symbolic value with an attribute of an object.
Formally, the process of measurement is the application of a measurement scale to associate a value with a particular attribute of a specific object. While this may seem a bit abstract, we engage in the process of measurement all the time. For instance, we step on a bathroom scale to determine our weight, we classify someone as male or female, or we count the number of chairs in a room to see if there will be enough to seat all the people coming to a meeting. In all these cases, the “physical value” of an attribute of an object is mapped to a numerical or symbolic value.
With this background, we can discuss the type of an attribute, a concept that is important in determining if a particular data analysis technique is consistent with a specific type of attribute.
The Type of an Attribute It is common to refer to the type of an attribute as the type of a measurement scale. It should be apparent from the previous discussion that an attribute can be described using different measurement scales and that the properties of an attribute need not be the same as the properties of the values used to measure it. In other words, the values used to represent an attribute can have properties that are not properties of the attribute itself, and vice versa. This is illustrated with two examples.
Example 2.3 (Employee Age and ID Number). Two attributes that might be associated with an employee are ID and age (in years). Both of these attributes can be represented as integers. However, while it is reasonable to talk about the average age of an employee, it makes no sense to talk about the average employee ID. Indeed, the only aspect of
employees that we want to capture with the ID attribute is that they are distinct. Consequently, the only valid operation for employee IDs is to test whether they are equal. There is no hint of this limitation, however, when integers are used to represent the employee ID attribute. For the age attribute, the properties of the integers used to represent age are very much the properties of the attribute. Even so, the correspondence is not complete because, for example, ages have a maximum, while integers do not.
Example 2.4 (Length of Line Segments). Consider Figure 2.1, which shows some objects—line segments—and how the length attribute of these objects can be mapped to numbers in two different ways. Each successive line segment, going from the top to the bottom, is formed by appending the topmost line segment to itself. Thus, the second line segment from the top is formed by appending the topmost line segment to itself twice, the third line segment from the top is formed by appending the topmost line segment to itself three times, and so forth. In a very real (physical) sense, all the line segments are multiples of the first. This fact is captured by the measurements on the right side of the figure, but not by those on the left side. More specifically, the measurement scale on the left side captures only the ordering of the length attribute, while the scale on the right side captures both the ordering and additivity properties. Thus, an attribute can be measured in a way that does not capture all the properties of the attribute.
Figure 2.1. The measurement of the length of line segments on two different scales of measurement.
Figure 2.1. Full Alternative Text
Knowing the type of an attribute is important because it tells us which properties of the measured values are consistent with the underlying properties of the attribute, and therefore, it allows us to avoid foolish actions, such as computing the average employee ID.
The Different Types of Attributes A useful (and simple) way to specify the type of an attribute is to identify the properties of numbers that correspond to underlying properties of the
attribute. For example, an attribute such as length has many of the properties of numbers. It makes sense to compare and order objects by length, as well as to talk about the differences and ratios of length. The following properties (operations) of numbers are typically used to describe attributes.
1. Distinctness = and ≠
2. Order <, ≤, >, and ≥
3. Addition + and −
4. Multiplication × and /
Given these properties, we can define four types of attributes: nominal , ordinal, interval , and ratio. Table 2.2 gives the definitions of these types, along with information about the statistical operations that are valid for each type. Each attribute type possesses all of the properties and operations of the attribute types above it. Consequently, any property or operation that is valid for nominal, ordinal, and interval attributes is also valid for ratio attributes. In other words, the definition of the attribute types is cumulative. However, this does not mean that the statistical operations appropriate for one attribute type are appropriate for the attribute types above it.
Table 2.2. Different attribute types. Attribute Type Description Examples Operations
Nominal
The values of a nominal attribute are just different names; i.e., nominal values provide only enough information to
zip codes, employee ID numbers, eye color, gender
mode, entropy, contingency correlation, χ2 test
Categorical (Qualitative)
distinguish one object from another. (=, ≠)
Ordinal
The values of an ordinal attribute provide enough information to order objects. (<, >)
hardness of minerals, {good, better, best}, grades, street numbers
median, percentiles, rank correlation, run tests, sign tests
Numeric (Quantitative)
Interval
For interval attributes, the differences between values are meaningful, i.e., a unit of measurement exists. (+, −)
calendar dates, temperature in Celsius or Fahrenheit
mean, standard deviation, Pearson’s correlation, t and F tests
Ratio
For ratio variables, both differences and ratios are meaningful. (×, /)
temperature in Kelvin, monetary quantities, counts, age, mass, length, electrical current
geometric mean, harmonic mean, percent variation
Nominal and ordinal attributes are collectively referred to as categorical or qualitative attributes. As the name suggests, qualitative attributes, such as employee ID, lack most of the properties of numbers. Even if they are represented by numbers, i.e., integers, they should be treated more like symbols. The remaining two types of attributes, interval and ratio, are collectively referred to as quantitative or numeric attributes. Quantitative attributes are represented by numbers and have most of the properties of numbers. Note that quantitative attributes can be integer-valued or
continuous.
The types of attributes can also be described in terms of transformations that do not change the meaning of an attribute. Indeed, S. Smith Stevens, the psychologist who originally defined the types of attributes shown in Table 2.2, defined them in terms of these permissible transformations. For example, the meaning of a length attribute is unchanged if it is measured in meters instead of feet.
The statistical operations that make sense for a particular type of attribute are those that will yield the same results when the attribute is transformed by using a transformation that preserves the attribute’s meaning. To illustrate, the average length of a set of objects is different when measured in meters rather than in feet, but both averages represent the same length. Table 2.3 shows the meaning-preserving transformations for the four attribute types of Table 2.2.
Table 2.3. Transformations that define attribute levels. Attribute Type Transformation Comment
Categorical (Qualitative)
Nominal Any one-to-one mapping, e.g., a permutation of values
If all employee ID numbers are reassigned, it will not make any difference.
Ordinal
An order-preserving change of values, i.e., new_value=f(old_value), where f is a monotonic
An attribute encompassing the notion of good, better, best can be represented
function. equally well by the values {1, 2, 3} or by {0.5, 1, 10}.
Numeric (Quantitative)
Interval new_value=a×old_value+b,a and b constants.
The Fahrenheit and Celsius temperature scales differ in the location of their zero value and the size of a degree (unit).
Ratio new_value=a×old_value Length can be measured in meters or feet.
Example 2.5 (Temperature Scales). Temperature provides a good illustration of some of the concepts that have been described. First, temperature can be either an interval or a ratio attribute, depending on its measurement scale. When measured on the Kelvin scale, a temperature of 2
◦ is, in a physically meaningful way, twice that of a
temperature of 1 ◦ . This is not true when temperature is measured on either the
Celsius or Fahrenheit scales, because, physically, a temperature of 1 ◦
Fahrenheit (Celsius) is not much different than a temperature of 2 ◦ Fahrenheit
(Celsius). The problem is that the zero points of the Fahrenheit and Celsius scales are, in a physical sense, arbitrary, and therefore, the ratio of two Celsius or Fahrenheit temperatures is not physically meaningful.
Describing Attributes by the
Number of Values An independent way of distinguishing between attributes is by the number of values they can take.
Discrete A discrete attribute has a finite or countably infinite set of values. Such attributes can be categorical, such as zip codes or ID numbers, or numeric, such as counts. Discrete attributes are often represented using integer variables. Binary attributes are a special case of discrete attributes and assume only two values, e.g., true/false, yes/no, male/female, or 0/1. Binary attributes are often represented as Boolean variables, or as integer variables that only take the values 0 or 1.
Continuous A continuous attribute is one whose values are real numbers. Examples include attributes such as temperature, height, or weight. Continuous attributes are typically represented as floating-point variables. Practically, real values can be measured and represented only with limited precision.
In theory, any of the measurement scale types—nominal, ordinal, interval, and ratio—could be combined with any of the types based on the number of attribute values—binary, discrete, and continuous. However, some combinations occur only infrequently or do not make much sense. For instance, it is difficult to think of a realistic data set that contains a continuous binary attribute. Typically, nominal and ordinal attributes are binary or discrete, while interval and ratio attributes are continuous. However, count attributes , which are discrete, are also ratio attributes.
Asymmetric Attributes For asymmetric attributes, only presence—a non-zero attribute value—is regarded as important. Consider a data set in which each object is a student and each attribute records whether a student took a particular course at a university. For a specific student, an attribute has a value of 1 if the student took the course associated with that attribute and a value of 0 otherwise.
Because students take only a small fraction of all available courses, most of the values in such a data set would be 0. Therefore, it is more meaningful and more efficient to focus on the non-zero values. To illustrate, if students are compared on the basis of the courses they don’t take, then most students would seem very similar, at least if the number of courses is large. Binary attributes where only non-zero values are important are called asymmetric binary attributes. This type of attribute is particularly important for association analysis, which is discussed in Chapter 5. It is also possible to have discrete or continuous asymmetric features. For instance, if the number of credits associated with each course is recorded, then the resulting data set will consist of asymmetric discrete or continuous attributes.
General Comments on Levels of Measurement As described in the rest of this chapter, there are many diverse types of data. The previous discussion of measurement scales, while useful, is not complete and has some limitations. We provide the following comments and guidance.
Distinctness, order, and meaningful intervals and ratios are only four properties of data—many others are possible. For instance, some data is inherently cyclical, e.g., position on the surface of the Earth or time. As another example, consider set valued attributes, where each attribute value is a set of elements, e.g., the set of movies seen in the last year. Define one set of elements (movies) to be greater (larger) than a second set if the second set is a subset of the first. However, such a relationship defines only a partial order that does not match any of the attribute types just defined.
The numbers or symbols used to capture attribute values may not capture all the properties of the attributes or may suggest properties that are not there. An illustration of this for integers was presented in Example 2.3 , i.e., averages of IDs and out of range ages.
Data is often transformed for the purpose of analysis—see Section 2.3.7.
This often changes the distribution of the observed variable to a distribution that is easier to analyze, e.g., a Gaussian (normal) distribution. Often, such transformations only preserve the order of the original values, and other properties are lost. Nonetheless, if the desired outcome is a statistical test of differences or a predictive model, such a transformation is justified.
The final evaluation of any data analysis, including operations on attributes, is whether the results make sense from a domain point of view.
In summary, it can be challenging to determine which operations can be performed on a particular attribute or a collection of attributes without compromising the integrity of the analysis. Fortunately, established practice often serves as a reliable guide. Occasionally, however, standard practices are erroneous or have limitations.
2.1.2 Types of Data Sets There are many types of data sets, and as the field of data mining develops and matures, a greater variety of data sets become available for analysis. In this section, we describe some of the most common types. For convenience, we have grouped the types of data sets into three groups: record data, graph- based data, and ordered data. These categories do not cover all possibilities and other groupings are certainly possible.
General Characteristics of Data Sets Before providing details of specific kinds of data sets, we discuss three characteristics that apply to many data sets and have a significant impact on the data mining techniques that are used: dimensionality, distribution, and resolution.
Dimensionality The dimensionality of a data set is the number of attributes that the objects in the data set possess. Analyzing data with a small number of dimensions tends to be qualitatively different from analyzing moderate or high-dimensional data. Indeed, the difficulties associated with the analysis of high-dimensional data are sometimes referred to as the curse of dimensionality. Because of this, an important motivation in preprocessing the data is dimensionality reduction. These issues are discussed in more depth later in this chapter and in Appendix B.
Distribution The distribution of a data set is the frequency of occurrence of various values or sets of values for the attributes comprising data objects. Equivalently, the distribution of a data set can be considered as a description of the concentration of objects in various regions of the data space. Statisticians have enumerated many types of distributions, e.g., Gaussian (normal), and described their properties. (See Appendix C.) Although statistical approaches for describing distributions can yield powerful analysis techniques, many data sets have distributions that are not well captured by standard statistical distributions.
As a result, many data mining algorithms do not assume a particular statistical distribution for the data they analyze. However, some general aspects of distributions often have a strong impact. For example, suppose a categorical attribute is used as a class variable, where one of the categories occurs 95% of the time, while the other categories together occur only 5% of the time. This skewness in the distribution can make classification difficult as discussed in Section 4.11. (Skewness has other impacts on data analysis that are not discussed here.)
A special case of skewed data is sparsity. For sparse binary, count or continuous data, most attributes of an object have values of 0. In many cases, fewer than 1% of the values are non-zero. In practical terms, sparsity is an
advantage because usually only the non-zero values need to be stored and manipulated. This results in significant savings with respect to computation time and storage. Indeed, some data mining algorithms, such as the association rule mining algorithms described in Chapter 5, work well only for sparse data. Finally, note that often the attributes in sparse data sets are asymmetric attributes.
Resolution It is frequently possible to obtain data at different levels of resolution, and often the properties of the data are different at different resolutions. For instance, the surface of the Earth seems very uneven at a resolution of a few meters, but is relatively smooth at a resolution of tens of kilometers. The patterns in the data also depend on the level of resolution. If the resolution is too fine, a pattern may not be visible or may be buried in noise; if the resolution is too coarse, the pattern can disappear. For example, variations in atmospheric pressure on a scale of hours reflect the movement of storms and other weather systems. On a scale of months, such phenomena are not detectable.
Record Data Much data mining work assumes that the data set is a collection of records (data objects), each of which consists of a fixed set of data fields (attributes). See Figure 2.2(a). For the most basic form of record data, there is no explicit relationship among records or data fields, and every record (object) has the same set of attributes. Record data is usually stored either in flat files or in relational databases. Relational databases are certainly more than a collection of records, but data mining often does not use any of the additional information available in a relational database. Rather, the database serves as a convenient place to find records. Different types of record data are described below and are illustrated in Figure 2.2.
Figure 2.2. Different variations of record data.
Figure 2.2. Full Alternative Text
Transaction or Market Basket Data
Transaction data is a special type of record data, where each record (transaction) involves a set of items. Consider a grocery store. The set of products purchased by a customer during one shopping trip constitutes a transaction, while the individual products that were purchased are the items. This type of data is called market basket data because the items in each record are the products in a person’s “market basket.” Transaction data is a collection of sets of items, but it can be viewed as a set of records whose fields are asymmetric attributes. Most often, the attributes are binary, indicating whether an item was purchased, but more generally, the attributes can be discrete or continuous, such as the number of items purchased or the amount spent on those items. Figure 2.2(b) shows a sample transaction data set. Each row represents the purchases of a particular customer at a particular time.
The Data Matrix If all the data objects in a collection of data have the same fixed set of numeric attributes, then the data objects can be thought of as points (vectors) in a multidimensional space, where each dimension represents a distinct attribute describing the object. A set of such data objects can be interpreted as an m by n matrix, where there are m rows, one for each object, and n columns, one for each attribute. (A representation that has data objects as columns and attributes as rows is also fine.) This matrix is called a data matrix or a pattern matrix. A data matrix is a variation of record data, but because it consists of numeric attributes, standard matrix operation can be applied to transform and manipulate the data. Therefore, the data matrix is the standard data format for most statistical data. Figure 2.2(c) shows a sample data matrix.
The Sparse Data Matrix A sparse data matrix is a special case of a data matrix where the attributes are of the same type and are asymmetric; i.e., only non-zero values are important. Transaction data is an example of a sparse data matrix that has only 0–1
entries. Another common example is document data. In particular, if the order of the terms (words) in a document is ignored—the “bag of words” approach —then a document can be represented as a term vector, where each term is a component (attribute) of the vector and the value of each component is the number of times the corresponding term occurs in the document. This representation of a collection of documents is often called a document-term matrix. Figure 2.2(d) shows a sample document-term matrix. The documents are the rows of this matrix, while the terms are the columns. In practice, only the non-zero entries of sparse data matrices are stored.
Graph-Based Data A graph can sometimes be a convenient and powerful representation for data. We consider two specific cases: (1) the graph captures relationships among data objects and (2) the data objects themselves are represented as graphs.
Data with Relationships among Objects The relationships among objects frequently convey important information. In such cases, the data is often represented as a graph. In particular, the data objects are mapped to nodes of the graph, while the relationships among objects are captured by the links between objects and link properties, such as direction and weight. Consider web pages on the World Wide Web, which contain both text and links to other pages. In order to process search queries, web search engines collect and process web pages to extract their contents. It is well-known, however, that the links to and from each page provide a great deal of information about the relevance of a web page to a query, and thus, must also be taken into consideration. Figure 2.3(a) shows a set of linked web pages. Another important example of such graph data are the social networks, where data objects are people and the relationships among them are their interactions via social media.
Data with Objects That Are Graphs If objects have structure, that is, the objects contain subobjects that have relationships, then such objects are frequently represented as graphs. For example, the structure of chemical compounds can be represented by a graph, where the nodes are atoms and the links between nodes are chemical bonds. Figure 2.3(b) shows a ball-and-stick diagram of the chemical compound benzene, which contains atoms of carbon (black) and hydrogen (gray). A graph representation makes it possible to determine which substructures occur frequently in a set of compounds and to ascertain whether the presence of any of these substructures is associated with the presence or absence of certain chemical properties, such as melting point or heat of formation. Frequent graph mining, which is a branch of data mining that analyzes such data, is considered in Section 6.5.
Figure 2.3.
Different variations of graph data.
Figure 2.3. Full Alternative Text
Ordered Data For some types of data, the attributes have relationships that involve order in time or space. Different types of ordered data are described next and are shown in Figure 2.4.
Sequential Transaction Data Sequential transaction data can be thought of as an extension of transaction data, where each transaction has a time associated with it. Consider a retail transaction data set that also stores the time at which the transaction took place. This time information makes it possible to find patterns such as “candy sales peak before Halloween.” A time can also be associated with each attribute. For example, each record could be the purchase history of a customer, with a listing of items purchased at different times. Using this information, it is possible to find patterns such as “people who buy DVD players tend to buy DVDs in the period immediately following the purchase.”
Figure 2.4(a) shows an example of sequential transaction data. There are five different times—t1, t2, t3, t4, and t5; three different customers—C1, C2, and C3; and five different items—A, B, C, D, and E. In the top table, each row corresponds to the items purchased at a particular time by each customer. For instance, at time t3, customer C2 purchased items A and D. In the bottom table, the same information is displayed, but each row corresponds to a particular customer. Each row contains information about each transaction involving the customer, where a transaction is considered to be a set of items and the time at which those items were purchased. For example, customer C3 bought items A and C at time t2.
Time Series Data Time series data is a special type of ordered data where each record is a time series , i.e., a series of measurements taken over time. For example, a financial data set might contain objects that are time series of the daily prices of various stocks. As another example, consider Figure 2.4(c), which shows a time series of the average monthly temperature for Minneapolis during the years 1982 to 1994. When working with temporal data, such as time series, it is important to consider temporal autocorrelation; i.e., if two measurements are close in time, then the values of those measurements are often very similar.
Figure 2.4. Different variations of ordered data.
Figure 2.4. Full Alternative Text
Sequence Data Sequence data consists of a data set that is a sequence of individual entities, such as a sequence of words or letters. It is quite similar to sequential data, except that there are no time stamps; instead, there are positions in an ordered sequence. For example, the genetic information of plants and animals can be represented in the form of sequences of nucleotides that are known as genes. Many of the problems associated with genetic sequence data involve predicting similarities in the structure and function of genes from similarities in nucleotide sequences. Figure 2.4(b) shows a section of the human genetic code expressed using the four nucleotides from which all DNA is constructed: A, T, G, and C.
Spatial and Spatio-Temporal Data Some objects have spatial attributes, such as positions or areas, in addition to other types of attributes. An example of spatial data is weather data (precipitation, temperature, pressure) that is collected for a variety of geographical locations. Often such measurements are collected over time, and thus, the data consists of time series at various locations. In that case, we refer to the data as spatio-temporal data. Although analysis can be conducted separately for each specific time or location, a more complete analysis of spatio-temporal data requires consideration of both the spatial and temporal aspects of the data.
An important aspect of spatial data is spatial autocorrelation; i.e., objects that are physically close tend to be similar in other ways as well. Thus, two points on the Earth that are close to each other usually have similar values for temperature and rainfall. Note that spatial autocorrelation is analogous to temporal autocorrelation.
Important examples of spatial and spatio-temporal data are the science and
engineering data sets that are the result of measurements or model output taken at regularly or irregularly distributed points on a two- or three- dimensional grid or mesh. For instance, Earth science data sets record the temperature or pressure measured at points (grid cells) on latitude–longitude spherical grids of various resolutions, e.g., 1° by 1°. See Figure 2.4(d). As another example, in the simulation of the flow of a gas, the speed and direction of flow at various instants in time can be recorded for each grid point in the simulation. A different type of spatio-temporal data arises from tracking the trajectories of objects, e.g., vehicles, in time and space.
Handling Non-Record Data Most data mining algorithms are designed for record data or its variations, such as transaction data and data matrices. Record-oriented techniques can be applied to non-record data by extracting features from data objects and using these features to create a record corresponding to each object. Consider the chemical structure data that was described earlier. Given a set of common substructures, each compound can be represented as a record with binary attributes that indicate whether a compound contains a specific substructure. Such a representation is actually a transaction data set, where the transactions are the compounds and the items are the substructures.
In some cases, it is easy to represent the data in a record format, but this type of representation does not capture all the information in the data. Consider spatio-temporal data consisting of a time series from each point on a spatial grid. This data is often stored in a data matrix, where each row represents a location and each column represents a particular point in time. However, such a representation does not explicitly capture the time relationships that are present among attributes and the spatial relationships that exist among objects. This does not mean that such a representation is inappropriate, but rather that these relationships must be taken into consideration during the analysis. For example, it would not be a good idea to use a data mining technique that ignores the temporal autocorrelation of the attributes or the spatial autocorrelation of the data objects, i.e., the locations on the spatial grid.
2.2 Data Quality Data mining algorithms are often applied to data that was collected for another purpose, or for future, but unspecified applications. For that reason, data mining cannot usually take advantage of the significant benefits of “ad- dressing quality issues at the source.” In contrast, much of statistics deals with the design of experiments or surveys that achieve a prespecified level of data quality. Because preventing data quality problems is typically not an option, data mining focuses on (1) the detection and correction of data quality problems and (2) the use of algorithms that can tolerate poor data quality. The first step, detection and correction, is often called data cleaning.
The following sections discuss specific aspects of data quality. The focus is on measurement and data collection issues, although some application-related issues are also discussed.
2.2.1 Measurement and Data Collection Issues It is unrealistic to expect that data will be perfect. There may be problems due to human error, limitations of measuring devices, or flaws in the data collection process. Values or even entire data objects can be missing. In other cases, there can be spurious or duplicate objects; i.e., multiple data objects that all correspond to a single “real” object. For example, there might be two different records for a person who has recently lived at two different addresses. Even if all the data is present and “looks fine,” there may be inconsistencies—a person has a height of 2 meters, but weighs only 2 kilograms.
In the next few sections, we focus on aspects of data quality that are related to data measurement and collection. We begin with a definition of measurement and data collection errors and then consider a variety of problems that
involve measurement error: noise, artifacts, bias, precision, and accuracy. We conclude by discussing data quality issues that involve both measurement and data collection problems: outliers, missing and inconsistent values, and duplicate data.
Measurement and Data Collection Errors The term measurement error refers to any problem resulting from the measurement process. A common problem is that the value recorded differs from the true value to some extent. For continuous attributes, the numerical difference of the measured and true value is called the error. The term data collection error refers to errors such as omitting data objects or attribute values, or inappropriately including a data object. For example, a study of animals of a certain species might include animals of a related species that are similar in appearance to the species of interest. Both measurement errors and data collection errors can be either systematic or random.
We will only consider general types of errors. Within particular domains, certain types of data errors are commonplace, and well-developed techniques often exist for detecting and/or correcting these errors. For example, keyboard errors are common when data is entered manually, and as a result, many data entry programs have techniques for detecting and, with human intervention, correcting such errors.
Noise and Artifacts Noise is the random component of a measurement error. It typically involves the distortion of a value or the addition of spurious objects. Figure 2.5 shows a time series before and after it has been disrupted by random noise. If a bit more noise were added to the time series, its shape would be lost. Figure 2.6 shows a set of data points before and after some noise points (indicated by ‘+’s) have been added. Notice that some of the noise points are intermixed
with the non-noise points.
Figure 2.5. Noise in a time series context.
Figure 2.5. Full Alternative Text
Figure 2.6. Noise in a spatial context.
Figure 2.6. Full Alternative Text
The term noise is often used in connection with data that has a spatial or temporal component. In such cases, techniques from signal or image processing can frequently be used to reduce noise and thus, help to discover patterns (signals) that might be “lost in the noise.” Nonetheless, the elimination of noise is frequently difficult, and much work in data mining focuses on devising robust algorithms that produce acceptable results even when noise is present.
Data errors can be the result of a more deterministic phenomenon, such as a streak in the same place on a set of photographs. Such deterministic distortions of the data are often referred to as artifacts.
Precision, Bias, and Accuracy In statistics and experimental science, the quality of the measurement process and the resulting data are measured by precision and bias. We provide the standard definitions, followed by a brief discussion. For the following definitions, we assume that we make repeated measurements of the same underlying quantity.
Definition 2.3 (Precision). The closeness of repeated measurements (of the same quantity) to one another.
Definition 2.4 (Bias).
A systematic variation of measurements from the quantity being measured.
Precision is often measured by the standard deviation of a set of values, while bias is measured by taking the difference between the mean of the set of values and the known value of the quantity being measured. Bias can be determined only for objects whose measured quantity is known by means external to the current situation. Suppose that we have a standard laboratory weight with a mass of 1g and want to assess the precision and bias of our new laboratory scale. We weigh the mass five times, and obtain the following five values:{ 1.015, 0.990, 1.013, 1.001, 0.986}. The mean of these values is 1.001, and hence, the bias is 0.001. The precision, as measured by the standard deviation, is 0.013.
It is common to use the more general term, accuracy , to refer to the degree of measurement error in data.
Definition 2.5 (Accuracy) The closeness of measurements to the true value of the quantity being measured.
Accuracy depends on precision and bias, but there is no specific formula for accuracy in terms of these two quantities.
One important aspect of accuracy is the use of significant digits. The goal is to use only as many digits to represent the result of a measurement or calculation as are justified by the precision of the data. For example, if the length of an object is measured with a meter stick whose smallest markings are millimeters, then we should record the length of data only to the nearest millimeter. The precision of such a measurement would be ± 0.5mm. We do not review the details of working with significant digits because most readers will have encountered them in previous courses and they are covered in considerable depth in science, engineering, and statistics textbooks.
Issues such as significant digits, precision, bias, and accuracy are sometimes overlooked, but they are important for data mining as well as statistics and
science. Many times, data sets do not come with information about the precision of the data, and furthermore, the programs used for analysis return results without any such information. Nonetheless, without some understanding of the accuracy of the data and the results, an analyst runs the risk of committing serious data analysis blunders.
Outliers Outliers are either (1) data objects that, in some sense, have characteristics that are different from most of the other data objects in the data set, or (2) values of an attribute that are unusual with respect to the typical values for that attribute. Alternatively, they can be referred to as anomalous objects or values. There is considerable leeway in the definition of an outlier, and many different definitions have been proposed by the statistics and data mining communities. Furthermore, it is important to distinguish between the notions of noise and outliers. Unlike noise, outliers can be legitimate data objects or values that we are interested in detecting. For instance, in fraud and network intrusion detection, the goal is to find unusual objects or events from among a large number of normal ones. Chapter 9 discusses anomaly detection in more detail.
Missing Values It is not unusual for an object to be missing one or more attribute values. In some cases, the information was not collected; e.g., some people decline to give their age or weight. In other cases, some attributes are not applicable to all objects; e.g., often, forms have conditional parts that are filled out only when a person answers a previous question in a certain way, but for simplicity, all fields are stored. Regardless, missing values should be taken into account during the data analysis.
There are several strategies (and variations on these strategies) for dealing with missing data, each of which is appropriate in certain circumstances. These strategies are listed next, along with an indication of their advantages
and disadvantages.
Eliminate Data Objects or Attributes A simple and effective strategy is to eliminate objects with missing values. However, even a partially specified data object contains some information, and if many objects have missing values, then a reliable analysis can be difficult or impossible. Nonetheless, if a data set has only a few objects that have missing values, then it may be expedient to omit them. A related strategy is to eliminate attributes that have missing values. This should be done with caution, however, because the eliminated attributes may be the ones that are critical to the analysis.
Estimate Missing Values Sometimes missing data can be reliably estimated. For example, consider a time series that changes in a reasonably smooth fashion, but has a few, widely scattered missing values. In such cases, the missing values can be estimated (interpolated) by using the remaining values. As another example, consider a data set that has many similar data points. In this situation, the attribute values of the points closest to the point with the missing value are often used to estimate the missing value. If the attribute is continuous, then the average attribute value of the nearest neighbors is used; if the attribute is categorical, then the most commonly occurring attribute value can be taken. For a concrete illustration, consider precipitation measurements that are recorded by ground stations. For areas not containing a ground station, the precipitation can be estimated using values observed at nearby ground stations.
Ignore the Missing Value during
Analysis Many data mining approaches can be modified to ignore missing values. For example, suppose that objects are being clustered and the similarity between pairs of data objects needs to be calculated. If one or both objects of a pair have missing values for some attributes, then the similarity can be calculated by using only the attributes that do not have missing values. It is true that the similarity will only be approximate, but unless the total number of attributes is small or the number of missing values is high, this degree of inaccuracy may not matter much. Likewise, many classification schemes can be modified to work with missing values.
Inconsistent Values Data can contain inconsistent values. Consider an address field, where both a zip code and city are listed, but the specified zip code area is not contained in that city. It is possible that the individual entering this information transposed two digits, or perhaps a digit was misread when the information was scanned from a handwritten form. Regardless of the cause of the inconsistent values, it is important to detect and, if possible, correct such problems.
Some types of inconsistences are easy to detect. For instance, a person’s height should not be negative. In other cases, it can be necessary to consult an external source of information. For example, when an insurance company processes claims for reimbursement, it checks the names and addresses on the reimbursement forms against a database of its customers.
Once an inconsistency has been detected, it is sometimes possible to correct the data. A product code may have “check” digits, or it may be possible to double-check a product code against a list of known product codes, and then correct the code if it is incorrect, but close to a known code. The correction of an inconsistency requires additional or redundant information.
Example 2.6 (Inconsistent Sea Surface Temperature). This example illustrates an inconsistency in actual time series data that measures the sea surface temperature (SST) at various points on the ocean. SST data was originally collected using ocean-based measurements from ships or buoys, but more recently, satellites have been used to gather the data. To create a long-term data set, both sources of data must be used. However, because the data comes from different sources, the two parts of the data are subtly different. This discrepancy is visually displayed in Figure 2.7, which shows the correlation of SST values between pairs of years. If a pair of years has a positive correlation, then the location corresponding to the pair of years is colored white; otherwise it is colored black. (Seasonal variations were removed from the data since, otherwise, all the years would be highly correlated.) There is a distinct change in behavior where the data has been put together in 1983. Years within each of the two groups, 1958–1982 and 1983– 1999, tend to have a positive correlation with one another, but a negative correlation with years in the other group. This does not mean that this data should not be used, only that the analyst should consider the potential impact of such discrepancies on the data mining analysis.
Figure 2.7. Correlation of SST data between pairs of years. White areas indicate positive correlation. Black areas indicate negative correlation.
Figure 2.7. Full Alternative Text
Duplicate Data A data set can include data objects that are duplicates, or almost duplicates, of one another. Many people receive duplicate mailings because they appear in a database multiple times under slightly different names. To detect and
eliminate such duplicates, two main issues must be addressed. First, if there are two objects that actually represent a single object, then one or more values of corresponding attributes are usually different, and these inconsistent values must be resolved. Second, care needs to be taken to avoid accidentally combining data objects that are similar, but not duplicates, such as two distinct people with identical names. The term deduplication is often used to refer to the process of dealing with these issues.
In some cases, two or more objects are identical with respect to the attributes measured by the database, but they still represent different objects. Here, the duplicates are legitimate, but can still cause problems for some algorithms if the possibility of identical objects is not specifically accounted for in their design. An example of this is given in Exercise 13 on page 108.
2.2.2 Issues Related to Applications Data quality issues can also be considered from an application viewpoint as expressed by the statement “data is of high quality if it is suitable for its intended use.” This approach to data quality has proven quite useful, particularly in business and industry. A similar viewpoint is also present in statistics and the experimental sciences, with their emphasis on the careful design of experiments to collect the data relevant to a specific hypothesis. As with quality issues at the measurement and data collection level, many issues are specific to particular applications and fields. Again, we consider only a few of the general issues.
Timeliness Some data starts to age as soon as it has been collected. In particular, if the data provides a snapshot of some ongoing phenomenon or process, such as the purchasing behavior of customers or web browsing patterns, then this snapshot represents reality for only a limited time. If the data is out of date, then so are the models and patterns that are based on it.
Relevance The available data must contain the information necessary for the application. Consider the task of building a model that predicts the accident rate for drivers. If information about the age and gender of the driver is omitted, then it is likely that the model will have limited accuracy unless this information is indirectly available through other attributes.
Making sure that the objects in a data set are relevant is also challenging. A common problem is sampling bias, which occurs when a sample does not contain different types of objects in proportion to their actual occurrence in the population. For example, survey data describes only those who respond to the survey. (Other aspects of sampling are discussed further in Section 2.3.2.) Because the results of a data analysis can reflect only the data that is present, sampling bias will typically lead to erroneous results when applied to the broader population.
Knowledge about the Data Ideally, data sets are accompanied by documentation that describes different aspects of the data; the quality of this documentation can either aid or hinder the subsequent analysis. For example, if the documentation identifies several attributes as being strongly related, these attributes are likely to provide highly redundant information, and we usually decide to keep just one. (Consider sales tax and purchase price.) If the documentation is poor, however, and fails to tell us, for example, that the missing values for a particular field are indicated with a -9999, then our analysis of the data may be faulty. Other important characteristics are the precision of the data, the type of features (nominal, ordinal, interval, ratio), the scale of measurement (e.g., meters or feet for length), and the origin of the data.
2.3 Data Preprocessing In this section, we consider which preprocessing steps should be applied to make the data more suitable for data mining. Data preprocessing is a broad area and consists of a number of different strategies and techniques that are interrelated in complex ways. We will present some of the most important ideas and approaches, and try to point out the interrelationships among them. Specifically, we will discuss the following topics:
Aggregation
Sampling
Dimensionality reduction
Feature subset selection
Feature creation
Discretization and binarization
Variable transformation
Roughly speaking, these topics fall into two categories: selecting data objects and attributes for the analysis or for creating/changing the attributes. In both cases, the goal is to improve the data mining analysis with respect to time, cost, and quality. Details are provided in the following sections.
A quick note about terminology: In the following, we sometimes use synonyms for attribute, such as feature or variable, in order to follow common usage.
2.3.1 Aggregation
Sometimes “less is more,” and this is the case with aggregation , the combining of two or more objects into a single object. Consider a data set consisting of transactions (data objects) recording the daily sales of products in various store locations (Minneapolis, Chicago, Paris, …) for different days over the course of a year. See Table 2.4. One way to aggregate transactions for this data set is to replace all the transactions of a single store with a single storewide transaction. This reduces the hundreds or thousands of transactions that occur daily at a specific store to a single daily transaction, and the number of data objects per day is reduced to the number of stores.
Table 2.4. Data set containing information about customer purchases. Transaction ID Item Store Location Date Price …
⋮ ⋮ ⋮ ⋮ ⋮
101123 Watch Chicago 09/06/04 $25.99 … 101123 Battery Chicago 09/06/04 $5.99 … 101124 Shoes Minneapolis 09/06/04 $75.00 …
An obvious issue is how an aggregate transaction is created; i.e., how the values of each attribute are combined across all the records corresponding to a particular location to create the aggregate transaction that represents the sales of a single store or date. Quantitative attributes, such as price, are typically aggregated by taking a sum or an average. A qualitative attribute, such as item, can either be omitted or summarized in terms of a higher level category, e.g., televisions versus electronics.
The data in Table 2.4 can also be viewed as a multidimensional array, where each attribute is a dimension. From this viewpoint, aggregation is the process of eliminating attributes, such as the type of item, or reducing the number of values for a particular attribute; e.g., reducing the possible values for date from 365 days to 12 months. This type of aggregation is commonly used in
Online Analytical Processing (OLAP). References to OLAP are given in the bibliographic Notes.
There are several motivations for aggregation. First, the smaller data sets resulting from data reduction require less memory and processing time, and hence, aggregation often enables the use of more expensive data mining algorithms. Second, aggregation can act as a change of scope or scale by providing a high-level view of the data instead of a low-level view. In the previous example, aggregating over store locations and months gives us a monthly, per store view of the data instead of a daily, per item view. Finally, the behavior of groups of objects or attributes is often more stable than that of individual objects or attributes. This statement reflects the statistical fact that aggregate quantities, such as averages or totals, have less variability than the individual values being aggregated. For totals, the actual amount of variation is larger than that of individual objects (on average), but the percentage of the variation is smaller, while for means, the actual amount of variation is less than that of individual objects (on average). A disadvantage of aggregation is the potential loss of interesting details. In the store example, aggregating over months loses information about which day of the week has the highest sales.
Example 2.7 (Australian Precipitation). This example is based on precipitation in Australia from the period 1982– 1993. Figure 2.8(a) shows a histogram for the standard deviation of average monthly precipitation for 3,030 0.5° by 0.5° grid cells in Australia, while Figure 2.8(b) shows a histogram for the standard deviation of the average yearly precipitation for the same locations. The average yearly precipitation has less variability than the average monthly precipitation. All precipitation measurements (and their standard deviations) are in centimeters.
Figure 2.8. Histograms of standard deviation for monthly and yearly precipitation in Australia for the period 1982–1993.
Figure 2.8. Full Alternative Text
2.3.2 Sampling Sampling is a commonly used approach for selecting a subset of the data objects to be analyzed. In statistics, it has long been used for both the preliminary investigation of the data and the final data analysis. Sampling can also be very useful in data mining. However, the motivations for sampling in statistics and data mining are often different. Statisticians use sampling because obtaining the entire set of data of interest is too expensive or time consuming, while data miners usually sample because it is too computationally expensive in terms of the memory or time required to process all the data. In some cases, using a sampling algorithm can reduce the
data size to the point where a better, but more computationally expensive algorithm can be used.
The key principle for effective sampling is the following: Using a sample will work almost as well as using the entire data set if the sample is representative. In turn, a sample is representative if it has approximately the same property (of interest) as the original set of data. If the mean (average) of the data objects is the property of interest, then a sample is representative if it has a mean that is close to that of the original data. Because sampling is a statistical process, the representativeness of any particular sample will vary, and the best that we can do is choose a sampling scheme that guarantees a high probability of getting a representative sample. As discussed next, this involves choosing the appropriate sample size and sampling technique.
Sampling Approaches There are many sampling techniques, but only a few of the most basic ones and their variations will be covered here. The simplest type of sampling is simple random sampling. For this type of sampling, there is an equal probability of selecting any particular object. There are two variations on random sampling (and other sampling techniques as well): (1) sampling without replacement —as each object is selected, it is removed from the set of all objects that together constitute the population , and (2) sampling with replacement —objects are not removed from the population as they are selected for the sample. In sampling with replacement, the same object can be picked more than once. The samples produced by the two methods are not much different when samples are relatively small compared to the data set size, but sampling with replacement is simpler to analyze because the probability of selecting any object remains constant during the sampling process.
When the population consists of different types of objects, with widely different numbers of objects, simple random sampling can fail to adequately represent those types of objects that are less frequent. This can cause problems when the analysis requires proper representation of all object types. For example, when building classification models for rare classes, it is critical
that the rare classes be adequately represented in the sample. Hence, a sampling scheme that can accommodate differing frequencies for the object types of interest is needed. Stratified sampling , which starts with prespecified groups of objects, is such an approach. In the simplest version, equal numbers of objects are drawn from each group even though the groups are of different sizes. In another variation, the number of objects drawn from each group is proportional to the size of that group.
Example 2.8 (Sampling and Loss of Information). Once a sampling technique has been selected, it is still necessary to choose the sample size. Larger sample sizes increase the probability that a sample will be representative, but they also eliminate much of the advantage of sampling. Conversely, with smaller sample sizes, patterns can be missed or erroneous patterns can be detected. Figure 2.9(a) shows a data set that contains 8000 two-dimensional points, while Figures 2.9(b) and 2.9(c) show samples from this data set of size 2000 and 500, respectively. Although most of the structure of this data set is present in the sample of 2000 points, much of the structure is missing in the sample of 500 points.
Figure 2.9. Example of the loss of structure with sampling.
Example 2.9 (Determining the Proper Sample Size). To illustrate that determining the proper sample size requires a methodical approach, consider the following task.
Given a set of data consisting of a small number of almost equalsized groups, find at least one representative point for each of the groups. Assume that the objects in each group are highly similar to each other, but not very similar to objects in different groups. Figure 2.10(a) shows an idealized set of clusters (groups) from which these points might be drawn.
Figure 2.10. Finding representative points from 10 groups.
Figure 2.10. Full Alternative Text
This problem can be efficiently solved using sampling. One approach is to take a small sample of data points, compute the pairwise similarities between points, and then form groups of points that are highly similar. The desired set of representative points is then obtained by taking one point from each of these groups. To follow this approach, however, we need to determine a sample size that would guarantee, with a high probability, the desired outcome; that is, that at least one point will be obtained from each cluster. Figure 2.10(b) shows the probability of getting one object from each of the 10 groups as the sample size runs from 10 to 60. Interestingly, with a sample size of 20, there is little chance (20%) of getting a sample that includes all 10 clusters. Even with a sample size of 30, there is still a moderate chance (almost 40%) of getting a sample that doesn’t contain objects from all 10 clusters. This issue is further explored in the context of clustering by Exercise 4 on page 603.
Progressive Sampling The proper sample size can be difficult to determine, so adaptive or progressive sampling schemes are sometimes used. These approaches start with a small sample, and then increase the sample size until a sample of sufficient size has been obtained. While this technique eliminates the need to determine the correct sample size initially, it requires that there be a way to evaluate the sample to judge if it is large enough.
Suppose, for instance, that progressive sampling is used to learn a predictive model. Although the accuracy of predictive models increases as the sample size increases, at some point the increase in accuracy levels off. We want to stop increasing the sample size at this leveling-off point. By keeping track of the change in accuracy of the model as we take progressively larger samples,
and by taking other samples close to the size of the current one, we can get an estimate of how close we are to this leveling-off point, and thus, stop sampling.
2.3.3 Dimensionality Reduction Data sets can have a large number of features. Consider a set of documents, where each document is represented by a vector whose components are the frequencies with which each word occurs in the document. In such cases, there are typically thousands or tens of thousands of attributes (components), one for each word in the vocabulary. As another example, consider a set of time series consisting of the daily closing price of various stocks over a period of 30 years. In this case, the attributes, which are the prices on specific days, again number in the thousands.
There are a variety of benefits to dimensionality reduction. A key benefit is that many data mining algorithms work better if the dimensionality—the number of attributes in the data—is lower. This is partly because dimensionality reduction can eliminate irrelevant features and reduce noise and partly because of the curse of dimensionality, which is explained below. Another benefit is that a reduction of dimensionality can lead to a more understandable model because the model usually involves fewer attributes. Also, dimensionality reduction may allow the data to be more easily visualized. Even if dimensionality reduction doesn’t reduce the data to two or three dimensions, data is often visualized by looking at pairs or triplets of attributes, and the number of such combinations is greatly reduced. Finally, the amount of time and memory required by the data mining algorithm is reduced with a reduction in dimensionality.
The term dimensionality reduction is often reserved for those techniques that reduce the dimensionality of a data set by creating new attributes that are a combination of the old attributes. The reduction of dimensionality by selecting attributes that are a subset of the old is known as feature subset selection or feature selection. It will be discussed in Section 2.3.4.
In the remainder of this section, we briefly introduce two important topics:
the curse of dimensionality and dimensionality reduction techniques based on linear algebra approaches such as principal components analysis (PCA). More details on dimensionality reduction can be found in Appendix B.
The Curse of Dimensionality The curse of dimensionality refers to the phenomenon that many types of data analysis become significantly harder as the dimensionality of the data increases. Specifically, as dimensionality increases, the data becomes increasingly sparse in the space that it occupies. Thus, the data objects we observe are quite possibly not a representative sample of all possible objects. For classification, this can mean that there are not enough data objects to allow the creation of a model that reliably assigns a class to all possible objects. For clustering, the differences in density and in the distances between points, which are critical for clustering, become less meaningful. (This is discussed further in Sections 8.1.2, 8.4.6, and 8.4.8.) As a result, many clustering and classification algorithms (and other data analysis algorithms) have trouble with high-dimensional data leading to reduced classification accuracy and poor quality clusters.
Linear Algebra Techniques for Dimensionality Reduction Some of the most common approaches for dimensionality reduction, particularly for continuous data, use techniques from linear algebra to project the data from a high-dimensional space into a lower-dimensional space. Principal Components Analysis (PCA) is a linear algebra technique for continuous attributes that finds new attributes (principal components) that (1) are linear combinations of the original attributes, (2) are orthogonal (perpendicular) to each other, and (3) capture the maximum amount of variation in the data. For example, the first two principal components capture as much of the variation in the data as is possible with two orthogonal attributes that are linear combinations of the original attributes. Singular
Value Decomposition (SVD) is a linear algebra technique that is related to PCA and is also commonly used for dimensionality reduction. For additional details, see Appendices A and B.
2.3.4 Feature Subset Selection Another way to reduce the dimensionality is to use only a subset of the features. While it might seem that such an approach would lose information, this is not the case if redundant and irrelevant features are present. Redundant features duplicate much or all of the information contained in one or more other attributes. For example, the purchase price of a product and the amount of sales tax paid contain much of the same information. Irrelevant features contain almost no useful information for the data mining task at hand. For instance, students’ ID numbers are irrelevant to the task of predicting students’ grade point averages. Redundant and irrelevant features can reduce classification accuracy and the quality of the clusters that are found.
While some irrelevant and redundant attributes can be eliminated immediately by using common sense or domain knowledge, selecting the best subset of features frequently requires a systematic approach. The ideal approach to feature selection is to try all possible subsets of features as input to the data mining algorithm of interest, and then take the subset that produces the best results. This method has the advantage of reflecting the objective and bias of the data mining algorithm that will eventually be used. Unfortunately, since the number of subsets involving n attributes is 2
n , such
an approach is impractical in most situations and alternative strategies are needed. There are three standard approaches to feature selection: embedded, filter, and wrapper.
Embedded approaches Feature selection occurs naturally as part of the data mining algorithm. Specifically, during the operation of the data mining algorithm, the algorithm
itself decides which attributes to use and which to ignore. Algorithms for building decision tree classifiers, which are discussed in Chapter 3, often operate in this manner.
Filter approaches Features are selected before the data mining algorithm is run, using some approach that is independent of the data mining task. For example, we might select sets of attributes whose pairwise correlation is as low as possible so that the attributes are non-redundant.
Wrapper approaches These methods use the target data mining algorithm as a black box to find the best subset of attributes, in a way similar to that of the ideal algorithm described above, but typically without enumerating all possible subsets.
Because the embedded approaches are algorithm-specific, only the filter and wrapper approaches will be discussed further here.
An Architecture for Feature Subset Selection It is possible to encompass both the filter and wrapper approaches within a common architecture. The feature selection process is viewed as consisting of four parts: a measure for evaluating a subset, a search strategy that controls the generation of a new subset of features, a stopping criterion, and a validation procedure. Filter methods and wrapper methods differ only in the way in which they evaluate a subset of features. For a wrapper method, subset evaluation uses the target data mining algorithm, while for a filter approach, the evaluation technique is distinct from the target data mining algorithm. The following discussion provides some details of this approach,
which is summarized in Figure 2.11.
Figure 2.11. Flowchart of a feature subset selection process.
Figure 2.11. Full Alternative Text
Conceptually, feature subset selection is a search over all possible subsets of features. Many different types of search strategies can be used, but the search strategy should be computationally inexpensive and should find optimal or near optimal sets of features. It is usually not possible to satisfy both requirements, and thus, trade-offs are necessary.
An integral part of the search is an evaluation step to judge how the current subset of features compares to others that have been considered. This requires an evaluation measure that attempts to determine the goodness of a subset of attributes with respect to a particular data mining task, such as classification or clustering. For the filter approach, such measures attempt to predict how
well the actual data mining algorithm will perform on a given set of attributes. For the wrapper approach, where evaluation consists of actually running the target data mining algorithm, the subset evaluation function is simply the criterion normally used to measure the result of the data mining.
Because the number of subsets can be enormous and it is impractical to examine them all, some sort of stopping criterion is necessary. This strategy is usually based on one or more conditions involving the following: the number of iterations, whether the value of the subset evaluation measure is optimal or exceeds a certain threshold, whether a subset of a certain size has been obtained, and whether any improvement can be achieved by the options available to the search strategy.
Finally, once a subset of features has been selected, the results of the target data mining algorithm on the selected subset should be validated. A straightforward validation approach is to run the algorithm with the full set of features and compare the full results to results obtained using the subset of features. Hopefully, the subset of features will produce results that are better than or almost as good as those produced when using all features. Another validation approach is to use a number of different feature selection algorithms to obtain subsets of features and then compare the results of running the data mining algorithm on each subset.
Feature Weighting Feature weighting is an alternative to keeping or eliminating features. More important features are assigned a higher weight, while less important features are given a lower weight. These weights are sometimes assigned based on domain knowledge about the relative importance of features. Alternatively, they can sometimes be determined automatically. For example, some classification schemes, such as support vector machines (Chapter 4), produce classification models in which each feature is given a weight. Features with larger weights play a more important role in the model. The normalization of objects that takes place when computing the cosine similarity (Section 2.4.5) can also be regarded as a type of feature weighting.
2.3.5 Feature Creation It is frequently possible to create, from the original attributes, a new set of attributes that captures the important information in a data set much more effectively. Furthermore, the number of new attributes can be smaller than the number of original attributes, allowing us to reap all the previously described benefits of dimensionality reduction. Two related methodologies for creating new attributes are described next: feature extraction and mapping the data to a new space.
Feature Extraction The creation of a new set of features from the original raw data is known as feature extraction. Consider a set of photographs, where each photograph is to be classified according to whether it contains a human face. The raw data is a set of pixels, and as such, is not suitable for many types of classification algorithms. However, if the data is processed to provide higher-level features, such as the presence or absence of certain types of edges and areas that are highly correlated with the presence of human faces, then a much broader set of classification techniques can be applied to this problem.
Unfortunately, in the sense in which it is most commonly used, feature extraction is highly domain-specific. For a particular field, such as image processing, various features and the techniques to extract them have been developed over a period of time, and often these techniques have limited applicability to other fields. Consequently, whenever data mining is applied to a relatively new area, a key task is the development of new features and feature extraction methods.
Although feature extraction is often complicated, Example 2.10 illustrates that it can be relatively straightforward.
Example 2.10 (Density).
Consider a data set consisting of information about historical artifacts, which, along with other information, contains the volume and mass of each artifact. For simplicity, assume that these artifacts are made of a small number of materials (wood, clay, bronze, gold) and that we want to classify the artifacts with respect to the material of which they are made. In this case, a density feature constructed from the mass and volume features, i.e., density =mass/volume , would most directly yield an accurate classification. Although there have been some attempts to automatically perform such simple feature extraction by exploring basic mathematical combinations of existing attributes, the most common approach is to construct features using domain expertise.
Mapping the Data to a New Space A totally different view of the data can reveal important and interesting features. Consider, for example, time series data, which often contains periodic patterns. If there is only a single periodic pattern and not much noise, then the pattern is easily detected. If, on the other hand, there are a number of periodic patterns and a significant amount of noise, then these patterns are hard to detect. Such patterns can, nonetheless, often be detected by applying a Fourier transform to the time series in order to change to a representation in which frequency information is explicit. In Example 2.11, it will not be necessary to know the details of the Fourier transform. It is enough to know that, for each time series, the Fourier transform produces a new data object whose attributes are related to frequencies.
Example 2.11 (Fourier Analysis). The time series presented in Figure 2.12(b) is the sum of three other time series, two of which are shown in Figure 2.12(a) and have frequencies of 7 and 17 cycles per second, respectively. The third time series is random noise. Figure 2.12(c) shows the power spectrum that can be computed after applying a Fourier transform to the original time series. (Informally, the power spectrum is proportional to the square of each frequency attribute.) In
spite of the noise, there are two peaks that correspond to the periods of the two original, non-noisy time series. Again, the main point is that better features can reveal important aspects of the data.
Figure 2.12. Application of the Fourier transform to identify the underlying frequencies in time series data.
Figure 2.12. Full Alternative Text
Many other sorts of transformations are also possible. Besides the Fourier transform, the wavelet transform has also proven very useful for time series and other types of data.
2.3.6 Discretization and Binarization Some data mining algorithms, especially certain classification algorithms, require that the data be in the form of categorical attributes. Algorithms that
find association patterns require that the data be in the form of binary attributes. Thus, it is often necessary to transform a continuous attribute into a categorical attribute (discretization), and both continuous and discrete attributes may need to be transformed into one or more binary attributes (binarization). Additionally, if a categorical attribute has a large number of values (categories), or some values occur infrequently, then it can be beneficial for certain data mining tasks to reduce the number of categories by combining some of the values.
As with feature selection, the best discretization or binarization approach is the one that “produces the best result for the data mining algorithm that will be used to analyze the data.” It is typically not practical to apply such a criterion directly. Consequently, discretization or binarization is performed in a way that satisfies a criterion that is thought to have a relationship to good performance for the data mining task being considered. In general, the best discretization depends on the algorithm being used, as well as the other attributes being considered. Typically, however, the discretization of each attribute is considered in isolation.
Binarization A simple technique to binarize a categorical attribute is the following: If there are m categorical values, then uniquely assign each original value to an integer in the interval [0, m−1]. If the attribute is ordinal, then order must be maintained by the assignment. (Note that even if the attribute is originally represented using integers, this process is necessary if the integers are not in the interval [0, m−1].) Next, convert each of these m integers to a binary number. Since n=[log2(m)] binary digits are required to represent these integers, represent these binary numbers using n binary attributes. To illustrate, a categorical variable with 5 values {awful, poor, OK, good, great} would require three binary variables x1, x2, and x3. The conversion is shown in Table 2.5.
Table 2.5. Conversion of a
categorical attribute to three binary attributes. Categorical Value Integer Value x1 x2 x3
awful 0 0 0 0 poor 1 0 0 1 OK 2 0 1 0
good 3 0 1 1 great 4 1 0 0
Such a transformation can cause complications, such as creating unintended relationships among the transformed attributes. For example, in Table 2.5, attributes x2 and x3 are correlated because information about the good value is encoded using both attributes. Furthermore, association analysis requires asymmetric binary attributes, where only the presence of the attribute (value =1). is important. For association problems, it is therefore necessary to introduce one asymmetric binary attribute for each categorical value, as shown in Table 2.6. If the number of resulting attributes is too large, then the techniques described in the following sections can be used to reduce the number of categorical values before binarization.
Table 2.6. Conversion of a categorical attribute to five asymmetric binary attributes. Categorical Value Integer Value x1 x2 x3 x4 x5
awful 0 1 0 0 0 0 poor 1 0 1 0 0 0 OK 2 0 0 1 0 0
good 3 0 0 0 1 0 great 4 0 0 0 0 1
Likewise, for association problems, it can be necessary to replace a single binary attribute with two asymmetric binary attributes. Consider a binary attribute that records a person’s gender, male or female. For traditional association rule algorithms, this information needs to be transformed into two asymmetric binary attributes, one that is a 1 only when the person is male and one that is a 1 only when the person is female. (For asymmetric binary attributes, the information representation is somewhat inefficient in that two bits of storage are required to represent each bit of information.)
Discretization of Continuous Attributes Discretization is typically applied to attributes that are used in classification or association analysis. Transformation of a continuous attribute to a categorical attribute involves two subtasks: deciding how many categories,n , to have and determining how to map the values of the continuous attribute to these categories. In the first step, after the values of the continuous attribute are sorted, they are then divided into n intervals by specifying n−1 split points. In the second, rather trivial step, all the values in one interval are mapped to the same categorical value. Therefore, the problem of discretization is one of deciding how many split points to choose and where to place them. The result can be represented either as a set of intervals {(x0, x1], (x1, x2],…, (xn−1, xn)}, where x0 and xn can be +∞ or −∞, respectively, or equivalently, as a series of inequalities x0<x≤x1, …, xn −1<x<xn.
Unsupervised Discretization A basic distinction between discretization methods for classification is whether class information is used (supervised) or not (unsupervised). If class
information is not used, then relatively simple approaches are common. For instance, the equal width approach divides the range of the attribute into a user-specified number of intervals each having the same width. Such an approach can be badly affected by outliers, and for that reason, an equal frequency (equal depth) approach, which tries to put the same number of objects into each interval, is often preferred. As another example of unsupervised discretization, a clustering method, such as K-means (see Chapter 7), can also be used. Finally, visually inspecting the data can sometimes be an effective approach.
Example 2.12 (Discretization Techniques). This example demonstrates how these approaches work on an actual data set. Figure 2.13(a) shows data points belonging to four different groups, along with two outliers—the large dots on either end. The techniques of the previous paragraph were applied to discretize the x values of these data points into four categorical values. (Points in the data set have a random y component to make it easy to see how many points are in each group.) Visually inspecting the data works quite well, but is not automatic, and thus, we focus on the other three approaches. The split points produced by the techniques equal width, equal frequency, and K-means are shown in Figures 2.13(b), 2.13(c), and 2.13(d), respectively. The split points are represented as dashed lines.
Figure 2.13.
Different discretization techniques.
Figure 2.13. Full Alternative Text
In this particular example, if we measure the performance of a discretization technique by the extent to which different objects that clump together have the same categorical value, then K-means performs best, followed by equal frequency, and finally, equal width. More generally, the best discretization will depend on the application and often involves domain-specific discretization. For example, the discretization of people into low income, middle income, and high income is based on economic factors.
Supervised Discretization If classification is our application and class labels are known for some data objects, then discretization approaches that use class labels often produce better classification. This should not be surprising, since an interval constructed with no knowledge of class labels often contains a mixture of class labels. A conceptually simple approach is to place the splits in a way that maximizes the purity of the intervals, i.e., the extent to which an interval contains a single class label. In practice, however, such an approach requires potentially arbitrary decisions about the purity of an interval and the minimum size of an interval.
To overcome such concerns, some statistically based approaches start with each attribute value in a separate interval and create larger intervals by merging adjacent intervals that are similar according to a statistical test. An alternative to this bottom-up approach is a top-down approach that starts by bisecting the initial values so that the resulting two intervals give minimum entropy. This technique only needs to consider each value as a possible split point, because it is assumed that intervals contain ordered sets of values. The splitting process is then repeated with another interval, typically choosing the interval with the worst (highest) entropy, until a user-specified number of intervals is reached, or a stopping criterion is satisfied.
Entropy-based approaches are one of the most promising approaches to
discretization, whether bottom-up or top-down. First, it is necessary to define entropy. Let k be the number of different class labels, m
i be the number of
values in the ith interval of a partition, and mij be the number of values of class j in interval i. Then the entropy ei of the ith interval is given by the equation
ei=−∑j=1kpijlog2 pij,
where pij=mij/mi is the probability (fraction of values) of class j in the ith interval. The total entropy, e, of the partition is the weighted average of the individual interval entropies, i.e.,
e=∑i=1nwiei,
where m is the number of values, wi=mi/m is the fraction of values in the ith interval, and n is the number of intervals. Intuitively, the entropy of an interval is a measure of the purity of an interval. If an interval contains only values of one class (is perfectly pure), then the entropy is 0 and it contributes nothing to the overall entropy. If the classes of values in an interval occur equally often (the interval is as impure as possible), then the entropy is a maximum.
Example 2.13 (Discretization of Two Attributes). The top-down method based on entropy was used to independently discretize both the x and y attributes of the two-dimensional data shown in Figure 2.14. In the first discretization, shown in Figure 2.14(a), the x and y attributes were both split into three intervals. (The dashed lines indicate the split points.) In the second discretization, shown in Figure 2.14(b), the x and y attributes were both split into five intervals.
Figure 2.14. Discretizing x and y attributes for four groups (classes) of points.
Figure 2.14. Full Alternative Text
This simple example illustrates two aspects of discretization. First, in two dimensions, the classes of points are well separated, but in one dimension, this is not so. In general, discretizing each attribute separately often guarantees suboptimal results. Second, five intervals work better than three, but six intervals do not improve the discretization much, at least in terms of entropy. (Entropy values and results for six intervals are not shown.) Consequently, it is desirable to have a stopping criterion that automatically finds the right number of partitions.
Categorical Attributes with Too Many Values
Categorical attributes can sometimes have too many values. If the categorical attribute is an ordinal attribute, then techniques similar to those for continuous attributes can be used to reduce the number of categories. If the categorical attribute is nominal, however, then other approaches are needed. Consider a university that has a large number of departments. Consequently, a department name attribute might have dozens of different values. In this situation, we could use our knowledge of the relationships among different departments to combine departments into larger groups, such as engineering, social sciences, or biological sciences. If domain knowledge does not serve as a useful guide or such an approach results in poor classification performance, then it is necessary to use a more empirical approach, such as grouping values together only if such a grouping results in improved classification accuracy or achieves some other data mining objective.
2.3.7 Variable Transformation A variable transformation refers to a transformation that is applied to all the values of a variable. (We use the term variable instead of attribute to adhere to common usage, although we will also refer to attribute transformation on occasion.) In other words, for each object, the transformation is applied to the value of the variable for that object. For example, if only the magnitude of a variable is important, then the values of the variable can be transformed by taking the absolute value. In the following section, we discuss two important types of variable transformations: simple functional transformations and normalization.
Simple Functions For this type of variable transformation, a simple mathematical function is applied to each value individually. If x is a variable, then examples of such transformations include xk, log x, ex, x, 1/x, sin x, or |x|. In statistics, variable transformations, especially sqrt, log, and 1/x, are often used to transform data that does not have a Gaussian (normal) distribution into data that does. While this can be important, other reasons often take precedence in data mining.
Suppose the variable of interest is the number of data bytes in a session, and the number of bytes ranges from 1 to 1 billion. This is a huge range, and it can be advantageous to compress it by using a log 10transformation. In this case, sessions that transferred 108 and 109 bytes would be more similar to each other than sessions that transferred 10 and 1000 bytes (9−8=1 versus 3−1=3). For some applications, such as network intrusion detection, this may be what is desired, since the first two sessions most likely represent transfers of large files, while the latter two sessions could be two quite distinct types of sessions.
Variable transformations should be applied with caution because they change the nature of the data. While this is what is desired, there can be problems if the nature of the transformation is not fully appreciated. For instance, the transformation 1/x reduces the magnitude of values that are 1 or larger, but increases the magnitude of values between 0 and 1. To illustrate, the values {1, 2, 3} go to { 1, 12, 13 }, but the values { 1, 12, 13 } go to {1, 2, 3}. Thus, for all sets of values, the transformation 1/x reverses the order. To help clarify the effect of a transformation, it is important to ask questions such as the following: What is the desired property of the transformed attribute? Does the order need to be maintained? Does the transformation apply to all values, especially negative values and 0? What is the effect of the transformation on the values between 0 and 1? Exercise 17 on page 109 explores other aspects of variable transformation.
Normalization or Standardization The goal of standardization or normalization is to make an entire set of values have a particular property. A traditional example is that of “standardizing a variable” in statistics. If x¯ is the mean (average) of the attribute values and sx is their standard deviation, then the transformation x′=(x−x¯)/sx creates a new variable that has a mean of 0 and a standard deviation of 1. If different variables are to be used together, e.g., for clustering, then such a transformation is often necessary to avoid having a variable with large values dominate the results of the analysis. To illustrate, consider comparing people based on two variables: age and income. For any two people, the difference in income will likely be much higher in absolute terms (hundreds or
thousands of dollars) than the difference in age (less than 150). If the differences in the range of values of age and income are not taken into account, then the comparison between people will be dominated by differences in income. In particular, if the similarity or dissimilarity of two people is calculated using the similarity or dissimilarity measures defined later in this chapter, then in many cases, such as that of Euclidean distance, the income values will dominate the calculation.
The mean and standard deviation are strongly affected by outliers, so the above transformation is often modified. First, the mean is replaced by the median, i.e., the middle value. Second, the standard deviation is replaced by the absolute standard deviation. Specifically, if x is a variable, then the absolute standard deviation of x is given by σA=∑i=1m|xi−μ|, where xi is the ith value of the variable, m is the number of objects, and μ is either the mean or median. Other approaches for computing estimates of the location (center) and spread of a set of values in the presence of outliers are described in statistics books. These more robust measures can also be used to define a standardization transformation.
2.4 Measures of Similarity and Dissimilarity Similarity and dissimilarity are important because they are used by a number of data mining techniques, such as clustering, nearest neighbor classification, and anomaly detection. In many cases, the initial data set is not needed once these similarities or dissimilarities have been computed. Such approaches can be viewed as transforming the data to a similarity (dissimilarity) space and then performing the analysis. Indeed, kernel methods are a powerful realization of this idea. These methods are introduced in Section 2.4.7 and are discussed more fully in the context of classification in Section 4.9.4.
We begin with a discussion of the basics: high-level definitions of similarity and dissimilarity, and a discussion of how they are related. For convenience, the term proximity is used to refer to either similarity or dissimilarity. Since the proximity between two objects is a function of the proximity between the corresponding attributes of the two objects, we first describe how to measure the proximity between objects having only one attribute.
We then consider proximity measures for objects with multiple attributes. This includes measures such as the Jaccard and cosine similarity measures, which are useful for sparse data, such as documents, as well as correlation and Euclidean distance, which are useful for non-sparse (dense) data, such as time series or multi-dimensional points. We also consider mutual information, which can be applied to many types of data and is good for detecting nonlinear relationships. In this discussion, we restrict ourselves to objects with relatively homogeneous attribute types, typically binary or continuous.
Next, we consider several important issues concerning proximity measures. This includes how to compute proximity between objects when they have heterogeneous types of attributes, and approaches to account for differences of scale and correlation among variables when computing distance between numerical objects. The section concludes with a brief discussion of how to
select the right proximity measure.
Although this section focuses on the computation of proximity between data objects, proximity can also be computed between attributes. For example, for the document-term matrix of Figure 2.2(d), the cosine measure can be used to compute similarity between a pair of documents or a pair of terms (words). Knowing that two variables are strongly related can, for example, be helpful for eliminating redundancy. In particular, the correlation and mutual information measures discussed later are often used for that purpose.
2.4.1 Basics
Definitions Informally, the similarity between two objects is a numerical measure of the degree to which the two objects are alike. Consequently, similarities are higher for pairs of objects that are more alike. Similarities are usually non- negative and are often between 0 (no similarity) and 1 (complete similarity).
The dissimilarity between two objects is a numerical measure of the degree to which the two objects are different. Dissimilarities are lower for more similar pairs of objects. Frequently, the term distance is used as a synonym for dissimilarity, although, as we shall see, distance often refers to a special class of dissimilarities. Dissimilarities sometimes fall in the interval [0, 1], but it is also common for them to range from 0 to ∞.
Transformations Transformations are often applied to convert a similarity to a dissimilarity, or vice versa, or to transform a proximity measure to fall within a particular range, such as [0,1]. For instance, we may have similarities that range from 1 to 10, but the particular algorithm or software package that we want to use may be designed to work only with dissimilarities, or it may work only with
similarities in the interval [0,1]. We discuss these issues here because we will employ such transformations later in our discussion of proximity. In addition, these issues are relatively independent of the details of specific proximity measures.
Frequently, proximity measures, especially similarities, are defined or transformed to have values in the interval [0,1]. Informally, the motivation for this is to use a scale in which a proximity value indicates the fraction of similarity (or dissimilarity) between two objects. Such a transformation is often relatively straightforward. For example, if the similarities between objects range from 1 (not at all similar) to 10 (completely similar), we can make them fall within the range [0, 1] by using the transformation s′=(s−1)/9, where s and s′ are the original and new similarity values, respectively. In the more general case, the transformation of similarities to the interval [0, 1] is given by the expression s′=(s−min_s)/(max_s−min_s), where max_s and min_s are the maximum and minimum similarity values, respectively. Likewise, dissimilarity measures with a finite range can be mapped to the interval [0,1] by using the formula d′=(d−min_d)/(max_d−min_d). This is an example of a linear transformation, which preserves the relative distances between points. In other words, if points, x1 and x2, are twice as far apart as points, x3 and x4, the same will be true after a linear transformation.
However, there can be complications in mapping proximity measures to the interval [0, 1] using a linear transformation. If, for example, the proximity measure originally takes values in the interval [0,∞], then max_d is not defined and a nonlinear transformation is needed. Values will not have the same relationship to one another on the new scale. Consider the transformation d=d/(1+d) for a dissimilarity measure that ranges from 0 to ∞. The dissimilarities 0, 0.5, 2, 10, 100, and 1000 will be transformed into the new dissimilarities 0, 0.33, 0.67, 0.90, 0.99, and 0.999, respectively. Larger values on the original dissimilarity scale are compressed into the range of values near 1, but whether this is desirable depends on the application.
Note that mapping proximity measures to the interval [0, 1] can also change the meaning of the proximity measure. For example, correlation, which is discussed later, is a measure of similarity that takes values in the interval [−1, 1]. Mapping these values to the interval [0,1] by taking the absolute
value loses information about the sign, which can be important in some applications. See Exercise 22 on page 111.
Transforming similarities to dissimilarities and vice versa is also relatively straightforward, although we again face the issues of preserving meaning and changing a linear scale into a nonlinear scale. If the similarity (or dissimilarity) falls in the interval [0,1], then the dissimilarity can be defined as d=1−s(s=1−d). Another simple approach is to define similarity as the negative of the dissimilarity (or vice versa). To illustrate, the dissimilarities 0, 1, 10, and 100 can be transformed into the similarities 0, −1, −10, and −100, respectively.
The similarities resulting from the negation transformation are not restricted to the range [0, 1], but if that is desired, then transformations such as s=1d+1, s=e−d, or s=1−d−min_dmax_d−min_d can be used. For the transformation s=1d+1, the dissimilarities 0, 1, 10, 100 are transformed into 1, 0.5, 0.09, 0.01, respectively. For s=e−d, they become 1.00, 0.37, 0.00, 0.00, respectively, while for s=1−d−min_dmax_d−min_d they become 1.00, 0.99, 0.90, 0.00, respectively. In this discussion, we have focused on converting dissimilarities to similarities. Conversion in the opposite direction is considered in Exercise 23 on page 111.
In general, any monotonic decreasing function can be used to convert dissimilarities to similarities, or vice versa. Of course, other factors also must be considered when transforming similarities to dissimilarities, or vice versa, or when transforming the values of a proximity measure to a new scale. We have mentioned issues related to preserving meaning, distortion of scale, and requirements of data analysis tools, but this list is certainly not exhaustive.
2.4.2 Similarity and Dissimilarity between Simple Attributes The proximity of objects with a number of attributes is typically defined by combining the proximities of individual attributes, and thus, we first discuss proximity between objects having a single attribute. Consider objects
described by one nominal attribute. What would it mean for two such objects to be similar? Because nominal attributes convey only information about the distinctness of objects, all we can say is that two objects either have the same value or they do not. Hence, in this case similarity is traditionally defined as 1 if attribute values match, and as 0 otherwise. A dissimilarity would be defined in the opposite way: 0 if the attribute values match, and 1 if they do not.
For objects with a single ordinal attribute, the situation is more complicated because information about order should be taken into account. Consider an attribute that measures the quality of a product, e.g., a candy bar, on the scale {poor, fair, OK, good, wonderful}. It would seem reasonable that a product, P1, which is rated wonderful, would be closer to a product P2, which is rated good, than it would be to a product P3, which is rated OK. To make this observation quantitative, the values of the ordinal attribute are often mapped to successive integers, beginning at 0 or 1, e.g., {poor=0, fair=1, OK=2, good=3, wonderful=4}. Then, d(P1, P2)=3−2=1 or, if we want the dissimilarity to fall between 0 and d(P1, P2)=3−24=0.25. A similarity for ordinal attributes can then be defined as s=1−d.
This definition of similarity (dissimilarity) for an ordinal attribute should make the reader a bit uneasy since this assumes equal intervals between successive values of the attribute, and this is not necessarily so. Otherwise, we would have an interval or ratio attribute. Is the difference between the values fair and good really the same as that between the values OK and wonderful? Probably not, but in practice, our options are limited, and in the absence of more information, this is the standard approach for defining proximity between ordinal attributes.
For interval or ratio attributes, the natural measure of dissimilarity between two objects is the absolute difference of their values. For example, we might compare our current weight and our weight a year ago by saying “I am ten pounds heavier.” In cases such as these, the dissimilarities typically range from 0 to ∞, rather than from 0 to 1. The similarity of interval or ratio attributes is typically expressed by transforming a dissimilarity into a similarity, as previously described.
Table 2.7 summarizes this discussion. In this table, x and y are two objects
that have one attribute of the indicated type. Also, d(x, y) and s(x, y) are the dissimilarity and similarity between x and y, respectively. Other approaches are possible; these are the most common ones.
Table 2.7. Similarity and dissimilarity for simple attributes Attribute Type Dissimilarity Similarity
Nominal d={0if x=y1if x≠y s={ 1if x=y0if x≠y
Ordinal
d=|x−y|/(n−1) (values mapped to integers 0 to n −1, where n is the number of values)
s=1−d
Interval or Ratio d=|x−y| s=−d, s=11+d, s=e−d,s=1−d−min_dmax_d−min_
The following two sections consider more complicated measures of proximity between objects that involve multiple attributes: (1) dissimilarities between data objects and (2) similarities between data objects. This division allows us to more naturally display the underlying motivations for employing various proximity measures. We emphasize, however, that similarities can be transformed into dissimilarities and vice versa using the approaches described earlier.
2.4.3 Dissimilarities between Data
Objects In this section, we discuss various kinds of dissimilarities. We begin with a discussion of distances, which are dissimilarities with certain properties, and then provide examples of more general kinds of dissimilarities.
Distances We first present some examples, and then offer a more formal description of distances in terms of the properties common to all distances. The Euclidean distance ,d , between two points, x and y , in one-, two-, three-, or higher- dimensional space, is given by the following familiar formula:
d(x,y)=∑k=1n(xk−yk)2, (2.1)
where n is the number of dimensions and xk and yk are, respectively, the kth attributes (components) of x and y. We illustrate this formula with Figure 2.15 and Tables 2.8 and 2.9, which show a set of points, the x and y coordinates of these points, and the distance matrix containing the pairwise distances of these points.
Figure 2.15.
Four two-dimensional points.
The Euclidean distance measure given in Equation 2.1 is generalized by the Minkowski distance metric shown in Equation 2.2,
d(x,y)=(∑k=1n|xk−yk|r)1/r, (2.2)
where r is a parameter. The following are the three most common examples of Minkowski distances.
r=1. City block (Manhattan, taxicab, L1 norm) distance. A common example is the Hamming distance , which is the number of bits that is different between two objects that have only binary attributes, i.e., between two binary vectors.
r=2. Euclidean distance (L2 norm).
r=∞. Supremum (Lmax or L∞ norm) distance. This is the maximum difference between any attribute of the objects. More formally, the L∞ distance is defined by Equation 2.3
d(x, y)=limr→∞(∑k=1n|xk−yk|r)1/r. (2.3)
The r parameter should not be confused with the number of dimensions (at- tributes) n. The Euclidean, Manhattan, and supremum distances are defined for all values of n: 1, 2, 3, …, and specify different ways of combining the differences in each dimension (attribute) into an overall distance.
Tables 2.10 and 2.11, respectively, give the proximity matrices for the L1 and L∞ distances using data from Table 2.8. Notice that all these distance matrices are symmetric; i.e., the ijth entry is the same as the jith entry. In Table 2.9, for instance, the fourth row of the first column and the fourth column of the first row both contain the value 5.1.
Table 2.8. x and y coordinates of four points.
point x coordinate y coordinate p1 0 2 p2 2 0 p3 3 1 p4 5 1
Table 2.9. Euclidean distance matrix for Table 2.8.
p1 p2 p3 p4 p1 0.0 2.8 3.2 5.1 p2 2.8 0.0 1.4 3.2 p3 3.2 1.4 0.0 2.0 p4 5.1 3.2 2.0 0.0
Table 2.10. L1 distance matrix for Table 2.8. L1 p1 p2 p3 p4 p1 0.0 4.0 4.0 6.0 p2 4.0 0.0 2.0 4.0 p3 4.0 2.0 0.0 2.0 p4 6.0 4.0 2.0 0.0
Table 2.11. L∞ distance matrix for Table 2.8.
L∞ p1 p2 p3 p4 p1 0.0 2.0 3.0 5.0 p2 2.0 0.0 1.0 3.0 p3 3.0 1.0 0.0 2.0 p4 5.0 3.0 2.0 0.0
Distances, such as the Euclidean distance, have some well-known properties. If d(x, y) is the distance between two points, x and y, then the following properties hold.
1. Positivity
1. d(x, y)≥0 for all x and y,
2. d(x, y)=0 only if x=y.
2. Symmetry d(x, y)=d(y, x) for all x and y.
3. Triangle Inequality d(x, z)≤d(x, y)+d(y, z) for all points x , y , and z.
Measures that satisfy all three properties are known as metrics. Some people use the term distance only for dissimilarity measures that satisfy these properties, but that practice is often violated. The three properties described here are useful, as well as mathematically pleasing. Also, if the triangle inequality holds, then this property can be used to increase the efficiency of techniques (including clustering) that depend on distances possessing this property. (See Exercise 25.) Nonetheless, many dissimilarities do not satisfy one or more of the metric properties. Example 2.14 illustrates such a measure.
Example 2.14 (Non-metric Dissimilarities: Set Differences). This example is based on the notion of the difference of two sets, as defined
in set theory. Given two sets A and B, A−B is the set of elements of A that are not in
B. For example, if A={1, 2, 3, 4} and B={2, 3, 4}, then A−B={1} and B −A=∅, the empty set. We can define the distance d between two sets A and B as d(A, B)=size(A−B), where size is a function returning the number of elements in a set. This distance measure, which is an integer value greater than or equal to 0, does not satisfy the second part of the positivity property, the symmetry property, or the triangle inequality. However, these properties can be made to hold if the dissimilarity measure is modified as follows: d(A, B)=size(A−B)+size(B−A). See Exercise 21 on page 110.
2.4.4 Similarities between Data Objects For similarities, the triangle inequality (or the analogous property) typically does not hold, but symmetry and positivity typically do. To be explicit, if s(x, y) is the similarity between points x and y, then the typical properties of similarities are the following:
1. s(x, y)=1 only if x=y. (0≤s≤1)
2. s(x, y)=s(y, x) for all x and y. (Symmetry)
There is no general analog of the triangle inequality for similarity measures. It is sometimes possible, however, to show that a similarity measure can easily be converted to a metric distance. The cosine and Jaccard similarity measures, which are discussed shortly, are two examples. Also, for specific similarity measures, it is possible to derive mathematical bounds on the similarity between two objects that are similar in spirit to the triangle inequality.
Example 2.15 (A Non-symmetric
Similarity Measure). Consider an experiment in which people are asked to classify a small set of characters as they flash on a screen. The confusion matrix for this experiment records how often each character is classified as itself, and how often each is classified as another character. Using the confusion matrix, we can define a similarity measure between a character x and a character y as the number of times that x is misclassified asy , but note that this measure is not symmetric. For example, suppose that “0” appeared 200 times and was classified as a “0” 160 times, but as an “o” 40 times. Likewise, suppose that “o” appeared 200 times and was classified as an “o” 170 times, but as “0” only 30 times. Then, s(0,o)=40, but s(o, 0)=30. In such situations, the similarity measure can be made symmetric by setting s′=(x,y)=s′(x,y)= (s(x,y+s(y,x))/2, where s indicates the new similarity measure.
2.4.5 Examples of Proximity Measures This section provides specific examples of some similarity and dissimilarity measures.
Similarity Measures for Binary Data Similarity measures between objects that contain only binary attributes are called similarity coefficients , and typically have values between 0 and 1. A value of 1 indicates that the two objects are completely similar, while a value of 0 indicates that the objects are not at all similar. There are many rationales for why one coefficient is better than another in specific instances.
Let x and y be two objects that consist of n binary attributes. The comparison
of two such objects, i.e., two binary vectors, leads to the following four quantities (frequencies):
f00=the number of attributes where x is 0 and y is 0f01= the number of attributes where
Simple Matching Coefficient One commonly used similarity coefficient is the simple matching coefficient (SMC), which is defined as
SMC=number of matching attribute valuesnumber of attributes=f11+f00f01+f (2.4)
This measure counts both presences and absences equally. Consequently, the SMC could be used to find students who had answered questions similarly on a test that consisted only of true/false questions.
Jaccard Coefficient Suppose that x and y are data objects that represent two rows (two transactions) of a transaction matrix (see Section 2.1.2). If each asymmetric binary attribute corresponds to an item in a store, then a 1 indicates that the item was purchased, while a 0 indicates that the product was not purchased. Because the number of products not purchased by any customer far outnumbers the number of products that were purchased, a similarity measure such as SMC would say that all transactions are very similar. As a result, the Jaccard coefficient is frequently used to handle objects consisting of asymmetric binary attributes. The Jaccard coefficient , which is often symbolized by j, is given by the following equation:
J=number of matching presencesnumber of attributes not involved in 00 matches (2.5)
Example 2.16 (The SMC and
Jaccard Similarity Coefficients). To illustrate the difference between these two similarity measures, we calculate SMC and j for the following two binary vectors.
x = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0)y = (0, 0, 0, 0, 0, 0, 1, 0, 0, 1)
f01=2the number of attributes where x was 0 and y was 1f10=1the number of attributes where
SMC=f11+f00f01+f10+f11+f00=0+72+1+0+7=0.7
J=f11f01+f10+f11=02+1+0=0
Cosine Similarity Documents are often represented as vectors, where each component (attribute) represents the frequency with which a particular term (word) occurs in the document. Even though documents have thousands or tens of thousands of attributes (terms), each document is sparse since it has relatively few nonzero attributes. Thus, as with transaction data, similarity should not depend on the number of shared 0 values because any two documents are likely to “not contain” many of the same words, and therefore, if 0–0 matches are counted, most documents will be highly similar to most other documents. Therefore, a similarity measure for documents needs to ignores 0–0 matches like the Jaccard measure, but also must be able to handle non-binary vectors. The cosine similarity , defined next, is one of the most common measures of document similarity. If x and y are two document vectors, then
cos(x, y)=⟨ x, y ⟩∥x∥∥y∥=x′y∥x∥∥y∥, (2.6)
where ′ indicates vector or matrix transpose and ⟨ x, y ⟩ indicates the inner product of the two vectors,
⟨ x, y ⟩=∑k=1nxkyk=x′y, (2.7)
and ∥x∥ is the length of vector x, ∥x∥=∑k=1nxk2=⟨ x, x ⟩=x′x.
The inner product of two vectors works well for asymmetric attributes since it depends only on components that are non-zero in both vectors. Hence, the similarity between two documents depends only upon the words that appear in both of them.
Example 2.17 (Cosine Similarity between Two Document Vectors). This example calculates the cosine similarity for the following two data objects, which might represent document vectors:
x=(3, 2, 0, 5, 0, 0, 0, 2, 0, 0)y=(1, 0, 0, 0, 0, 0, 0, 1, 0, 2)
⟨ x, y ⟩=3×1+2×0+0×0+5×0+0×0+0×0+0×0+2×1+0×0×2=5∥x∥=3×3+2×2+0×0+5
As indicated by Figure 2.16, cosine similarity really is a measure of the (cosine of the) angle between x and y. Thus, if the cosine similarity is 1, the angle between x and y is 0°, and x and y are the same except for length. If the cosine similarity is 0, then the angle between x and y is 90°, and they do not share any terms (words).
Figure 2.16.
Geometric illustration of the cosine measure.
Equation 2.6 also can be written as Equation 2.8.
cos(x, y)=⟨ x∥x∥, y∥y∥ ⟩=⟨ x′, y′ ⟩, (2.8)
where x′=x/∥x∥ and y′=y/∥y∥. Dividing x and y by their lengths normalizes them to have a length of 1. This means that cosine similarity does not take the length of the two data objects into account when computing similarity. (Euclidean distance might be a better choice when length is important.) For vectors with a length of 1, the cosine measure can be calculated by taking a simple inner product. Consequently, when many cosine similarities between objects are being computed, normalizing the objects to have unit length can reduce the time required.
Extended Jaccard Coefficient (Tanimoto Coefficient) The extended Jaccard coefficient can be used for document data and that reduces to the Jaccard coefficient in the case of binary attributes. This coefficient, which we shall represent as EJ, is defined by the following equation:
EJ(x, y)=⟨ x, y ⟩ǁ x ǁ2+ǁ y ǁ2−⟨ x, y ⟩=x′yǁ x ǁ2+ǁ y ǁ2−x′y. (2.9)
Correlation Correlation is frequently used to measure the linear relationship between two sets of values that are observed together. Thus, correlation can measure the relationship between two variables (height and weight) or between two objects (a pair of temperature time series). Correlation is used much more frequently to measure the similarity between attributes since the values in two data objects come from different attributes, which can have very different attribute types and scales. There are many types of correlation, and indeed
correlation is sometimes used in a general sense to mean the relationship between two sets of values that are observed together. In this discussion, we will focus on a measure appropriate for numerical values.
Specifically, Pearson’s correlation between two sets of numerical values, i.e., two vectors, x and y, is defined by the following equation:
corr(x, y)=covariance(x, y)standard_deviation(x)×standard_deviation(y)=sxys (2.10)
where we use the following standard statistical notation and definitions:
covariance(x, y)=sxy=1n−1∑k=1n(xk−x¯)(yk−y¯) (2.11) standard_deviation(x)=sx=1n−1∑k=1n(xk−x¯)2 standard_deviation(y)=sy=1n−1∑k=1n(yk−y¯)2
x¯=1n∑k=1nxk is the mean of x y¯=1n∑k=1nyk is the mean of y
Example 2.18 (Perfect Correlation). Correlation is always in the range −1 to 1. A correlation of 1 (−1) means that x and y have a perfect positive (negative) linear relationship; that is, xk=ayk+b, where a and b are constants. The following two vectors x and y illustrate cases where the correlation is −1 and +1, respectively. In the first case, the means of x and y were chosen to be 0, for simplicity.
x=(−3, 6, 0, 3, −6)y=(1, −2, 0, −1, 2)corr(x, y)=−1xk=−3yk x=(3, 6, 0, 3, 6)y=(1, 2, 0, 1, 2)corr(x, y)=1xk=3yk
Example 2.19 (Nonlinear Relationships). If the correlation is 0, then there is no linear relationship between the two sets
of values. However, nonlinear relationships can still exist. In the following example, yk=xk2, but their correlation is 0.
x=(−3, −2, −1, 0, 1, 2, 3)y=(9, 4, 1, 0, 1, 4, 9)
Example 2.20 (Visualizing Correlation). It is also easy to judge the correlation between two vectors x and y by plotting pairs of corresponding values of x and y in a scatter plot. Figure 2.17 shows a number of these scatter plots when x and y consist of a set of 30 pairs of values that are randomly generated (with a normal distribution) so that the correlation of x and y ranges from −1 to 1. Each circle in a plot represents one of the 30 pairs of x and y values; its x coordinate is the value of that pair for x, while its y coordinate is the value of the same pair for y.
Figure 2.17. Scatter plots illustrating correlations from −1 to 1.
Figure 2.17. Full Alternative Text
If we transform x and y by subtracting off their means and then normalizing them so that their lengths are 1, then their correlation can be calculated by taking the dot product. Let us refer to these transformed vectors of x and y as x′ and y′, respectively. (Notice that this transformation is not the same as the standardization used in other contexts, where we subtract the means and divide by the standard deviations, as discussed in Section 2.3.7.) This transformation highlights an interesting relationship between the correlation measure and the cosine measure. Specifically, the correlation between x and
y is identical to the cosine between x′ and y′. However, the cosine between x and y is not the same as the cosine between x′ and y′, even though they both have the same correlation measure. In general, the correlation between two vectors is equal to the cosine measure only in the special case when the means of the two vectors are 0.
Differences Among Measures For Continuous Attributes In this section, we illustrate the difference among the three proximity measures for continuous attributes that we have just defined: cosine, correlation, and Minkowski distance. Specifically, we consider two types of data transformations that are commonly used, namely, scaling (multiplication) by a constant factor and translation (addition) by a constant value. A proximity measure is considered to be invariant to a data transformation if its value remains unchanged even after performing the transformation. Table 2.12 compares the behavior of cosine, correlation, and Minkowski distance measures regarding their invariance to scaling and translation operations. It can be seen that while correlation is invariant to both scaling and translation, cosine is only invariant to scaling but not to translation. Minkowski distance measures, on the other hand, are sensitive to both scaling and translation and are thus invariant to neither.
Table 2.12. Properties of cosine, correlation, and Minkowski distance measures.
Property Cosine Correlation MinkowskiDistance Invariant to scaling (multiplication) Yes Yes No
Invariant to translation (addition) No Yes No
Let us consider an example to demonstrate the significance of these differences among different proximity measures.
Example 2.21 (Comparing proximity measures). Consider the following two vectors x and y with seven numeric attributes.
x=(1, 2, 4, 3, 0, 0, 0)y=(1, 2, 3, 4, 0, 0, 0)
It can be seen that both x and y have 4 non-zero values, and the values in the two vectors are mostly the same, except for the third and the fourth components. The cosine, correlation, and Euclidean distance between the two vectors can be computed as follows.
cos(x, y)=2930×30=0.9667correlation(x, y)=2.35711.5811×1.5811=0.9429Euclidean distance x−y ǁ=1.4142
Not surprisingly, x and y have a cosine and correlation measure close to 1, while the Euclidean distance between them is small, indicating that they are quite similar. Now let us consider the vector ys, which is a scaled version of y (multiplied by a constant factor of 2), and the vector yt, which is constructed by translating y by 5 units as follows.
ys=2×y=(2, 4, 6, 8, 0, 0, 0)
yt=y+5=(6, 7, 8, 9, 5, 5, 5)
We are interested in finding whether ys and yt show the same proximity with x as shown by the original vector y. Table 2.13 shows the different measures of proximity computed for the pairs (x, y), (x, ys), and (x, yt). It can be seen that the value of correlation between x and y remains unchanged even after
replacing y with ys or yt. However, the value of cosine remains equal to 0.9667 when computed for (x, y) and (x, ys), but significantly reduces to 0.7940 when computed for (x, yt). This highlights the fact that cosine is invariant to the scaling operation but not to the translation operation, in contrast with the correlation measure. The Euclidean distance, on the other hand, shows different values for all three pairs of vectors, as it is sensitive to both scaling and translation.
Table 2.13. Similarity between (x, y), (x, ys), and (x, yt).
Measure (x, y) (x, ys) (x, yt) Cosine 0.9667 0.9667 0.7940
Correlation 0.9429 0.9429 0.9429 Euclidean Distance 1.4142 5.8310 14.2127
We can observe from this example that different proximity measures behave differently when scaling or translation operations are applied on the data. The choice of the right proximity measure thus depends on the desired notion of similarity between data objects that is meaningful for a given application. For example, if x and y represented the frequencies of different words in a document-term matrix, it would be meaningful to use a proximity measure that remains unchanged when y is replaced by ys, because ys is just a scaled version of y with the same distribution of words occurring in the document. However, yt is different from y, since it contains a large number of words with non-zero frequencies that do not occur in y. Because cosine is invariant to scaling but not to translation, it will be an ideal choice of proximity measure for this application.
Consider a different scenario in which x represents a location’s temperature measured on the Celsius scale for seven days. Let y, ys, and yt be the temperatures measured on those days at a different location, but using three different measurement scales. Note that different units of temperature have different offsets (e.g. Celsius and Kelvin) and different scaling factors (e.g.
Celsius and Fahrenheit). It is thus desirable to use a proximity measure that captures the proximity between temperature values without being affected by the measurement scale. Correlation would then be the ideal choice of proximity measure for this application, as it is invariant to both scaling and translation.
As another example, consider a scenario where x represents the amount of precipitation (in cm) measured at seven locations. Let y, ys, and yt be estimates of the precipitation at these locations, which are predicted using three different models. Ideally, we would like to choose a model that accurately reconstructs the measurements in x without making any error. It is evident that y provides a good approximation of the values in x, whereas ys and yt provide poor estimates of precipitation, even though they do capture the trend in precipitation across locations. Hence, we need to choose a proximity measure that penalizes any difference in the model estimates from the actual observations, and is sensitive to both the scaling and translation operations. The Euclidean distance satisfies this property and thus would be the right choice of proximity measure for this application. Indeed, the Euclidean distance is commonly used in computing the accuracy of models, which will be discussed later in Chapter 3.
2.4.6 Mutual Information Like correlation, mutual information is used as a measure of similarity between two sets of paired values that is sometimes used as an alternative to correlation, particularly when a nonlinear relationship is suspected between the pairs of values. This measure comes from information theory, which is the study of how to formally define and quantify information. Indeed, mutual information is a measure of how much information one set of values provides about another, given that the values come in pairs, e.g., height and weight. If the two sets of values are independent, i.e., the value of one tells us nothing about the other, then their mutual information is 0. On the other hand, if the two sets of values are completely dependent, i.e., knowing the value of one tells us the value of the other and vice-versa, then they have maximum mutual information. Mutual information does not have a maximum value, but we will define a normalized version of it that ranges between 0 and 1.
To define mutual information, we consider two sets of values, X and Y , which occur in pairs (X, Y). We need to measure the average information in a single set of values, i.e., either in X or in Y , and in the pairs of their values. This is commonly measured by entropy. More specifically, assume X and Y are discrete, that is, X can take m distinct values,u1, u2, …, um and Y can take n distinct values,v1, v2, …, vn. Then their individual and joint entropy can be defined in terms of the probabilities of each value and pair of values as follows:
H(X)=−∑j=1mP(X=uj)log2 P(X=uj) (2.12) H(Y)=−∑k=1nP(Y=vk)log2 P(Y=vk) (2.13) H(X, Y)=−∑j=1m∑k=1nP(X=uj, Y=vk)log2 P(X=uj, Y=vk) (2.14)
where if the probability of a value or combination of values is 0, then 0 log2(0) is conventionally taken to be 0.
The mutual information of X and Y can now be defined straightforwardly:
I(X, Y)=H(X)+H(Y)−H(X, Y) (2.15)
Note that H(X, Y) is symmetric, i.e., H(X, Y)=H(Y, X), and thus mutual information is also symmetric, i.e., I(X, Y)=I(Y).
Practically, X and Y are either the values in two attributes or two rows of the same data set. In Example 2.22, we will represent those values as two vectors x and y and calculate the probability of each value or pair of values from the frequency with which values or pairs of values occur in x, y and (xi, yi), where xi is the ith component of x and yi is the ith component of y. Let us illustrate using a previous example.
Example 2.22 (Evaluating Nonlinear Relationships with Mutual Information).
Recall Example 2.19 where yk=xk2, but their correlation was 0.
x=(−3, −2, −1, 0, 1, 2, 3)y=(9, 4, 1, 0, 1, 4, 9)
From Figure 2.22, I(x, y)=H(x)+H(y)−H(x, y)=1.9502. Although a variety of approaches to normalize mutual information are possible—see Bibliographic Notes—for this example, we will apply one that divides the mutual information by log2(min(m, n)) and produces a result between 0 and 1. This yields a value of 1.9502/log2(4))=0.9751. Thus, we can see that x and y are strongly related. They are not perfectly related because given a value of y there is, except for y=0, some ambiguity about the value of x. Notice that for y=−x, the normalized mutual information would be 1.
Figure 2.18. Computation of mutual information.
Table 2.14. Entropy for x xj P(x=xj) −P(x=xj)log2 P(x=xj) −3 1/7 0.4011 −2 1/7 0.4011 −1 1/7 0.4011 0 1/7 0.4011 1 1/7 0.4011 2 1/7 0.4011 3 1/7 0.4011
H(x) 2.8074
Table 2.15. Entropy for y
yk P(y=yk) −P(y=yk)log2 P(y=yk) 9 2/7 0.5164 4 2/7 0.5164 1 2/7 0.5164 0 1/7 0.4011
H(y) 1.9502
Table 2.16. Joint entropy for x and y xj yk P(x=xj, y=xk) −P(x=xj, y=xk)log2 P(x=xj, y=xk) −3 9 1/7 0.4011 −2 4 1/7 0.4011 −1 1 1/7 0.4011 0 0 1/7 0.4011 1 1 1/7 0.4011 2 4 1/7 0.4011 3 9 1/7 0.4011
H(x, y) 2.8074
2.4.7 Kernel Functions* It is easy to understand how similarity and distance might be useful in an application such as clustering, which tries to group similar objects together. What is much less obvious is that many other data analysis tasks, including predictive modeling and dimensionality reduction, can be expressed in terms of pairwise “proximities” of data objects. More specifically, many data analysis problems can be mathematically formulated to take as input, a kernel matrix, K, which can be considered a type of proximity matrix. Thus, an initial preprocessing step is used to convert the input data into a kernel matrix, which is the input to the data analysis algorithm.
More formally, if a data set has m data objects, then K is an m by m matrix. If xi and xj are the ith and jth data objects, respectively, then kij, the ijth entry of K, is computed by a kernel function:
kij=κ(xi, xj) (2.16)
As we will see in the material that follows, the use of a kernel matrix allows both wider applicability of an algorithm to various kinds of data and an ability to model nonlinear relationships with algorithms that are designed only for detecting linear relationships.
Kernels make an algorithm data independent If an algorithm uses a kernel matrix, then it can be used with any type of data for which a kernel function can be designed. This is illustrated by Algorithm 2.1. Although only some data analysis algorithms can be modified to use a kernel matrix as input, this approach is extremely powerful because it allows such an algorithm to be used with almost any type of data for which an appropriate kernel function can be defined. Thus, a classification algorithm can be used, for example, with record data, string data, or graph data. If an algorithm can be reformulated to use a kernel matrix, then its applicability to different types of data increases dramatically. As we will see in later chapters, many clustering, classification, and anomaly detection algorithms work only with similarities or distances, and thus, can be easily modified to work with kernels.
Algorithm 2.1 Basic kernel algorithm.
1. Read in the m data objects in the data set.
2. Compute the kernel matrix, K by applying the kernel function, κ, to each pair of data objects.
3. Run the data analysis algorithm with K as input.
4. Return the analysis result, e.g., predicted class or cluster labels.
Mapping data into a higher dimensional data space can allow modeling of nonlinear relationships There is yet another, equally important, aspect of kernel based data algorithms—their ability to model nonlinear relationships with algorithms that model only linear relationships. Typically, this works by first transforming (mapping) the data from a lower dimensional data space to a higher dimensional space.
Example 2.23 (Mapping Data to a Higher Dimensional Space). Consider the relationship between two variables x and y given by the following equation, which defines an ellipse in two dimensions (Figure 2.19(a)):
Figure 2.19. Mapping data to a higher dimensional space: two to three dimensions.
Figure 2.19. Full Alternative Text 4x2+9xy+7y2=10 (2.17)
We can map our two dimensional data to three dimensions by creating three new variables, u, v, and w, which are defined as follows:
w=x2u=xyv=y2
As a result, we can now express Equation 2.17 as a linear one. This equation describes a plane in three dimensions. Points on the ellipse will lie on that plane, while points inside and outside the ellipse will lie on opposite sides of the plane. See Figure 2.19(b). The viewpoint of this 3D plot is along the surface of the separating plane so that the plane appears as a line.
4u+9v+7w=10 (2.18)
The Kernel Trick The approach illustrated above shows the value in mapping data to higher dimensional space, an operation that is integral to kernel-based methods. Conceptually, we first define a function φ that maps data points x and y to data points φ(x) and φ(y) in a higher dimensional space such that the inner product ⟨x, y⟩ gives the desired measure of proximity of x and y. It may seem that we have potentially sacrificed a great deal by using such an approach, because we can greatly expand the size of our data, increase the computational complexity of our analysis, and encounter problems with the curse of dimensionality by computing similarity in a high-dimensional space. However, this is not the case since these problems can be avoided by defining a kernel function κ that can compute the same similarity value, but with the data points in the original space, i.e., κ(x, y)=⟨ φ(x), φ(y) ⟩. This is known as the kernel trick. Despite the name, the kernel trick has a very solid mathematical foundation and is a remarkably powerful approach for data analysis.
Not every function of a pair of data objects satisfies the properties needed for a kernel function, but it has been possible to design many useful kernels for a wide variety of data types. For example, three common kernel functions are the polynomial, Gaussian (radial basis function (RBF)), and sigmoid kernels. If x and y are two data objects, specifically, two data vectors, then these two kernel functions can be expressed as follows, respectively:
κ(x, y)−(x′y+c)d (2.19) κ(x, y)=exp(−ǁ x−y ǁ/2σ2) (2.20) κ(x, y)=tanh(αx′y+c) (2.21)
where α and c≥0 are constants, d is an integer parameter that gives the polynomial degree, ǁ x−y ǁ is the length of the vector x−y and σ>0 is a parameter that governs the “spread” of a Gaussian.
Example 2.24 (The Polynomial Kernel). Note that the kernel functions presented in the previous section are computing the same similarity value as would be computed if we actually mapped the data to a higher dimensional space and then computed an inner product there. For example, for the polynomial kernel of degree 2, let φ be the function that maps a two-dimensional data vector x=(x1, x2) to the higher dimensional space. Specifically, let
φ(x)=(x12, x22, 2x1x2, 2cx1, 2cx2, c). (2.22)
For the higher dimensional space, let the proximity be defined as the inner product of φ(x) and φ(y), i.e., ⟨ φ(x), φ(y) ⟩. Then, as previously mentioned, it can be shown that
κ(x, y)=⟨ φ(x), φ(y) ⟩ (2.23)
where κ is defined by Equation 2.19 above. Specifically, if x=(x1, x2) and y= (y1, y2), then
κ(x, y)=⟨ x, y ⟩=x′y=(x12y12, x22y22, 2x1x2y1y2, 2cx1y1, 2cx2y2, c2). (2.24)
More generally, the kernel trick depends on defining κ and φ so that Equation 2.23 holds. This has been done for a wide variety of kernels.
This discussion of kernel-based approaches was intended only to provide a brief introduction to this topic and has omitted many details. A fuller discussion of the kernel-based approach is provided in Section 4.9.4, which discusses these issues in the context of nonlinear support vector machines for classification. More general references for the kernel based analysis can be found in the Bibliographic Notes of this chapter.
2.4.8 Bregman Divergence* This section provides a brief description of Bregman divergences, which are a family of proximity functions that share some common properties. As a result, it is possible to construct general data mining algorithms, such as clustering algorithms, that work with any Bregman divergence. A concrete example is the K-means clustering algorithm (Section 7.2). Note that this section requires knowledge of vector calculus.
Bregman divergences are loss or distortion functions. To understand the idea of a loss function, consider the following. Let x and y be two points, where y is regarded as the original point and x is some distortion or approximation of it. For example, x may be a point that was generated by adding random noise to y. The goal is to measure the resulting distortion or loss that results if y is approximated by x. Of course, the more similar x and y are, the smaller the loss or distortion. Thus, Bregman divergences can be used as dissimilarity functions.
More formally, we have the following definition.
Definition 2.6 (Bregman divergence) Given a strictly convex function ϕ (with a few modest restrictions that are generally satisfied), the Bregman divergence (loss function) D(x, y) generated by that function is given by the following equation:
D(x, y)=ϕ(x)−ϕ(y)−⟨ ∇ϕ(y), (x−y) ⟩ (2.25)
where ∇ϕ(y) is the gradient of ϕ evaluated at y, x−y, is the vector difference between x and y, and ⟨ ∇ϕ(y), (x−y) ⟩ is the inner product between ∇ϕ(y) and (x−y). For points in Euclidean space, the inner product is just the dot product.
D(x, y) can be written as D(x, y)=ϕ(x)−L(x), where L(x)=ϕ(y)+⟨ ∇ϕ(y), (x −y) ⟩ and represents the equation of a plane that is tangent to the function ϕ at
y. Using calculus terminology, L(x) is the linearization of ϕ around the point y, and the Bregman divergence is just the difference between a function and a linear approximation to that function. Different Bregman divergences are obtained by using different choices for ϕ.
Example 2.25. We provide a concrete example using squared Euclidean distance, but restrict ourselves to one dimension to simplify the mathematics. Let x and y be real numbers and ϕ(t) be the real-valued function, ϕ(t)=t2. In that case, the gradient reduces to the derivative, and the dot product reduces to multiplication. Specifically, Equation 2.25 becomes Equation 2.26.
D(x,y)=x2−y2−2y(x−y)=(x−y)2 (2.26)
The graph for this example, with y=1, is shown in Figure 2.20. The Bregman divergence is shown for two values of x: x=2 and x=3.
Figure 2.20. Illustration of Bregman divergence.
Figure 2.20. Full Alternative Text
2.4.9 Issues in Proximity Calculation This section discusses several important issues related to proximity measures: (1) how to handle the case in which attributes have different scales and/or are correlated, (2) how to calculate proximity between objects that are composed of different types of attributes, e.g., quantitative and qualitative, (3) and how
to handle proximity calculations when attributes have different weights; i.e., when not all attributes contribute equally to the proximity of objects.
Standardization and Correlation for Distance Measures An important issue with distance measures is how to handle the situation when attributes do not have the same range of values. (This situation is often described by saying that “the variables have different scales.”) In a previous example, Euclidean distance was used to measure the distance between people based on two attributes: age and income. Unless these two attributes are standardized, the distance between two people will be dominated by income.
A related issue is how to compute distance when there is correlation between some of the attributes, perhaps in addition to differences in the ranges of values. A generalization of Euclidean distance, the Mahalanobis distance, is useful when attributes are correlated, have different ranges of values (different variances), and the distribution of the data is approximately Gaussian (normal). Correlated variables have a large impact on standard distance measures since a change in any of the correlated variables is reflected in a change in all the correlated variables. Specifically, the Mahalanobis distance between two objects (vectors) x and y is defined as
Mahalanobis(x, y)=(x−y)′∑−1(x−y), (2.27)
where ∑−1 is the inverse of the covariance matrix of the data. Note that the covariance matrix ∑ is the matrix whose ijth entry is the covariance of the ith and jth attributes as defined by Equation 2.11.
Example 2.26. In Figure 2.21, there are 1000 points, whose x and y attributes have a correlation of 0.6. The distance between the two large points at the opposite
ends of the long axis of the ellipse is 14.7 in terms of Euclidean distance, but only 6 with respect to Mahalanobis distance. This is because the Mahalanobis distance gives less emphasis to the direction of largest variance. In practice, computing the Mahalanobis distance is expensive, but can be worthwhile for data whose attributes are correlated. If the attributes are relatively uncorrelated, but have different ranges, then standardizing the variables is sufficient.
Figure 2.21. Set of two-dimensional points. The Mahalanobis distance between the two points represented by large dots is 6; their Euclidean distance is 14.7.
Combining Similarities for Heterogeneous Attributes The previous definitions of similarity were based on approaches that assumed all the attributes were of the same type. A general approach is needed when the attributes are of different types. One straightforward approach is to compute the similarity between each attribute separately using Table 2.7, and then combine these similarities using a method that results in a similarity between 0 and 1. One possible approach is to define the overall similarity as the average of all the individual attribute similarities. Unfortunately, this approach does not work well if some of the attributes are asymmetric attributes. For example, if all the attributes are asymmetric binary attributes, then the similarity measure suggested previously reduces to the simple matching coefficient, a measure that is not appropriate for asymmetric binary attributes. The easiest way to fix this problem is to omit asymmetric attributes from the similarity calculation when their values are 0 for both of the objects whose similarity is being computed. A similar approach also works well for handling missing values.
In summary, Algorithm 2.2 is effective for computing an overall similarity between two objects, x and y, with different types of attributes. This procedure can be easily modified to work with dissimilarities.
Algorithm 2.2 Similarities of heterogeneous objects.
1. 1: For the kth attribute, compute a similarity, sk(x, y), in the range [0, 1].
2. 2: Define an indicator variable, δk, for the kth attribute as follows:
δk={ 0if the kth attribute is an asymmetric attribute andboth objects have a value of
3. 3: Compute the overall similarity between the two objects using the following formula:
similarity (x, y)=∑k=1nδksk(x, y)∑k=1nδk (2.28)
Using Weights In much of the previous discussion, all attributes were treated equally when computing proximity. This is not desirable when some attributes are more important to the definition of proximity than others. To address these situations, the formulas for proximity can be modified by weighting the contribution of each attribute.
With attribute weights, wk, (2.28) becomes
similarity (x, y)=∑k=1nwkδksk(x, y)∑k=1nwkδk. (2.29)
The definition of the Minkowski distance can also be modified as follows:
d (x, y)=(∑k=1nwk|xk−yk|r)1/r. (2.30)
2.4.10 Selecting the Right Proximity Measure A few general observations may be helpful. First, the type of proximity measure should fit the type of data. For many types of dense, continuous data, metric distance measures such as Euclidean distance are often used. Proximity between continuous attributes is most often expressed in terms of differences, and distance measures provide a well-defined way of combining these differences into an overall proximity measure. Although attributes can have different scales and be of differing importance, these issues can often be dealt with as described earlier, such as normalization and weighting of attributes.
For sparse data, which often consists of asymmetric attributes, we typically employ similarity measures that ignore 0–0 matches. Conceptually, this reflects the fact that, for a pair of complex objects, similarity depends on the number of characteristics they both share, rather than the number of characteristics they both lack. The cosine, Jaccard, and extended Jaccard measures are appropriate for such data.
There are other characteristics of data vectors that often need to be considered. Invariance to scaling (multiplication) and to translation (addition) were previously discussed with respect to Euclidean distance and the cosine and correlation measures. The practical implications of such considerations are that, for example, cosine is more suitable for sparse document data where only scaling is important, while correlation works better for time series, where both scaling and translation are important. Euclidean distance or other types of Minkowski distance are most appropriate when two data vectors are to match as closely as possible across all components (features).
In some cases, transformation or normalization of the data is needed to obtain a proper similarity measure. For instance, time series can have trends or periodic patterns that significantly impact similarity. Also, a proper computation of similarity often requires that time lags be taken into account. Finally, two time series may be similar only over specific periods of time. For example, there is a strong relationship between temperature and the use of natural gas, but only during the heating season.
Practical consideration can also be important. Sometimes, one or more proximity measures are already in use in a particular field, and thus, others will have answered the question of which proximity measures should be used. Other times, the software package or clustering algorithm being used can drastically limit the choices. If efficiency is a concern, then we may want to choose a proximity measure that has a property, such as the triangle inequality, that can be used to reduce the number of proximity calculations. (See Exercise 25.)
However, if common practice or practical restrictions do not dictate a choice, then the proper choice of a proximity measure can be a time-consuming task that requires careful consideration of both domain knowledge and the purpose for which the measure is being used. A number of different similarity
measures may need to be evaluated to see which ones produce results that make the most sense.
2.5 Bibliographic Notes It is essential to understand the nature of the data that is being analyzed, and at a fundamental level, this is the subject of measurement theory. In particular, one of the initial motivations for defining types of attributes was to be precise about which statistical operations were valid for what sorts of data. We have presented the view of measurement theory that was initially described in a classic paper by S. S. Stevens [112]. (Tables 2.2 and 2.3 are derived from those presented by Stevens [113].) While this is the most common view and is reasonably easy to understand and apply, there is, of course, much more to measurement theory. An authoritative discussion can be found in a three-volume series on the foundations of measurement theory [88, 94, 114]. Also of interest is a wide-ranging article by Hand [77], which discusses measurement theory and statistics, and is accompanied by comments from other researchers in the field. Numerous critiques and extensions of the approach of Stevens have been made [66, 97, 117]. Finally, many books and articles describe measurement issues for particular areas of science and engineering.
Data quality is a broad subject that spans every discipline that uses data. Discussions of precision, bias, accuracy, and significant figures can be found in many introductory science, engineering, and statistics textbooks. The view of data quality as “fitness for use” is explained in more detail in the book by Redman [103]. Those interested in data quality may also be interested in MIT’s Information Quality (MITIQ) Program [95, 118]. However, the knowledge needed to deal with specific data quality issues in a particular domain is often best obtained by investigating the data quality practices of researchers in that field.
Aggregation is a less well-defined subject than many other preprocessing tasks. However, aggregation is one of the main techniques used by the database area of Online Analytical Processing (OLAP) [68, 76, 102]. There has also been relevant work in the area of symbolic data analysis (Bock and Diday [64]). One of the goals in this area is to summarize traditional record data in terms of symbolic data objects whose attributes are more complex
than traditional attributes. Specifically, these attributes can have values that are sets of values (categories), intervals, or sets of values with weights (histograms). Another goal of symbolic data analysis is to be able to perform clustering, classification, and other kinds of data analysis on data that consists of symbolic data objects.
Sampling is a subject that has been well studied in statistics and related fields. Many introductory statistics books, such as the one by Lindgren [90], have some discussion about sampling, and entire books are devoted to the subject, such as the classic text by Cochran [67]. A survey of sampling for data mining is provided by Gu and Liu [74], while a survey of sampling for databases is provided by Olken and Rotem [98]. There are a number of other data mining and database-related sampling references that may be of interest, including papers by Palmer and Faloutsos [100], Provost et al. [101], Toivonen [115], and Zaki et al. [119].
In statistics, the traditional techniques that have been used for dimensionality reduction are multidimensional scaling (MDS) (Borg and Groenen [65], Kruskal and Uslaner [89]) and principal component analysis (PCA) (Jolliffe [80]), which is similar to singular value decomposition (SVD) (Demmel [70]). Dimensionality reduction is discussed in more detail in Appendix B.
Discretization is a topic that has been extensively investigated in data mining. Some classification algorithms work only with categorical data, and association analysis requires binary data, and thus, there is a significant motivation to investigate how to best binarize or discretize continuous attributes. For association analysis, we refer the reader to work by Srikant and Agrawal [111], while some useful references for discretization in the area of classification include work by Dougherty et al. [71], Elomaa and Rousu [72], Fayyad and Irani [73], and Hussain et al. [78].
Feature selection is another topic well investigated in data mining. A broad coverage of this topic is provided in a survey by Molina et al. [96] and two books by Liu and Motada [91, 92]. Other useful papers include those by Blum and Langley [63], Kohavi and John [87], and Liu et al. [93].
It is difficult to provide references for the subject of feature transformations because practices vary from one discipline to another. Many statistics books
have a discussion of transformations, but typically the discussion is restricted to a particular purpose, such as ensuring the normality of a variable or making sure that variables have equal variance. We offer two references: Osborne [99] and Tukey [116].
While we have covered some of the most commonly used distance and similarity measures, there are hundreds of such measures and more are being created all the time. As with so many other topics in this chapter, many of these measures are specific to particular fields, e.g., in the area of time series see papers by Kalpakis et al. [81] and Keogh and Pazzani [83]. Clustering books provide the best general discussions. In particular, see the books by Anderberg [62], Jain and Dubes [79], Kaufman and Rousseeuw [82], and Sneath and Sokal [109].
Information-based measures of similarity have become more popular lately despite the computational difficulties and expense of calculating them. A good introduction to information theory is provided by Cover and Thomas [69]. Computing the mutual information for continuous variables can be straightforward if they follow a well-know distribution, such as Gaussian. However, this is often not the case, and many techniques have been developed. As one example, the article by Khan, et al. [85] compares various methods in the context of comparing short time series. See also the information and mutual information packages for R and Matlab. Mutual information has been the subject of considerable recent attention due to paper by Reshef, et al. [104, 105] that introduced an alternative measure, albeit one based on mutual information, which was claimed to have superior properties. Although this approach had some early support, e.g., [110], others have pointed out various limitations [75, 86, 108].
Two popular books on the topic of kernel methods are [106] and [107]. The latter also has a website with links to kernel-related materials [84]. In addition, many current data mining, machine learning, and statistical learning textbooks have some material about kernel methods. Further references for kernel methods in the context of support vector machine classifiers are provided in the bibliographic Notes of Section 4.9.4.
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2.6 Exercises 1. 1. In the initial example of Chapter 2, the statistician says, “Yes, fields 2
and 3 are basically the same.” Can you tell from the three lines of sample data that are shown why she says that?
2. 2. Classify the following attributes as binary, discrete, or continuous. Also classify them as qualitative (nominal or ordinal) or quantitative (interval or ratio). Some cases may have more than one interpretation, so briefly indicate your reasoning if you think there may be some ambiguity.
Example: Age in years. Answer: Discrete, quantitative, ratio
1. Time in terms of AM or PM.
2. Brightness as measured by a light meter.
3. Brightness as measured by people’s judgments.
4. Angles as measured in degrees between 0 and 360.
5. Bronze, Silver, and Gold medals as awarded at the Olympics.
6. Height above sea level.
7. Number of patients in a hospital.
8. ISBN numbers for books. (Look up the format on the Web.)
9. Ability to pass light in terms of the following values: opaque, translucent, transparent.
10. Military rank.
11. Distance from the center of campus.
12. Density of a substance in grams per cubic centimeter.
13. Coat check number. (When you attend an event, you can often give your coat to someone who, in turn, gives you a number that you can use to claim your coat when you leave.)
3. 3. You are approached by the marketing director of a local company, who believes that he has devised a foolproof way to measure customer satisfaction. He explains his scheme as follows: “It’s so simple that I can’t believe that no one has thought of it before. I just keep track of the number of customer complaints for each product. I read in a data mining book that counts are ratio attributes, and so, my measure of product satisfaction must be a ratio attribute. But when I rated the products based on my new customer satisfaction measure and showed them to my boss, he told me that I had overlooked the obvious, and that my measure was worthless. I think that he was just mad because our bestselling product had the worst satisfaction since it had the most complaints. Could you help me set him straight?”
1. Who is right, the marketing director or his boss? If you answered, his boss, what would you do to fix the measure of satisfaction?
2. What can you say about the attribute type of the original product satisfaction attribute?
4. 4. A few months later, you are again approached by the same marketing director as in Exercise 3. This time, he has devised a better approach to measure the extent to which a customer prefers one product over other similar products. He explains, “When we develop new products, we typically create several variations and evaluate which one customers prefer. Our standard procedure is to give our test subjects all of the product variations at one time and then ask them to rank the product variations in order of preference. However, our test subjects are very indecisive, especially when there are more than two products. As a result, testing takes forever. I suggested that we perform the comparisons in pairs and then use these comparisons to get the rankings. Thus, if we have three product variations, we have the customers compare variations 1 and 2, then 2 and 3, and finally 3 and 1. Our
testing time with my new procedure is a third of what it was for the old procedure, but the employees conducting the tests complain that they cannot come up with a consistent ranking from the results. And my boss wants the latest product evaluations, yesterday. I should also mention that he was the person who came up with the old product evaluation approach. Can you help me?”
1. Is the marketing director in trouble? Will his approach work for generating an ordinal ranking of the product variations in terms of customer preference? Explain.
2. Is there a way to fix the marketing director’s approach? More generally, what can you say about trying to create an ordinal measurement scale based on pairwise comparisons?
3. For the original product evaluation scheme, the overall rankings of each product variation are found by computing its average over all test subjects. Comment on whether you think that this is a reasonable approach. What other approaches might you take?
5. 5. Can you think of a situation in which identification numbers would be useful for prediction?
6. 6. An educational psychologist wants to use association analysis to analyze test results. The test consists of 100 questions with four possible answers each.
1. How would you convert this data into a form suitable for association analysis?
2. In particular, what type of attributes would you have and how many of them are there?
7. 7. Which of the following quantities is likely to show more temporal autocorrelation: daily rainfall or daily temperature? Why?
8. 8. Discuss why a document-term matrix is an example of a data set that has asymmetric discrete or asymmetric continuous features.
9. 9. Many sciences rely on observation instead of (or in addition to) designed experiments. Compare the data quality issues involved in observational science with those of experimental science and data mining.
10. 10. Discuss the difference between the precision of a measurement and the terms single and double precision, as they are used in computer science, typically to represent floating-point numbers that require 32 and 64 bits, respectively.
11. 11. Give at least two advantages to working with data stored in text files instead of in a binary format.
12. 12. Distinguish between noise and outliers. Be sure to consider the following questions.
1. Is noise ever interesting or desirable? Outliers?
2. Can noise objects be outliers?
3. Are noise objects always outliers?
4. Are outliers always noise objects?
5. Can noise make a typical value into an unusual one, or vice versa?
Algorithm 2.3 Algorithm for finding k-nearest neighbors.
1. 1: for i=1 to number of data objects do
2. 2: Find the distances of the ith object to all other objects.
3. 3: Sort these distances in decreasing order.
(Keep track of which object is associated with each distance.)
4. 4: return the objects associated with the first k distances of the sorted list
5. 5: end for
13. 13. Consider the problem of finding the K-nearest neighbors of a data object. A programmer designs Algorithm 2.3 for this task.
1. Describe the potential problems with this algorithm if there are duplicate objects in the data set. Assume the distance function will return a distance of 0 only for objects that are the same.
2. How would you fix this problem?
14. 14. The following attributes are measured for members of a herd of Asian elephants: weight, height, tusk length, trunk length, and ear area. Based on these measurements, what sort of proximity measure from Section 2.4 would you use to compare or group these elephants? Justify your answer and explain any special circumstances.
15. 15. You are given a set of m objects that is divided into k groups, where the ith group is of size mi. If the goal is to obtain a sample of size n<m, what is the difference between the following two sampling schemes? (Assume sampling with replacement.)
1. We randomly select n×mi/m elements from each group.
2. We randomly select n elements from the data set, without regard for the group to which an object belongs.
16. 16. Consider a document-term matrix, where tfij is the frequency of the ith word (term) in the jth document and m is the number of documents. Consider the variable transformation that is defined by
tfij′=tfij×logmdfi, (2.31)
where dfi is the number of documents in which the ith term appears, which is known as the document frequency of the term. This
transformation is known as the inverse document frequency transformation.
1. What is the effect of this transformation if a term occurs in one document? In every document?
2. What might be the purpose of this transformation?
17. 17. Assume that we apply a square root transformation to a ratio attribute x to obtain the new attribute x*. As part of your analysis, you identify an interval (a, b) in which x* has a linear relationship to another attributey.
1. What is the corresponding interval (A, B) in terms of x ?
2. Give an equation that relates y to x.
18. 18. This exercise compares and contrasts some similarity and distance measures.
1. For binary data, the L1 distance corresponds to the Hamming distance; that is, the number of bits that are different between two binary vectors. The Jaccard similarity is a measure of the similarity between two binary vectors. Compute the Hamming distance and the Jaccard similarity between the following two binary vectors.
x=0101010001y=0100011000
2. Which approach, Jaccard or Hamming distance, is more similar to the Simple Matching Coefficient, and which approach is more similar to the cosine measure? Explain. (Note: The Hamming measure is a distance, while the other three measures are similarities, but don’t let this confuse you.)
3. Suppose that you are comparing how similar two organisms of different species are in terms of the number of genes they share. Describe which measure, Hamming or Jaccard, you think would be more appropriate for comparing the genetic makeup of two
organisms. Explain. (Assume that each animal is represented as a binary vector, where each attribute is 1 if a particular gene is present in the organism and 0 otherwise.)
4. If you wanted to compare the genetic makeup of two organisms of the same species, e.g., two human beings, would you use the Hamming distance, the Jaccard coefficient, or a different measure of similarity or distance? Explain. (Note that two human beings share >99.9% of the same genes.)
19. 19. For the following vectors, x and y, calculate the indicated similarity or distance measures.
1. x=(1, 1, 1, 1), y=(2, 2, 2, 2) cosine, correlation, Euclidean
2. x=(0, 1, 0, 1), y=(1, 0, 1, 0) cosine, correlation, Euclidean, Jaccard
3. x=(0, −1, 0, 1), y=(1, 0, −1, 0) cosine, correlation, Euclidean
4. x=(1, 1, 0, 1, 0, 1), y=(1, 1, 1, 0, 0, 1) cosine, correlation, Jaccard
5. x=(2, −1, 0, 2, 0, −3), y=( −1, 1, −1, 0, 0, −1) cosine, correlation
20. 20. Here, we further explore the cosine and correlation measures.
1. What is the range of values possible for the cosine measure?
2. If two objects have a cosine measure of 1, are they identical? Explain.
3. What is the relationship of the cosine measure to correlation, if any? (Hint: Look at statistical measures such as mean and standard deviation in cases where cosine and correlation are the same and different.)
4. Figure 2.22(a) shows the relationship of the cosine measure to Euclidean distance for 100,000 randomly generated points that have been normalized to have an L2 length of 1. What general observation can you make about the relationship between
Euclidean distance and cosine similarity when vectors have an L2 norm of 1?
Figure 2.22. Graphs for Exercise 20.
Figure 2.22. Full Alternative Text
5. Figure 2.22(b) shows the relationship of correlation to Euclidean distance for 100,000 randomly generated points that have been standardized to have a mean of 0 and a standard deviation of 1. What general observation can you make about the relationship between Euclidean distance and correlation when the vectors have been standardized to have a mean of 0 and a standard deviation of 1?
6. Derive the mathematical relationship between cosine similarity and Euclidean distance when each data object has an L2 length of 1.
7. Derive the mathematical relationship between correlation and Euclidean distance when each data point has been been standardized by subtracting its mean and dividing by its standard deviation.
21. 21. Show that the set difference metric given by
d(A, B)=size(A−B)+size(B−A) (2.32)
satisfies the metric axioms given on page 77. A and B are sets and A−B is the set difference.
22. 22. Discuss how you might map correlation values from the interval [−1, 1] to the interval [0, 1]. Note that the type of transformation that you use might depend on the application that you have in mind. Thus, consider two applications: clustering time series and predicting the behavior of one time series given another.
23. 23. Given a similarity measure with values in the interval [0, 1], describe two ways to transform this similarity value into a dissimilarity value in the interval [0, ∞].
24. 24. Proximity is typically defined between a pair of objects.
1. Define two ways in which you might define the proximity among a group of objects.
2. How might you define the distance between two sets of points in Euclidean space?
3. How might you define the proximity between two sets of data objects? (Make no assumption about the data objects, except that a proximity measure is defined between any pair of objects.)
25. 25. You are given a set of points s in Euclidean space, as well as the distance of each point in s to a point x. (It does not matter if x∈S.)
1. If the goal is to find all points within a specified distance ε of point y, y≠x, explain how you could use the triangle inequality and the
already calculated distances to x to potentially reduce the number of distance calculations necessary? Hint: The triangle inequality, d(x, z)≤d(x, y)+d(y, x), can be rewritten as d(x, y)≥d(x, z)−d(y, z).
2. In general, how would the distance between x and y affect the number of distance calculations?
3. Suppose that you can find a small subset of points S′, from the original data set, such that every point in the data set is within a specified distance ε of at least one of the points in S′, and that you also have the pairwise distance matrix for S′. Describe a technique that uses this information to compute, with a minimum of distance calculations, the set of all points within a distance of β of a specified point from the data set.
26. 26. Show that 1 minus the Jaccard similarity is a distance measure between two data objects, x and y, that satisfies the metric axioms given on page 77. Specifically, d(x, y)=1−J(x, y).
27. 27. Show that the distance measure defined as the angle between two data vectors, x and y, satisfies the metric axioms given on page 77. Specifically, d(x, y)=arccos(cos(x, y)).
28. 28. Explain why computing the proximity between two attributes is often simpler than computing the similarity between two objects.
3 Classification: Basic Concepts and Techniques Humans have an innate ability to classify things into categories, e.g., mundane tasks such as filtering spam email messages or more specialized tasks such as recognizing celestial objects in telescope images (see Figure 3.1). While manual classification often suffices for small and simple data sets with only a few attributes, larger and more complex data sets require an automated solution.
Figure 3.1. Classification of galaxies from telescope images taken from the NASA website.
This chapter introduces the basic concepts of classification and describes some of its key issues such as model overfitting, model selection, and model evaluation. While these topics are illustrated using a classification technique
known as decision tree induction, most of the discussion in this chapter is also applicable to other classification techniques, many of which are covered in Chapter 4.
3.1 Basic Concepts Figure 3.2 illustrates the general idea behind classification. The data for a classification task consists of a collection of instances (records). Each such instance is characterized by the tuple (x, y), where x is the set of attribute values that describe the instance and y is the class label of the instance. The attribute set x can contain attributes of any type, while the class label y must be categorical.
Figure 3.2. A schematic illustration of a classification task.
A classification model is an abstract representation of the relationship between the attribute set and the class label. As will be seen in the next two chapters, the model can be represented in many ways, e.g., as a tree, a probability table, or simply, a vector of real-valued parameters. More formally, we can express it mathematically as a target function f that takes as input the attribute set x and produces an output corresponding to the predicted class label. The model is said to classify an instance (x, y) correctly if f(x)=y.
Table 3.1 shows examples of attribute sets and class labels for various classification tasks. Spam filtering and tumor identification are examples of binary classification problems, in which each data instance can be categorized into one of two classes. If the number of classes is larger than 2, as in the galaxy classification example, then it is called a multiclass classification problem.
Table 3.1. Examples of classification tasks.
Task Attribute set Class label Spam filtering
Features extracted from email message header and content spam or non-spam
Tumor identification
Features extracted from magnetic resonance imaging (MRI) scans
malignant or benign
Galaxy classification
Features extracted from telescope images
elliptical, spiral, or irregular- shaped
We illustrate the basic concepts of classification in this chapter with the following two examples.
3.1. Example Vertebrate Classification Table 3.2 shows a sample data set for classifying vertebrates into mammals, reptiles, birds, fishes, and amphibians. The attribute set includes characteristics of the vertebrate such as its body temperature, skin cover, and ability to fly. The data set can also be used for a binary classification task such as mammal classification, by grouping the reptiles, birds, fishes, and amphibians into a single category called non-mammals.
Table 3.2. A sample data for the vertebrate classification problem.
Vertebrate Name
Body Temperature
Skin Cover
Gives Birth
Aquatic Creature
Aerial Creature
Has Legs Hibernates
human warm-blooded hair yes no no yes no
python cold-blooded scales no no no no yes salmon cold-blooded scales no yes no no no
whale warm-blooded hair yes yes no no no
frog cold-blooded none no semi no yes yes komodo cold-blooded scales no no no yes no dragon
bat warm-blooded hair yes no yes yes yes
pigeon warm-blooded feathers no no yes yes no
cat warm-blooded fur yes no no yes no
leopard cold-blooded scales yes yes no no no shark turtle cold-blooded scales no semi no yes no
penguin warm-blooded feathers no semi no yes no
porcupine warm-blooded quills yes no no yes yes
eel cold-blooded scales no yes no no no salamander cold-blooded none no semi no yes yes
3.2. Example Loan Borrower Classification Consider the problem of predicting whether a loan borrower will repay the
loan or default on the loan payments. The data set used to build the classification model is shown in Table 3.3. The attribute set includes personal information of the borrower such as marital status and annual income, while the class label indicates whether the borrower had defaulted on the loan payments.
Table 3.3. A sample data for the loan borrower classification problem. ID Home Owner Marital Status Annual Income Defaulted? 1 Yes Single 125000 No 2 No Married 100000 No 3 No Single 70000 No 4 Yes Married 120000 No 5 No Divorced 95000 Yes 6 No Single 60000 No 7 Yes Divorced 220000 No 8 No Single 85000 Yes 9 No Married 75000 No 10 No Single 90000 Yes
A classification model serves two important roles in data mining. First, it is used as a predictive model to classify previously unlabeled instances. A good classification model must provide accurate predictions with a fast response time. Second, it serves as a descriptive model to identify the characteristics that distinguish instances from different classes. This is particularly useful for critical applications, such as medical diagnosis, where it is insufficient to have a model that makes a prediction without justifying how it reaches such a decision.
For example, a classification model induced from the vertebrate data set
shown in Table 3.2 can be used to predict the class label of the following vertebrate:
Vertebrate Name
Body Temperature
Skin Cover
Gives Birth
Aquatic Creature
Aerial Creature
Has Legs Hibernates
gila monster cold-blooded scales no no no yes yes
In addition, it can be used as a descriptive model to help determine characteristics that define a vertebrate as a mammal, a reptile, a bird, a fish, or an amphibian. For example, the model may identify mammals as warm- blooded vertebrates that give birth to their young.
There are several points worth noting regarding the previous example. First, although all the attributes shown in Table 3.2 are qualitative, there are no restrictions on the type of attributes that can be used as predictor variables. The class label, on the other hand, must be of nominal type. This distinguishes classification from other predictive modeling tasks such as regression, where the predicted value is often quantitative. More information about regression can be found in Appendix D.
Another point worth noting is that not all attributes may be relevant to the classification task. For example, the average length or weight of a vertebrate may not be useful for classifying mammals, as these attributes can show same value for both mammals and non-mammals. Such an attribute is typically discarded during preprocessing. The remaining attributes might not be able to distinguish the classes by themselves, and thus, must be used in concert with other attributes. For instance, the Body Temperature attribute is insufficient to distinguish mammals from other vertebrates. When it is used together with Gives Birth, the classification of mammals improves significantly. However, when additional attributes, such as Skin Cover are included, the model becomes overly specific and no longer covers all mammals. Finding the optimal combination of attributes that best discriminates instances from different classes is the key challenge in building classification models.
3.2 General Framework for Classification Classification is the task of assigning labels to unlabeled data instances and a classifier is used to perform such a task. A classifier is typically described in terms of a model as illustrated in the previous section. The model is created using a given a set of instances, known as the training set, which contains attribute values as well as class labels for each instance. The systematic approach for learning a classification model given a training set is known as a learning algorithm. The process of using a learning algorithm to build a classification model from the training data is known as induction. This process is also often described as “learning a model” or “building a model.” This process of applying a classification model on unseen test instances to predict their class labels is known as deduction. Thus, the process of classification involves two steps: applying a learning algorithm to training data to learn a model, and then applying the model to assign labels to unlabeled instances. Figure 3.3 illustrates the general framework for classification.
Figure 3.3. General framework for building a classification model.
Figure 3.3. Full Alternative Text
A classification technique refers to a general approach to classification, e.g., the decision tree technique that we will study in this chapter. This classification technique like most others, consists of a family of related models and a number of algorithms for learning these models. In Chapter 4, we will study additional classification techniques, including neural networks
and support vector machines.
A couple notes on terminology. First, the terms “classifier” and “model” are often taken to be synonymous. If a classification technique builds a single, global model, then this is fine. However, while every model defines a classifier, not every classifier is defined by a single model. Some classifiers, such as k-nearest neighbor classifiers, do not build an explicit model (Section 4.3), while other classifiers, such as ensemble classifiers, combine the output of a collection of models (Section 4.10). Second, the term “classifier” is often used in a more general sense to refer to a classification technique. Thus, for example, “decision tree classifier” can refer to the decision tree classification technique or a specific classifier built using that technique. Fortunately, the meaning of “classifier” is usually clear from the context.
In the general framework shown in Figure 3.3, the induction and deduction steps should be performed separately. In fact, as will be discussed later in Section 3.6, the training and test sets should be independent of each other to ensure that the induced model can accurately predict the class labels of instances it has never encountered before. Models that deliver such predictive insights are said to have good generalization performance. The performance of a model (classifier) can be evaluated by comparing the predicted labels against the true labels of instances. This information can be summarized in a table called a confusion matrix. Table 3.4 depicts the confusion matrix for a binary classification problem. Each entry fij denotes the number of instances from class i predicted to be of class j. For example, f01 is the number of instances from class 0 incorrectly predicted as class 1. The number of correct predictions made by the model is (f11+f00) and the number of incorrect predictions is (f10+f01).
Table 3.4. Confusion matrix for a binary classification problem.
Predicted Class Class=1 Class=0
Actual Class Class=1 f11 f10 Class=0 f01 f00
Although a confusion matrix provides the information needed to determine how well a classification model performs, summarizing this information into a single number makes it more convenient to compare the relative performance of different models. This can be done using an evaluation metric such as accuracy, which is computed in the following way:
Accuracy =
Accuracy=Number of correct predictionsTotal number of predictions. (3.1)
For binary classification problems, the accuracy of a model is given by
Accuracy=f11+f00f11+f10+f01+f00. (3.2)
Error rate is another related metric, which is defined as follows for binary classification problems:
Error rate=Number of wrong predictionsTotal number of predictions=f10+f01 (3.3)
The learning algorithms of most classification techniques are designed to learn models that attain the highest accuracy, or equivalently, the lowest error rate when applied to the test set. We will revisit the topic of model evaluation in Section 3.6.
3.3 Decision Tree Classifier This section introduces a simple classification technique known as the decision tree classifier. To illustrate how a decision tree works, consider the classification problem of distinguishing mammals from non-mammals using the vertebrate data set shown in Table 3.2. Suppose a new species is discovered by scientists. How can we tell whether it is a mammal or a non- mammal? One approach is to pose a series of questions about the characteristics of the species. The first question we may ask is whether the species is cold- or warm-blooded. If it is cold-blooded, then it is definitely not a mammal. Otherwise, it is either a bird or a mammal. In the latter case, we need to ask a follow-up question: Do the females of the species give birth to their young? Those that do give birth are definitely mammals, while those that do not are likely to be non-mammals (with the exception of egg-laying mammals such as the platypus and spiny anteater).
The previous example illustrates how we can solve a classification problem by asking a series of carefully crafted questions about the attributes of the test instance. Each time we receive an answer, we could ask a follow-up question until we can conclusively decide on its class label. The series of questions and their possible answers can be organized into a hierarchical structure called a decision tree. Figure 3.4 shows an example of the decision tree for the mammal classification problem. The tree has three types of nodes:
A root node, with no incoming links and zero or more outgoing links.
Internal nodes, each of which has exactly one incoming link and two or more outgoing links.
Leaf or terminal nodes, each of which has exactly one incoming link and no outgoing links.
Every leaf node in the decision tree is associated with a class label. The non- terminal nodes, which include the root and internal nodes, contain attribute test conditions that are typically defined using a single attribute. Each
possible outcome of the attribute test condition is associated with exactly one child of this node. For example, the root node of the tree shown in Figure 3.4 uses the attribute Body Temperature to define an attribute test condition that has two outcomes, warm and cold, resulting in two child nodes.
Figure 3.4. A decision tree for the mammal classification problem.
Figure 3.4. Full Alternative Text
Given a decision tree, classifying a test instance is straightforward. Starting from the root node, we apply its attribute test condition and follow the appropriate branch based on the outcome of the test. This will lead us either to another internal node, for which a new attribute test condition is applied, or to a leaf node. Once a leaf node is reached, we assign the class label associated with the node to the test instance. As an illustration, Figure 3.5 traces the path used to predict the class label of a flamingo. The path terminates at a leaf node labeled as Non-mammals.
Figure 3.5. Classifying an unlabeled vertebrate. The dashed lines represent the outcomes of applying various attribute test conditions on the unlabeled vertebrate. The vertebrate is eventually assigned to the Non-mammals class.
Figure 3.5. Full Alternative Text
3.3.1 A Basic Algorithm to Build a Decision Tree Many possible decision trees that can be constructed from a particular data set. While some trees are better than others, finding an optimal one is computationally expensive due to the exponential size of the search space. Efficient algorithms have been developed to induce a reasonably accurate,
albeit suboptimal, decision tree in a reasonable amount of time. These algorithms usually employ a greedy strategy to grow the decision tree in a top-down fashion by making a series of locally optimal decisions about which attribute to use when partitioning the training data. One of the earliest method is Hunt's algorithm, which is the basis for many current implementations of decision tree classifiers, including ID3, C4.5, and CART. This subsection presents Hunt's algorithm and describes some of the design issues that must be considered when building a decision tree.
Hunt's Algorithm In Hunt's algorithm, a decision tree is grown in a recursive fashion. The tree initially contains a single root node that is associated with all the training instances. If a node is associated with instances from more than one class, it is expanded using an attribute test condition that is determined using a splitting criterion. A child leaf node is created for each outcome of the attribute test condition and the instances associated with the parent node are distributed to the children based on the test outcomes. This node expansion step can then be recursively applied to each child node, as long as it has labels of more than one class. If all the instances associated with a leaf node have identical class labels, then the node is not expanded any further. Each leaf node is assigned a class label that occurs most frequently in the training instances associated with the node.
To illustrate how the algorithm works, consider the training set shown in Table 3.3 for the loan borrower classification problem. Suppose we apply Hunt's algorithm to fit the training data. The tree initially contains only a single leaf node as shown in Figure 3.6(a). This node is labeled as Defaulted = No, since the majority of the borrowers did not default on their loan payments. The training error of this tree is 30% as three out of the ten training instances have the class label Defaulted = Yes. The leaf node can therefore be further expanded because it contains training instances from more than one class.
Figure 3.6. Hunt's algorithm for building decision trees.
Figure 3.6. Full Alternative Text
Let Home Owner be the attribute chosen to split the training instances. The justification for choosing this attribute as the attribute test condition will be discussed later. The resulting binary split on the Home Owner attribute is shown in Figure 3.6(b). All the training instances for which Home Owner = Yes are propagated to the left child of the root node and the rest are propagated to the right child. Hunt's algorithm is then recursively applied to
each child. The left child becomes a leaf node labeled Defaulted = No, since all instances associated with this node have identical class label Defaulted = No. The right child has instances from each class label. Hence, we split it further. The resulting subtrees after recursively expanding the right child are shown in Figures 3.6(c) and (d).
Hunt's algorithm, as described above, makes some simplifying assumptions that are often not true in practice. In the following, we describe these assumptions and briefly discuss some of the possible ways for handling them.
1. Some of the child nodes created in Hunt's algorithm can be empty if none of the training instances have the particular attribute values. One way to handle this is by declaring each of them as a leaf node with a class label that occurs most frequently among the training instances associated with their parent nodes.
2. If all training instances associated with a node have identical attribute values but different class labels, it is not possible to expand this node any further. One way to handle this case is to declare it a leaf node and assign it the class label that occurs most frequently in the training instances associated with this node.
Design Issues of Decision Tree Induction Hunt's algorithm is a generic procedure for growing decision trees in a greedy fashion. To implement the algorithm, there are two key design issues that must be addressed.
1. What is the splitting criterion? At each recursive step, an attribute must be selected to partition the training instances associated with a node into smaller subsets associated with its child nodes. The splitting criterion determines which attribute is chosen as the test condition and how the training instances should be distributed to the child nodes. This will be discussed in Sections 3.3.2 and 3.3.3.
2. What is the stopping criterion? The basic algorithm stops expanding a node only when all the training instances associated with the node have the same class labels or have identical attribute values. Although these conditions are sufficient, there are reasons to stop expanding a node much earlier even if the leaf node contains training instances from more than one class. This process is called early termination and the condition used to determine when a node should be stopped from expanding is called a stopping criterion. The advantages of early termination are discussed in Section 3.4.
3.3.2 Methods for Expressing Attribute Test Conditions Decision tree induction algorithms must provide a method for expressing an attribute test condition and its corresponding outcomes for different attribute types.
Binary Attributes The test condition for a binary attribute generates two potential outcomes, as shown in Figure 3.7.
Figure 3.7. Attribute test condition for a binary attribute.
Nominal Attributes Since a nominal attribute can have many values, its attribute test condition can be expressed in two ways, as a multiway split or a binary split as shown in Figure 3.8. For a multiway split (Figure 3.8(a)), the number of outcomes depends on the number of distinct values for the corresponding attribute. For example, if an attribute such as marital status has three distinct values— single, married, or divorced—its test condition will produce a three-way split. It is also possible to create a binary split by partitioning all values taken by the nominal attribute into two groups. For example, some decision tree algorithms, such as CART, produce only binary splits by considering all 2k −1−1 ways of creating a binary partition of k attribute values. Figure 3.8(b) illustrates three different ways of grouping the attribute values for marital status into two subsets.
Figure 3.8. Attribute test conditions for nominal attributes.
Figure 3.8. Full Alternative Text
Ordinal Attributes Ordinal attributes can also produce binary or multi-way splits. Ordinal attribute values can be grouped as long as the grouping does not violate the order property of the attribute values. Figure 3.9 illustrates various ways of
splitting training records based on the Shirt Size attribute. The groupings shown in Figures 3.9(a) and (b) preserve the order among the attribute values, whereas the grouping shown in Figure 3.9(c) violates this property because it combines the attribute values Small and Large into the same partition while Medium and Extra Large are combined into another partition.
Figure 3.9. Different ways of grouping ordinal attribute values.
Figure 3.9. Full Alternative Text
Continuous Attributes For continuous attributes, the attribute test condition can be expressed as a comparison test (e.g., A<v) producing a binary split, or as a range query of the form vi≤A<vi+1, for i=1, …, k, producing a multiway split. The difference between these approaches is shown in Figure 3.10. For the binary split, any possible value v between the minimum and maximum attribute values in the training data can be used for constructing the comparison test A<v. However, it is sufficient to only consider distinct attribute values in the training set as candidate split positions. For the multiway split, any possible
collection of attribute value ranges can be used, as long as they are mutually exclusive and cover the entire range of attribute values between the minimum and maximum values observed in the training set. One approach for constructing multiway splits is to apply the discretization strategies described in Section 2.3.6 on page 63. After discretization, a new ordinal value is assigned to each discretized interval, and the attribute test condition is then defined using this newly constructed ordinal attribute.
Figure 3.10. Test condition for continuous attributes.
Figure 3.10. Full Alternative Text
3.3.3 Measures for Selecting an Attribute Test Condition There are many measures that can be used to determine the goodness of an attribute test condition. These measures try to give preference to attribute test conditions that partition the training instances into purer subsets in the child nodes, which mostly have the same class labels. Having purer nodes is useful since a node that has all of its training instances from the same class does not
need to be expanded further. In contrast, an impure node containing training instances from multiple classes is likely to require several levels of node expansions, thereby increasing the depth of the tree considerably. Larger trees are less desirable as they are more susceptible to model overfitting, a condition that may degrade the classification performance on unseen instances, as will be discussed in Section 3.4. They are also difficult to interpret and incur more training and test time as compared to smaller trees.
In the following, we present different ways of measuring the impurity of a node and the collective impurity of its child nodes, both of which will be used to identify the best attribute test condition for a node.
Impurity Measure for a Single Node The impurity of a node measures how dissimilar the class labels are for the data instances belonging to a common node. Following are examples of measures that can be used to evaluate the impurity of a node t:
Entropy=−∑i=0c−1pi(t) log2pi(t), (3.4) Gini index=1−∑i=0c−1pi(t)2, (3.5) Classification error=1−maxi[pi(t)], (3.6)
where pi(t) is the relative frequency of training instances that belong to class i at node t, c is the total number of classes, and 0 log2 0=0 in entropy calculations. All three measures give a zero impurity value if a node contains instances from a single class and maximum impurity if the node has equal proportion of instances from multiple classes.
Figure 3.11 compares the relative magnitude of the impurity measures when applied to binary classification problems. Since there are only two classes, p0(t)+p1(t)=1. The horizontal axis p refers to the fraction of instances that belong to one of the two classes. Observe that all three measures attain their maximum value when the class distribution is uniform (i.e., p0(t)+p1(t)=0.5) and minimum value when all the instances belong to a single class (i.e., either p0(t) or p1(t) equals to 1). The following examples illustrate how the values of the impurity measures vary as we alter the class distribution.
Figure 3.11. Comparison among the impurity measures for binary classification problems.
Figure 3.11. Full Alternative Text Node N1 Count Gini=1−(0/6)2−(6/6)2=0 Class=0 0 Entropy=−(0/6) log2(0/6)−(6/6) log2(6/6)=0 Class=1 6 Error=1−max[0/6, 6/6]=0 Node N2 Count Gini=1−(1/6)2−(5/6)2=0.278 Class=0 1 Entropy=−(1/6) log2(1/6)−(5/6) log2(5/6)=0.650 Class=1 5 Error=1−max[1/6, 5/6]=0.167 Node N3 Count Gini=1−(3/6)2−(3/6)2=0.5 Class=0 3 Entropy=−(3/6) log2(3/6)−(3/6) log2(3/6)=1
Class=1 3 Error=1−max[6/6, 3/6]=0.5
Based on these calculations, node N1 has the lowest impurity value, followed by N2 and N3. This example, along with Figure 3.11, shows the consistency among the impurity measures, i.e., if a node N1 has lower entropy than node N2, then the Gini index and error rate of N1 will also be lower than that of N2. Despite their agreement, the attribute chosen as splitting criterion by the impurity measures can still be different (see Exercise 6 on page 187).
Collective Impurity of Child Nodes Consider an attribute test condition that splits a node containing N training instances into k children, {v1, v2, ⋯ ,vk}, where every child node represents a partition of the data resulting from one of the k outcomes of the attribute test condition. Let N(vj) be the number of training instances associated with a child node vj, whose impurity value is I(vj). Since a training instance in the parent node reaches node vj for a fraction of N(vj)/N times, the collective impurity of the child nodes can be computed by taking a weighted sum of the impurities of the child nodes, as follows:
I(children)=∑j=1kN(vj)NI(vj), (3.7)
3.3. Example Weighted Entropy Consider the candidate attribute test condition shown in Figures 3.12(a) and (b) for the loan borrower classification problem. Splitting on the Home Owner attribute will generate two child nodes
Figure 3.12. Examples of candidate attribute test conditions.
Figure 3.12. Full Alternative Text
whose weighted entropy can be calculated as follows:
I(Home Owner=yes)=03log203−33log233=0I(Home Owner=no)= −37log237−47log247=0.985I(Home Owner=310×0+710×0.985=0.690
Splitting on Marital Status, on the other hand, leads to three child nodes with a weighted entropy given by
I(Marital Status=Single)= −25log225−35log235=0.971I(Marital Status=Married)= −03log203−33log233=0I(Marital Status=Divorced)= −12log212−12log212=1.000I(Marital Status)=510×0.971+310×0+210×1=0.686
Thus, Marital Status has a lower weighted entropy than Home Owner.
Identifying the best attribute test condition
To determine the goodness of an attribute test condition, we need to compare the degree of impurity of the parent node (before splitting) with the weighted degree of impurity of the child nodes (after splitting). The larger their difference, the better the test condition. This difference, Δ, also termed as the gain in purity of an attribute test condition, can be defined as follows:
Δ=I(parent)−I(children), (3.8)
Figure 3.13. Splitting criteria for the loan borrower classification problem using Gini index.
Figure 3.13. Full Alternative Text
where I(parent) is the impurity of a node before splitting and I(children) is the weighted impurity measure after splitting. It can be shown that the gain is non-negative since I(parent)≥I(children) for any reasonable measure such as those presented above. The higher the gain, the purer are the classes in the
child nodes relative to the parent node. The splitting criterion in the decision tree learning algorithm selects the attribute test condition that shows the maximum gain. Note that maximizing the gain at a given node is equivalent to minimizing the weighted impurity measure of its children since I(parent) is the same for all candidate attribute test conditions. Finally, when entropy is used as the impurity measure, the difference in entropy is commonly known as information gain, Δinfo.
In the following, we present illustrative approaches for identifying the best attribute test condition given qualitative or quantitative attributes.
Splitting of Qualitative Attributes Consider the first two candidate splits shown in Figure 3.12 involving qualitative attributes Home Owner and Marital Status. The initial class distribution at the parent node is (0.3, 0.7), since there are 3 instances of class Yes and 7 instances of class No in the training data. Thus,
I(parent)=−310log2310−710log2710=0.881
The information gains for Home Owner and Marital Status are each given by
Δinfo(Home Owner)=0.881−0.690=0.191Δinfo(Marital Status)=0.881−0.686=
The information gain for Marital Status is thus higher due to its lower weighted entropy, which will thus be considered for splitting.
Binary Splitting of Qualitative Attributes Consider building a decision tree using only binary splits and the Gini index as the impurity measure. Figure 3.13 shows examples of four candidate splitting criteria for the Home Owner and Marital Status attributes. Since there are 3 borrowers in the training set who defaulted and 7 others who
repaid their loan (see Table in Figure 3.13), the Gini index of the parent node before splitting is
1−(310)2−(710)2=0.420.
If Home Owner is chosen as the splitting attribute, the Gini index for the child nodes N1 and N2 are 0 and 0.490, respectively. The weighted average Gini index for the children is
(3/10)×0+(7/10)×0.490=0.343,
where the weights represent the proportion of training instances assigned to each child. The gain using Home Owner as splitting attribute is 0.420−0.343=0.077. Similarly, we can apply a binary split on the Marital Status attribute. However, since Marital Status is a nominal attribute with three outcomes, there are three possible ways to group the attribute values into a binary split. The weighted average Gini index of the children for each candidate binary split is shown in Figure 3.13. Based on these results, Home Owner and the last binary split using Marital Status are clearly the best candidates, since they both produce the lowest weighted average Gini index. Binary splits can also be used for ordinal attributes, if the binary partitioning of the attribute values does not violate the ordering property of the values.
Binary Splitting of Quantitative Attributes Consider the problem of identifying the best binary split Annual Income≤τ for the preceding loan approval classification problem. As discussed previously, even though τ can take any value between the minimum and maximum values of annual income in the training set, it is sufficient to only consider the annual income values observed in the training set as candidate split positions. For each candidate τ, the training set is scanned once to count the number of borrowers with annual income less than or greater than τ along with their class proportions. We can then compute the Gini index at each candidate split position and choose the τ that produces the lowest value.
Computing the Gini index at each candidate split position requires O(N) operations, where N is the number of training instances. Since there are at most N possible candidates, the overall complexity of this brute-force method is O(N2). It is possible to reduce the complexity of this problem to O(N log N) by using a method described as follows (see illustration in Figure 3.14). In this method, we first sort the training instances based on their annual income, a one-time cost that requires O(N log N) operations. The candidate split positions are given by the midpoints between every two adjacent sorted values: $55,000, $65,000, $72,500, and so on. For the first candidate, since none of the instances has an annual income less than or equal to $55,000, the Gini index for the child node with Annual Income< $55,000 is equal to zero. In contrast, there are 3 training instances of class Yes and 7 instances of class No with annual income greater than $55,000. The Gini index for this node is 0.420. The weighted average Gini index for the first candidate split position, τ=$55,000, is equal to 0×0+1×0.420=0.420.
Figure 3.14. Splitting continuous attributes.
Figure 3.14. Full Alternative Text
For the next candidate, τ=$65,000, the class distribution of its child nodes can be obtained with a simple update of the distribution for the previous candidate. This is because, as τ increases from $55,000 to $65,000, there is only one training instance affected by the change. By examining the class
label of the affected training instance, the new class distribution is obtained. For example, as τ increases to $65,000, there is only one borrower in the training set, with an annual income of $60,000, affected by this change. Since the class label for the borrower is No, the count for class No increases from 0 to 1 (for Annual Income≤$65,000) and decreases from 7 to 6 (for Annual Income>$65,000), as shown in Figure 3.14. The distribution for the Yes class remains unaffected. The updated Gini index for this candidate split position is 0.400.
This procedure is repeated until the Gini index for all candidates are found. The best split position corresponds to the one that produces the lowest Gini index, which occurs at τ=$97,500. Since the Gini index at each candidate split position can be computed in O(1) time, the complexity of finding the best split position is O(N) once all the values are kept sorted, a one-time operation that takes O(N log N) time. The overall complexity of this method is thus O(N log N), which is much smaller than the O(N2) time taken by the brute-force method. The amount of computation can be further reduced by considering only candidate split positions located between two adjacent sorted instances with different class labels. For example, we do not need to consider candidate split positions located between $60,000 and $75,000 because all three instances with annual income in this range ($60,000, $70,000, and $75,000) have the same class labels. Choosing a split position within this range only increases the degree of impurity, compared to a split position located outside this range. Therefore, the candidate split positions at τ=$65,000 and τ=$72,500 can be ignored. Similarly, we do not need to consider the candidate split positions at $87,500, $92,500, $110,000, $122,500, and $172,500 because they are located between two adjacent instances with the same labels. This strategy reduces the number of candidate split positions to consider from 9 to 2 (excluding the two boundary cases τ=$55,000 and τ=$230,000).
Gain Ratio One potential limitation of impurity measures such as entropy and Gini index is that they tend to favor qualitative attributes with large number of distinct values. Figure 3.12 shows three candidate attributes for partitioning the data
set given in Table 3.3. As previously mentioned, the attribute Marital Status is a better choice than the attribute Home Owner, because it provides a larger information gain. However, if we compare them against Customer ID, the latter produces the purest partitions with the maximum information gain, since the weighted entropy and Gini index is equal to zero for its children. Yet, Customer ID is not a good attribute for splitting because it has a unique value for each instance. Even though a test condition involving Customer ID will accurately classify every instance in the training data, we cannot use such a test condition on new test instances with Customer ID values that haven't been seen before during training. This example suggests having a low impurity value alone is insufficient to find a good attribute test condition for a node. As we will see later in Section 3.4, having more number of child nodes can make a decision tree more complex and consequently more susceptible to overfitting. Hence, the number of children produced by the splitting attribute should also be taken into consideration while deciding the best attribute test condition.
There are two ways to overcome this problem. One way is to generate only binary decision trees, thus avoiding the difficulty of handling attributes with varying number of partitions. This strategy is employed by decision tree classifiers such as CART. Another way is to modify the splitting criterion to take into account the number of partitions produced by the attribute. For example, in the C4.5 decision tree algorithm, a measure known as gain ratio is used to compensate for attributes that produce a large number of child nodes. This measure is computed as follows:
Gain ratio=ΔinfoSplit Info=Entropy(Parent)−∑i=1kN(vi)NEntropy(vi) −∑i=1kN(vi)Nlog2N(vi)N (3.9)
where N(vi) is the number of instances assigned to node vi and k is the total number of splits. The split information measures the entropy of splitting a node into its child nodes and evaluates if the split results in a larger number of equally-sized child nodes or not. For example, if every partition has the same number of instances, then ∀i:N(vi)/N=1/k and the split information would be equal to log2 k. Thus, if an attribute produces a large number of splits, its split information is also large, which in turn, reduces the gain ratio.
3.4. Example Gain Ratio Consider the data set given in Exercise 2 on page 185. We want to select the best attribute test condition among the following three attributes: Gender, Car Type, and Customer ID. The entropy before splitting is
Entropy(parent)=−1020log21020−1020log21020=1.
If Gender is used as attribute test condition:
Entropy(children)=1020[−610log2610−410log2410 ]×2=0.971Gain Ratio=1−0.971−1020log21020−1020log21020=0.0291=0.029
If Car Type is used as attribute test condition:
Entropy(children)=420[−14log214−34log234 ]+820×0+820[−18log218−78log278 ]=0.380Gain Ratio=1−0.380−420log2420−820log2820−820log2820=0.6201.52
Finally, if Customer ID is used as attribute test condition:
Entropy(children)=120[−11log211−01log201 ]×20=0Gain Ratio=1−0−120log2120×20=14.32=0.23
Thus, even though Customer ID has the highest information gain, its gain ratio is lower than Car Type since it produces a larger number of splits.
3.3.4 Algorithm for Decision Tree Induction Algorithm 3.1 presents a pseudocode for decision tree induction algorithm. The input to this algorithm is a set of training instances E along with the attribute set F . The algorithm works by recursively selecting the best attribute to split the data (Step 7) and expanding the nodes of the tree (Steps
11 and 12) until the stopping criterion is met (Step 1). The details of this algorithm are explained below.
1. The createNode() function extends the decision tree by creating a new node. A node in the decision tree either has a test condition, denoted as node.test cond, or a class label, denoted as node.label.
2. The find best split() function determines the attribute test condition for partitioning the training instances associated with a node. The splitting attribute chosen depends on the impurity measure used. The popular measures include entropy and the Gini index.
3. The Classify() function determines the class label to be assigned to a leaf node. For each leaf node t, let p(i|t) denote the fraction of training instances from class i associated with the node t. The label assigned to the leaf node is typically the one that occurs most frequently in the training instances that are associated with this node.
Algorithm 3.1 A skeleton decision tree induction algorithm.
TreeGrowth (E, F) 1: if stopping cond(E,F ) = true then 2: leaf = createNode(). 3: leaf.label = Classify(E). 4: return leaf. 5: else 6: root = createNode(). 7: root.test cond = find best split(E, F ). 8: let V = {v|v is a possible outcome of root.test cond } 9: for each v ∈ V do 10: Ev = {e | root.test cond(e) = v and e ∈ E}. 11: child = TreeGrowth(Ev, F ). 12: add child as descendent of root and label the edge ( 13: end for 14: end if 15: return root.
leaf.label=argmaxi p(i|t), (3.10)
where the argmax operator returns the class i that maximizes p(i|t). Besides providing the information needed to determine the class label of a leaf node, p(i|t) can also be used as a rough estimate of the probability that an instance assigned to the leaf node t belongs to class i. Sections 4.11.2 and 4.11.4 in the next chapter describe how such probability estimates can be used to determine the performance of a decision tree under different cost functions.
4. The stopping cond() function is used to terminate the tree-growing process by checking whether all the instances have identical class label or attribute values. Since decision tree classifiers employ a top-down, recursive partitioning approach for building a model, the number of training instances associated with a node decreases as the depth of the tree increases. As a result, a leaf node may contain too few training instances to make a statistically significant decision about its class label. This is known as the data fragmentation problem. One way to avoid this problem is to disallow splitting of a node when the number of instances associated with the node fall below a certain threshold. A more systematic way to control the size of a decision tree (number of leaf nodes) will be discussed in Section 3.5.4.
3.3.5 Example Application: Web Robot Detection Consider the task of distinguishing the access patterns of web robots from those generated by human users. A web robot (also known as a web crawler) is a software program that automatically retrieves files from one or more websites by following the hyperlinks extracted from an initial set of seed URLs. These programs have been deployed for various purposes, from gathering web pages on behalf of search engines to more malicious activities such as spamming and committing click frauds in online advertisements.
Figure 3.15. Input data for web robot detection.
Figure 3.15. Full Alternative Text
The web robot detection problem can be cast as a binary classification task. The input data for the classification task is a web server log, a sample of
which is shown in Figure 3.15(a). Each line in the log file corresponds to a request made by a client (i.e., a human user or a web robot) to the web server. The fields recorded in the web log include the client's IP address, timestamp of the request, URL of the requested file, size of the file, and user agent, which is a field that contains identifying information about the client. For human users, the user agent field specifies the type of web browser or mobile device used to fetch the files, whereas for web robots, it should technically contain the name of the crawler program. However, web robots may conceal their true identities by declaring their user agent fields to be identical to known browsers. Therefore, user agent is not a reliable field to detect web robots.
The first step toward building a classification model is to precisely define a data instance and associated attributes. A simple approach is to consider each log entry as a data instance and use the appropriate fields in the log file as its attribute set. This approach, however, is inadequate for several reasons. First, many of the attributes are nominal-valued and have a wide range of domain values. For example, the number of unique client IP addresses, URLs, and referrers in a log file can be very large. These attributes are undesirable for building a decision tree because their split information is extremely high (see Equation (3.9)). In addition, it might not be possible to classify test instances containing IP addresses, URLs, or referrers that are not present in the training data. Finally, by considering each log entry as a separate data instance, we disregard the sequence of web pages retrieved by the client—a critical piece of information that can help distinguish web robot accesses from those of a human user.
A better alternative is to consider each web session as a data instance. A web session is a sequence of requests made by a client during a given visit to the website. Each web session can be modeled as a directed graph, in which the nodes correspond to web pages and the edges correspond to hyperlinks connecting one web page to another. Figure 3.15(b) shows a graphical representation of the first web session given in the log file. Every web session can be characterized using some meaningful attributes about the graph that contain discriminatory information. Figure 3.15(c) shows some of the attributes extracted from the graph, including the depth and breadth of its corresponding tree rooted at the entry point to the website. For example, the
depth and breadth of the tree shown in Figure 3.15(b) are both equal to two.
The derived attributes shown in Figure 3.15(c) are more informative than the original attributes given in the log file because they characterize the behavior of the client at the website. Using this approach, a data set containing 2916 instances was created, with equal numbers of sessions due to web robots (class 1) and human users (class 0). 10% of the data were reserved for training while the remaining 90% were used for testing. The induced decision tree is shown in Figure 3.16, which has an error rate equal to 3.8% on the training set and 5.3% on the test set. In addition to its low error rate, the tree also reveals some interesting properties that can help discriminate web robots from human users:
1. Accesses by web robots tend to be broad but shallow, whereas accesses by human users tend to be more focused (narrow but deep).
2. Web robots seldom retrieve the image pages associated with a web page.
3. Sessions due to web robots tend to be long and contain a large number of requested pages.
4. Web robots are more likely to make repeated requests for the same web page than human users since the web pages retrieved by human users are often cached by the browser.
3.3.6 Characteristics of Decision Tree Classifiers The following is a summary of the important characteristics of decision tree induction algorithms.
1. Applicability: Decision trees are a nonparametric approach for building classification models. This approach does not require any prior assumption about the probability distribution governing the class and attributes of the data, and thus, is applicable to a wide variety of data
sets. It is also applicable to both categorical and continuous data without requiring the attributes to be transformed into a common representation via binarization, normalization, or standardization. Unlike some binary classifiers described in Chapter 4, it can also deal with multiclass problems without the need to decompose them into multiple binary classification tasks. Another appealing feature of decision tree classifiers is that the induced trees, especially the shorter ones, are relatively easy to interpret. The accuracies of the trees are also quite comparable to other classification techniques for many simple data sets.
2. Expressiveness: A decision tree provides a universal representation for discrete-valued functions. In other words, it can encode any function of discrete-valued attributes. This is because every discrete-valued function can be represented as an assignment table, where every unique combination of discrete attributes is assigned a class label. Since every combination of attributes can be represented as a leaf in the decision tree, we can always find a decision tree whose label assignments at the leaf nodes matches with the assignment table of the original function. Decision trees can also help in providing compact representations of functions when some of the unique combinations of attributes can be represented by the same leaf node. For example, Figure 3.17 shows the assignment table of the Boolean function (A∧B)∨(C∧D) involving four binary attributes, resulting in a total of 24=16 possible assignments. The tree shown in Figure 3.17 shows a compressed encoding of this assignment table. Instead of requiring a fully-grown tree with 16 leaf nodes, it is possible to encode the function using a simpler tree with only 7 leaf nodes. Nevertheless, not all decision trees for discrete-valued attributes can be simplified. One notable example is the parity function, whose value is 1 when there is an even number of true values among its Boolean attributes, and 0 otherwise. Accurate modeling of such a function requires a full decision tree with 2d nodes, where d is the number of Boolean attributes (see Exercise 1 on page 185).
Figure 3.16. Decision tree model for web robot detection.
Figure 3.16. Full Alternative Text
Figure 3.17. Decision tree for the Boolean function (A∧B)∨(C∧D).
Figure 3.17. Full Alternative Text
3. Computational Efficiency: Since the number of possible decision trees can be very large, many decision tree algorithms employ a heuristic- based approach to guide their search in the vast hypothesis space. For example, the algorithm presented in Section 3.3.4 uses a greedy, top- down, recursive partitioning strategy for growing a decision tree. For many data sets, such techniques quickly construct a reasonably good decision tree even when the training set size is very large. Furthermore, once a decision tree has been built, classifying a test record is extremely fast, with a worst-case complexity of O(w), where w is the maximum depth of the tree.
4. Handling Missing Values: A decision tree classifier can handle missing attribute values in a number of ways, both in the training and the test sets. When there are missing values in the test set, the classifier must decide which branch to follow if the value of a splitting node attribute is missing for a given test instance. One approach, known as the probabilistic split method, which is employed by the C4.5 decision
tree classifier, distributes the data instance to every child of the splitting node according to the probability that the missing attribute has a particular value. In contrast, the CART algorithm uses the surrogate split method, where the instance whose splitting attribute value is missing is assigned to one of the child nodes based on the value of another non-missing surrogate attribute whose splits most resemble the partitions made by the missing attribute. Another approach, known as the separate class method is used by the CHAID algorithm, where the missing value is treated as a separate categorical value distinct from other values of the splitting attribute. Figure 3.18 shows an example of the three different ways for handling missing values in a decision tree classifier. Other strategies for dealing with missing values are based on data preprocessing, where the instance with missing value is either imputed with the mode (for categorical attribute) or mean (for continuous attribute) value or discarded before the classifier is trained.
Figure 3.18. Methods for handling missing attribute values in decision tree classifier.
Figure 3.18. Full Alternative Text
During training, if an attribute v has missing values in some of the
training instances associated with a node, we need a way to measure the gain in purity if v is used for splitting. One simple way is to exclude instances with missing values of v in the counting of instances associated with every child node, generated for every possible outcome of v.Further, if v is chosen as the attribute test condition at a node, training instances with missing values of v can be propagated to the child nodes using any of the methods described above for handling missing values in test instances.
5. Handling Interactions among Attributes: Attributes are considered interacting if they are able to distinguish between classes when used together, but individually they provide little or no information. Due to the greedy nature of the splitting criteria in decision trees, such attributes could be passed over in favor of other attributes that are not as useful. This could result in more complex decision trees than necessary. Hence, decision trees can perform poorly when there are interactions among attributes.
To illustrate this point, consider the three-dimensional data shown in Figure 3.19(a), which contains 2000 data points from one of two classes, denoted as + and ∘ in the diagram. Figure 3.19(b) shows the distribution of the two classes in the two-dimensional space involving attributes X and Y , which is a noisy version of the XOR Boolean function. We can see that even though the two classes are well-separated in this two-dimensional space, neither of the two attributes contain sufficient information to distinguish between the two classes when used alone. For example, the entropies of the following attribute test conditions: X≤10 and Y≤10, are close to 1, indicating that neither X nor Y provide any reduction in the impurity measure when used individually. X and Y thus represent a case of interaction among attributes. The data set also contains a third attribute, Z, in which both classes are distributed uniformly, as shown in Figures 3.19(c) and 3.19(d), and hence, the entropy of any split involving Z is close to 1. As a result, Z is as likely to be chosen for splitting as the interacting but useful attributes, X and Y . For further illustration of this issue, readers are referred to Example 3.7 in Section 3.4.1 and Exercise 7 at the end of this chapter.
Figure 3.19. Example of a XOR data involving X and Y , along with an irrelevant attribute Z.
Figure 3.19. Full Alternative Text
6. Handling Irrelevant Attributes: An attribute is irrelevant if it is not useful for the classification task. Since irrelevant attributes are poorly associated with the target class labels, they will provide little or no gain in purity and thus will be passed over by other more relevant features. Hence, the presence of a small number of irrelevant attributes will not impact the decision tree construction process. However, not all attributes that provide little to no gain are irrelevant (see Figure 3.19). Hence, if the classification problem is complex (e.g., involving interactions among attributes) and there are a large number of irrelevant attributes, then some of these attributes may be accidentally chosen during the tree- growing process, since they may provide a better gain than a relevant attribute just by random chance. Feature selection techniques can help to improve the accuracy of decision trees by eliminating the irrelevant attributes during preprocessing. We will investigate the issue of too many irrelevant attributes in Section 3.4.1.
7. Handling Redundant Attributes: An attribute is redundant if it is strongly correlated with another attribute in the data. Since redundant attributes show similar gains in purity if they are selected for splitting, only one of them will be selected as an attribute test condition in the decision tree algorithm. Decision trees can thus handle the presence of redundant attributes.
8. Using Rectilinear Splits: The test conditions described so far in this chapter involve using only a single attribute at a time. As a consequence, the tree-growing procedure can be viewed as the process of partitioning the attribute space into disjoint regions until each region contains records of the same class. The border between two neighboring regions of different classes is known as a decision boundary. Figure 3.20 shows
the decision tree as well as the decision boundary for a binary classification problem. Since the test condition involves only a single attribute, the decision boundaries are rectilinear; i.e., parallel to the coordinate axes. This limits the expressiveness of decision trees in representing decision boundaries of data sets with continuous attributes. Figure 3.21 shows a two-dimensional data set involving binary classes that cannot be perfectly classified by a decision tree whose attribute test conditions are defined based on single attributes. The binary classes in the data set are generated from two skewed Gaussian distributions, centered at (8,8) and (12,12), respectively. The true decision boundary is represented by the diagonal dashed line, whereas the rectilinear decision boundary produced by the decision tree classifier is shown by the thick solid line. In contrast, an oblique decision tree may overcome this limitation by allowing the test condition to be specified using more than one attribute. For example, the binary classification data shown in Figure 3.21 can be easily represented by an oblique decision tree with a single root node with test condition
x+y<20.
Figure 3.20. Example of a decision tree and its decision boundaries for a two-dimensional data set.
Figure 3.20. Full Alternative Text
Figure 3.21. Example of data set that cannot be partitioned optimally using a decision tree with single attribute test conditions. The true
decision boundary is shown by the dashed line.
Although an oblique decision tree is more expressive and can produce more compact trees, finding the optimal test condition is computationally more expensive.
9. Choice of Impurity Measure: It should be noted that the choice of impurity measure often has little effect on the performance of decision tree classifiers since many of the impurity measures are quite consistent with each other, as shown in Figure 3.11 on page 129. Instead, the strategy used to prune the tree has a greater impact on the final tree than the choice of impurity measure.
3.4 Model Overfitting Methods presented so far try to learn classification models that show the lowest error on the training set. However, as we will show in the following example, even if a model fits well over the training data, it can still show poor generalization performance, a phenomenon known as model overfitting.
Figure 3.22. Examples of training and test sets of a two-dimensional classification problem.
Figure 3.22. Full Alternative Text
Figure 3.23. Effect of varying tree size (number of leaf nodes) on training and test errors.
Figure 3.23. Full Alternative Text
3.5. Example Overfitting and Underfitting of Decision Trees Consider the two-dimensional data set shown in Figure 3.22(a). The data set contains instances that belong to two separate classes, represented as + and ∘, respectively, where each class has 5400 instances. All instances belonging to the ∘ class were generated from a uniform distribution. For the + class, 5000 instances were generated from a Gaussian distribution centered at (10,10) with unit variance, while the remaining 400 instances were sampled from the same uniform distribution as the ∘ class. We can see from Figure 3.22(a) that the + class can be largely distinguished from the ∘ class
by drawing a circle of appropriate size centered at (10,10). To learn a classifier using this two-dimensional data set, we randomly sampled 10% of the data for training and used the remaining 90% for testing. The training set, shown in Figure 3.22(b), looks quite representative of the overall data. We used Gini index as the impurity measure to construct decision trees of increasing sizes (number of leaf nodes), by recursively expanding a node into child nodes till every leaf node was pure, as described in Section 3.3.4.
Figure 3.23(a) shows changes in the training and test error rates as the size of the tree varies from 1 to 8. Both error rates are initially large when the tree has only one or two leaf nodes. This situation is known as model underfitting. Underfitting occurs when the learned decision tree is too simplistic, and thus, incapable of fully representing the true relationship between the attributes and the class labels. As we increase the tree size from 1 to 8, we can observe two effects. First, both the error rates decrease since larger trees are able to represent more complex decision boundaries. Second, the training and test error rates are quite close to each other, which indicates that the performance on the training set is fairly representative of the generalization performance. As we further increase the size of the tree from 8 to 150, the training error continues to steadily decrease till it eventually reaches zero, as shown in Figure 3.23(b). However, in a striking contrast, the test error rate ceases to decrease any further beyond a certain tree size, and then it begins to increase. The training error rate thus grossly under-estimates the test error rate once the tree becomes too large. Further, the gap between the training and test error rates keeps on widening as we increase the tree size. This behavior, which may seem counter-intuitive at first, can be attributed to the phenomena of model overfitting.
3.4.1 Reasons for Model Overfitting Model overfitting is the phenomena where, in the pursuit of minimizing the training error rate, an overly complex model is selected that captures specific patterns in the training data but fails to learn the true nature of relationships between attributes and class labels in the overall data. To illustrate this, Figure 3.24 shows decision trees and their corresponding decision boundaries (shaded rectangles represent regions assigned to the + class) for two trees of
sizes 5 and 50. We can see that the decision tree of size 5 appears quite simple and its decision boundaries provide a reasonable approximation to the ideal decision boundary, which in this case corresponds to a circle centered around the Gaussian distribution at (10, 10). Although its training and test error rates are non-zero, they are very close to each other, which indicates that the patterns learned in the training set should generalize well over the test set. On the other hand, the decision tree of size 50 appears much more complex than the tree of size 5, with complicated decision boundaries. For example, some of its shaded rectangles (assigned the + class) attempt to cover narrow regions in the input space that contain only one or two + training instances. Note that the prevalence of + instances in such regions is highly specific to the training set, as these regions are mostly dominated by - instances in the overall data. Hence, in an attempt to perfectly fit the training data, the decision tree of size 50 starts fine tuning itself to specific patterns in the training data, leading to poor performance on an independently chosen test set.
Figure 3.24.
Decision trees with different model complexities.
Figure 3.24. Full Alternative Text
Figure 3.25. Performance of decision trees using 20% data for training (twice the original training size).
Figure 3.25. Full Alternative Text
There are a number of factors that influence model overfitting. In the following, we provide brief descriptions of two of the major factors: limited training size and high model complexity. Though they are not exhaustive, the interplay between them can help explain most of the common model overfitting phenomena in real-world applications.
Limited Training Size
Note that a training set consisting of a finite number of instances can only provide a limited representation of the overall data. Hence, it is possible that the patterns learned from a training set do not fully represent the true patterns in the overall data, leading to model overfitting. In general, as we increase the size of a training set (number of training instances), the patterns learned from the training set start resembling the true patterns in the overall data. Hence, the effect of overfitting can be reduced by increasing the training size, as illustrated in the following example.
3.6 Example Effect of Training Size Suppose that we use twice the number of training instances than what we had used in the experiments conducted in Example 3.5. Specifically, we use 20% data for training and use the remainder for testing. Figure 3.25(b) shows the training and test error rates as the size of the tree is varied from 1 to 150. There are two major differences in the trends shown in this figure and those shown in Figure 3.23(b) (using only 10% of the data for training). First, even though the training error rate decreases with increasing tree size in both figures, its rate of decrease is much smaller when we use twice the training size. Second, for a given tree size, the gap between the training and test error rates is much smaller when we use twice the training size. These differences suggest that the patterns learned using 20% of data for training are more generalizable than those learned using 10% of data for training.
Figure 3.25(a) shows the decision boundaries for the tree of size 50, learned using 20% of data for training. In contrast to the tree of the same size learned using 10% data for training (see Figure 3.24(d)), we can see that the decision tree is not capturing specific patterns of noisy + instances in the training set. Instead, the high model complexity of 50 leaf nodes is being effectively used to learn the boundaries of the + instances centered at (10, 10).
High Model Complexity Generally, a more complex model has a better ability to represent complex
patterns in the data. For example, decision trees with larger number of leaf nodes can represent more complex decision boundaries than decision trees with fewer leaf nodes. However, an overly complex model also has a tendency to learn specific patterns in the training set that do not generalize well over unseen instances. Models with high complexity should thus be judiciously used to avoid overfitting.
One measure of model complexity is the number of “parameters” that need to be inferred from the training set. For example, in the case of decision tree induction, the attribute test conditions at internal nodes correspond to the parameters of the model that need to be inferred from the training set. A decision tree with larger number of attribute test conditions (and consequently more leaf nodes) thus involves more “parameters” and hence is more complex.
Given a class of models with a certain number of parameters, a learning algorithm attempts to select the best combination of parameter values that maximizes an evaluation metric (e.g., accuracy) over the training set. If the number of parameter value combinations (and hence the complexity) is large, the learning algorithm has to select the best combination from a large number of possibilities, using a limited training set. In such cases, there is a high chance for the learning algorithm to pick a spurious combination of parameters that maximizes the evaluation metric just by random chance. This is similar to the multiple comparisons problem (also referred as multiple testing problem) in statistics.
As an illustration of the multiple comparisons problem, consider the task of predicting whether the stock market will rise or fall in the next ten trading days. If a stock analyst simply makes random guesses, the probability that her prediction is correct on any trading day is 0.5. However, the probability that she will predict correctly at least nine out of ten times is
(109)+(1010)210=0.0107,
which is extremely low.
Suppose we are interested in choosing an investment advisor from a pool of 200 stock analysts. Our strategy is to select the analyst who makes the most
number of correct predictions in the next ten trading days. The flaw in this strategy is that even if all the analysts make their predictions in a random fashion, the probability that at least one of them makes at least nine correct predictions is
1−(1−0.0107)200=0.8847,
which is very high. Although each analyst has a low probability of predicting at least nine times correctly, considered together, we have a high probability of finding at least one analyst who can do so. However, there is no guarantee in the future that such an analyst will continue to make accurate predictions by random guessing.
How does the multiple comparisons problem relate to model overfitting? In the context of learning a classification model, each combination of parameter values corresponds to an analyst, while the number of training instances corresponds to the number of days. Analogous to the task of selecting the best analyst who makes the most accurate predictions on consecutive days, the task of a learning algorithm is to select the best combination of parameters that results in the highest accuracy on the training set. If the number of parameter combinations is large but the training size is small, it is highly likely for the learning algorithm to choose a spurious parameter combination that provides high training accuracy just by random chance. In the following example, we illustrate the phenomena of overfitting due to multiple comparisons in the context of decision tree induction.
Figure 3.26. Example of a two-dimensional (X-Y) data set.
Figure 3.27. Training and test error rates illustrating the effect of multiple comparisons problem on model overfitting.
Figure 3.27. Full Alternative Text
3.7. Example Multiple Comparisons and Overfitting Consider the two-dimensional data set shown in Figure 3.26 containing 500 + and 500 ∘ instances, which is similar to the data shown in Figure 3.19. In this data set, the distributions of both classes are well-separated in the two- dimensional (XY) attribute space, but none of the two attributes (X or Y) are sufficiently informative to be used alone for separating the two classes. Hence, splitting the data set based on any value of an X or Y attribute will provide close to zero reduction in an impurity measure. However, if X and Y
attributes are used together in the splitting criterion (e.g., splitting X at 10 and Y at 10), the two classes can be effectively separated.
Figure 3.28. Decision tree with 6 leaf nodes using X and Y as attributes. Splits have been numbered from 1 to 5 in order of other occurrence in the tree.
Figure 3.28. Full Alternative Text
Figure 3.27(a) shows the training and test error rates for learning decision trees of varying sizes, when 30% of the data is used for training and the remainder of the data for testing. We can see that the two classes can be separated using a small number of leaf nodes. Figure 3.28 shows the decision boundaries for the tree with six leaf nodes, where the splits have been numbered according to their order of appearance in the tree. Note that the
even though splits 1 and 3 provide trivial gains, their consequent splits (2, 4, and 5) provide large gains, resulting in effective discrimination of the two classes.
Assume we add 100 irrelevant attributes to the two-dimensional X-Y data. Learning a decision tree from this resultant data will be challenging because the number of candidate attributes to choose for splitting at every internal node will increase from two to 102. With such a large number of candidate attribute test conditions to choose from, it is quite likely that spurious attribute test conditions will be selected at internal nodes because of the multiple comparisons problem. Figure 3.27(b) shows the training and test error rates after adding 100 irrelevant attributes to the training set. We can see that the test error rate remains close to 0.5 even after using 50 leaf nodes, while the training error rate keeps on declining and eventually becomes 0.
3.5 Model Selection There are many possible classification models with varying levels of model complexity that can be used to capture patterns in the training data. Among these possibilities, we want to select the model that shows lowest generalization error rate. The process of selecting a model with the right level of complexity, which is expected to generalize well over unseen test instances, is known as model selection. As described in the previous section, the training error rate cannot be reliably used as the sole criterion for model selection. In the following, we present three generic approaches to estimate the generalization performance of a model that can be used for model selection. We conclude this section by presenting specific strategies for using these approaches in the context of decision tree induction.
3.5.1 Using a Validation Set Note that we can always estimate the generalization error rate of a model by using “out-of-sample” estimates, i.e. by evaluating the model on a separate validation set that is not used for training the model. The error rate on the validation set, termed as the validation error rate, is a better indicator of generalization performance than the training error rate, since the validation set has not been used for training the model. The validation error rate can be used for model selection as follows.
Given a training set D.train, we can partition D.train into two smaller subsets, D.tr and D.val, such that D.tr is used for training while D.val is used as the validation set. For example, two-thirds of D.train can be reserved as D.tr for training, while the remaining one-third is used as D.val for computing validation error rate. For any choice of classification model m that is trained on D.tr, we can estimate its validation error rate on D.val, errval(m). The model that shows the lowest value of errval(m) can then be selected as the preferred choice of model.
The use of validation set provides a generic approach for model selection. However, one limitation of this approach is that it is sensitive to the sizes of D.tr and D.val, obtained by partitioning D.train. If the size of D.tr is too small, it may result in the learning of a poor classification model with sub- standard performance, since a smaller training set will be less representative of the overall data. On the other hand, if the size of D.val is too small, the validation error rate might not be reliable for selecting models, as it would be computed over a small number of instances.
Figure 3.29. Class distribution of validation data for the two decision trees shown in Figure 3.30.
Figure 3.29. Full Alternative Text
3.8. Example Validation Error
In the following example, we illustrate one possible approach for using a validation set in decision tree induction. Figure 3.29 shows the predicted labels at the leaf nodes of the decision trees generated in Figure 3.30. The counts given beneath the leaf nodes represent the proportion of data instances in the validation set that reach each of the nodes. Based on the predicted labels of the nodes, the validation error rate for the left tree is errval(TL)=6/16=0.375, while the validation error rate for the right tree is errval(TR)=4/16=0.25. Based on their validation error rates, the right tree is preferred over the left one.
3.5.2 Incorporating Model Complexity Since the chance for model overfitting increases as the model becomes more complex, a model selection approach should not only consider the training error rate but also the model complexity. This strategy is inspired by a well- known principle known as Occam's razor or the principle of parsimony, which suggests that given two models with the same errors, the simpler model is preferred over the more complex model. A generic approach to account for model complexity while estimating generalization performance is formally described as follows.
Given a training set D.train, let us consider learning a classification model m that belongs to a certain class of models, M. For example, if M represents the set of all possible decision trees, then m can correspond to a specific decision tree learned from the training set. We are interested in estimating the generalization error rate of m, gen.error(m). As discussed previously, the training error rate of m, train.error(m, D.train), can under-estimate gen.error(m) when the model complexity is high. Hence, we represent gen.error(m) as a function of not just the training error rate but also the model complexity of M, complexity(M), as follows:
gen.error(m)=train.error(m, D.train)+α×complexity(M), (3.11)
where α is a hyper-parameter that strikes a balance between minimizing
training error and reducing model complexity. A higher value of α gives more emphasis to the model complexity in the estimation of generalization performance. To choose the right value of α, we can make use of the validation set in a similar way as described in 3.5.1. For example, we can iterate through a range of values of α and for every possible value, we can learn a model on a subset of the training set, D.tr, and compute its validation error rate on a separate subset, D.val. We can then select the value of α that provides the lowest validation error rate.
Equation 3.11 provides one possible approach for incorporating model complexity into the estimate of generalization performance. This approach is at the heart of a number of techniques for estimating generalization performance, such as the structural risk minimization principle, the Akaike's Information Criterion (AIC), and the Bayesian Information Criterion (BIC). The structural risk minimization principle serves as the building block for learning support vector machines, which will be discussed later in Chapter 4. For more details on AIC and BIC, see the Bibliographic Notes.
In the following, we present two different approaches for estimating the complexity of a model, complexity(M). While the former is specific to decision trees, the latter is more generic and can be used with any class of models.
Estimating the Complexity of Decision Trees In the context of decision trees, the complexity of a decision tree can be estimated as the ratio of the number of leaf nodes to the number of training instances. Let k be the number of leaf nodes and Ntrain be the number of training instances. The complexity of a decision tree can then be described as k/Ntrain. This reflects the intuition that for a larger training size, we can learn a decision tree with larger number of leaf nodes without it becoming overly complex. The generalization error rate of a decision tree T can then be computed using Equation 3.11 as follows:
errgen(T)=err(T)+Ω×kNtrain,
where err(T) is the training error of the decision tree and Ω is a hyper- parameter that makes a trade-off between reducing training error and minimizing model complexity, similar to the use of α in Equation 3.11. Ω can be viewed as the relative cost of adding a leaf node relative to incurring a training error. In the literature on decision tree induction, the above approach for estimating generalization error rate is also termed as the pessimistic error estimate. It is called pessimistic as it assumes the generalization error rate to be worse than the training error rate (by adding a penalty term for model complexity). On the other hand, simply using the training error rate as an estimate of the generalization error rate is called the optimistic error estimate or the resubstitution estimate.
3.9. Example Generalization Error Estimates Consider the two binary decision trees, TL and TR, shown in Figure 3.30. Both trees are generated from the same training data and TL is generated by expanding three leaf nodes of TR. The counts shown in the leaf nodes of the trees represent the class distribution of the training instances. If each leaf node is labeled according to the majority class of training instances that reach the node, the training error rate for the left tree will be err(TL)=4/24=0.167, while the training error rate for the right tree will be err(TR)=6/24=0.25. Based on their training error rates alone, TL would preferred over TR, even though TL is more complex (contains larger number of leaf nodes) than TR.
Figure 3.30. Example of two decision trees generated from the same training data.
Figure 3.30. Full Alternative Text
Now, assume that the cost associated with each leaf node is Ω=0.5. Then, the generalization error estimate for TL will be
errgen(TL)=424+0.5×724=7.524=0.3125
and the generalization error estimate for TR will be
errgen(TR) =624+0.5×424=824=0.3333.
Since TL has a lower generalization error rate, it will still be preferred over TR. Note that Ω=0.5 implies that a node should always be expanded into its two child nodes if it improves the prediction of at least one training instance, since expanding a node is less costly than misclassifying a training instance. On the other hand, if Ω=1, then the generalization error rate for TL is
errgen(TL)=11/24=0.458 and for TR is errgen(TR)=10/24=0.417. In this case, TR will be preferred over TL because it has a lower generalization error rate. This example illustrates that different choices of Ω can change our preference of decision trees based on their generalization error estimates. However, for a given choice of Ω, the pessimistic error estimate provides an approach for modeling the generalization performance on unseen test instances. The value of Ω can be selected with the help of a validation set.
Minimum Description Length Principle Another way to incorporate model complexity is based on an information- theoretic approach known as the minimum description length or MDL principle. To illustrate this approach, consider the example shown in Figure 3.31. In this example, both person A and person B are given a set of instances with known attribute values x. Assume person A knows the class label y for every instance, while person B has no such information. A would like to share the class information with B by sending a message containing the labels. The message would contain Θ(N) bits of information, where N is the number of instances.
Figure 3.31. An illustration of the minimum description length principle.
Figure 3.31. Full Alternative Text
Alternatively, instead of sending the class labels explicitly, A can build a classification model from the instances and transmit it to B. B can then apply the model to determine the class labels of the instances. If the model is 100% accurate, then the cost for transmission is equal to the number of bits required to encode the model. Otherwise, A must also transmit information about which instances are misclassified by the model so that B can reproduce the same class labels. Thus, the overall transmission cost, which is equal to the total description length of the message, is
Cost(model, data)=Cost(data|model)+α×Cost(model), (3.12)
where the first term on the right-hand side is the number of bits needed to encode the misclassified instances, while the second term is the number of
bits required to encode the model. There is also a hyper-parameter α that trades-off the relative costs of the misclassified instances and the model. Notice the familiarity between this equation and the generic equation for generalization error rate presented in Equation 3.11. A good model must have a total description length less than the number of bits required to encode the entire sequence of class labels. Furthermore, given two competing models, the model with lower total description length is preferred. An example showing how to compute the total description length of a decision tree is given in Exercise 10 on page 189.
3.5.3 Estimating Statistical Bounds Instead of using Equation 3.11 to estimate the generalization error rate of a model, an alternative way is to apply a statistical correction to the training error rate of the model that is indicative of its model complexity. This can be done if the probability distribution of training error is available or can be assumed. For example, the number of errors committed by a leaf node in a decision tree can be assumed to follow a binomial distribution. We can thus compute an upper bound limit to the observed training error rate that can be used for model selection, as illustrated in the following example.
3.10. Example Statistical Bounds on Training Error Consider the left-most branch of the binary decision trees shown in Figure 3.30. Observe that the left-most leaf node of TR has been expanded into two child nodes in TL. Before splitting, the training error rate of the node is 2/7=0.286. By approximating a binomial distribution with a normal distribution, the following upper bound of the training error rate e can be derived:
eupper(N, e, α)=e+zα/222N+zα/2e(1−e)N+zα/224N21+zα/22N, (3.13)
where α is the confidence level, zα/2 is the standardized value from a standard normal distribution, and N is the total number of training instances used to compute e. By replacing α=25%,N=7, and e=2/7, the upper bound for the error rate is eupper(7, 2/7, 0.25)=0.503, which corresponds to 7×0.503=3.521 errors. If we expand the node into its child nodes as shown in TL, the training error rates for the child nodes are 1/4=0.250 and 1/3=0.333, respectively. Using Equation (3.13), the upper bounds of these error rates are eupper(4, 1/4,0.25)=0.537 and eupper(3, 1/3, 0.25)=0.650, respectively. The overall training error of the child nodes is 4×0.537+3×0.650=4.098, which is larger than the estimated error for the corresponding node in TR, suggesting that it should not be split.
3.5.4 Model Selection for Decision Trees Building on the generic approaches presented above, we present two commonly used model selection strategies for decision tree induction.
Prepruning (Early Stopping Rule) In this approach, the tree-growing algorithm is halted before generating a fully grown tree that perfectly fits the entire training data. To do this, a more restrictive stopping condition must be used; e.g., stop expanding a leaf node when the observed gain in the generalization error estimate falls below a certain threshold. This estimate of the generalization error rate can be computed using any of the approaches presented in the preceding three subsections, e.g., by using pessimistic error estimates, by using validation error estimates, or by using statistical bounds. The advantage of prepruning is that it avoids the computations associated with generating overly complex subtrees that overfit the training data. However, one major drawback of this method is that, even if no significant gain is obtained using one of the existing splitting criterion, subsequent splitting may result in better subtrees. Such subtrees would not be reached if prepruning is used because of the
greedy nature of decision tree induction.
Post-pruning In this approach, the decision tree is initially grown to its maximum size. This is followed by a tree-pruning step, which proceeds to trim the fully grown tree in a bottom-up fashion. Trimming can be done by replacing a subtree with (1) a new leaf node whose class label is determined from the majority class of instances affiliated with the subtree (approach known as subtree replacement), or (2) the most frequently used branch of the subtree (approach known as subtree raising). The tree-pruning step terminates when no further improvement in the generalization error estimate is observed beyond a certain threshold. Again, the estimates of generalization error rate can be computed using any of the approaches presented in the previous three subsections. Post-pruning tends to give better results than prepruning because it makes pruning decisions based on a fully grown tree, unlike prepruning, which can suffer from premature termination of the tree-growing process. However, for post-pruning, the additional computations needed to grow the full tree may be wasted when the subtree is pruned.
Figure 3.32 illustrates the simplified decision tree model for the web robot detection example given in Section 3.3.5. Notice that the subtree rooted at depth=1 has been replaced by one of its branches corresponding to breadth<=7, width>3, and MultiP=1, using subtree raising. On the other hand, the subtree corresponding to depth>1 and MultiAgent=0 has been replaced by a leaf node assigned to class 0, using subtree replacement. The subtree for depth>1 and MultiAgent=1 remains intact.
Figure 3.32. Post-pruning of the decision tree for web robot detection.
Figure 3.32. Full Alternative Text
3.6 Model Evaluation The previous section discussed several approaches for model selection that can be used to learn a classification model from a training set D.train. Here we discuss methods for estimating its generalization performance, i.e. its performance on unseen instances outside of D.train. This process is known as model evaluation.
Note that model selection approaches discussed in Section 3.5 also compute an estimate of the generalization performance using the training set D.train. However, these estimates are biased indicators of the performance on unseen instances, since they were used to guide the selection of classification model. For example, if we use the validation error rate for model selection (as described in Section 3.5.1), the resulting model would be deliberately chosen to minimize the errors on the validation set. The validation error rate may thus under-estimate the true generalization error rate, and hence cannot be reliably used for model evaluation.
A correct approach for model evaluation would be to assess the performance of a learned model on a labeled test set has not been used at any stage of model selection. This can be achieved by partitioning the entire set of labeled instances D, into two disjoint subsets, D.train, which is used for model selection and D.test, which is used for computing the test error rate, errtest. In the following, we present two different approaches for partitioning D into D.train and D.test, and computing the test error rate, errtest.
3.6.1 Holdout Method The most basic technique for partitioning a labeled data set is the holdout method, where the labeled set D is randomly partitioned into two disjoint sets, called the training set D.train and the test set D.test. A classification model is then induced from D.train using the model selection approaches presented in Section 3.5, and its error rate on D.test, errtest, is used as an
estimate of the generalization error rate. The proportion of data reserved for training and for testing is typically at the discretion of the analysts, e.g., two- thirds for training and one-third for testing.
Similar to the trade-off faced while partitioning D.train into D.tr and D.val in Section 3.5.1, choosing the right fraction of labeled data to be used for training and testing is non-trivial. If the size of D.train is small, the learned classification model may be improperly learned using an insufficient number of training instances, resulting in a biased estimation of generalization performance. On the other hand, if the size of D.test is small, errtest may be less reliable as it would be computed over a small number of test instances. Moreover, errtest can have a high variance as we change the random partitioning of D into D.train and D.test.
The holdout method can be repeated several times to obtain a distribution of the test error rates, an approach known as random subsampling or repeated holdout method. This method produces a distribution of the error rates that can be used to understand the variance of errtest.
3.6.2 Cross-Validation Cross-validation is a widely-used model evaluation method that aims to make effective use of all labeled instances in D for both training and testing. To illustrate this method, suppose that we are given a labeled set that we have randomly partitioned into three equal-sized subsets, S1, S2, and S3, as shown in Figure 3.33. For the first run, we train a model using subsets S2 and S3 (shown as empty blocks) and test the model on subset S1. The test error rate on S1, denoted as err(S1), is thus computed in the first run. Similarly, for the second run, we use S1 and S3 as the training set and S2 as the test set, to compute the test error rate, err(S2), on S2. Finally, we use S1 and S3 for training in the third run, while S3 is used for testing, thus resulting in the test error rate err(S3) for S3. The overall test error rate is obtained by summing up the number of errors committed in each test subset across all runs and dividing it by the total number of instances. This approach is called three-fold cross-validation.
Figure 3.33. Example demonstrating the technique of 3-fold cross-validation.
Figure 3.33. Full Alternative Text
The k-fold cross-validation method generalizes this approach by segmenting the labeled data D (of size N) into k equal-sized partitions (or folds). During the ith run, one of the partitions of D is chosen as D.test(i) for testing, while the rest of the partitions are used as D.train(i) for training. A model m(i) is learned using D.train(i) and applied on D.test(i) to obtain the sum of test errors, errsum(i). This procedure is repeated k times. The total test error rate, errtest, is then computed as
errtest=∑i=1kerrsum(i)N.(3.14)
Every instance in the data is thus used for testing exactly once and for training exactly (k−1) times. Also, every run uses (k−1)/k fraction of the data for training and 1/k fraction for testing.
The right choice of k in k-fold cross-validation depends on a number of characteristics of the problem. A small value of k will result in a smaller training set at every run, which will result in a larger estimate of
generalization error rate than what is expected of a model trained over the entire labeled set. On the other hand, a high value of k results in a larger training set at every run, which reduces the bias in the estimate of generalization error rate. In the extreme case, when k=N, every run uses exactly one data instance for testing and the remainder of the data for testing. This special case of k-fold cross-validation is called the leave-one-out approach. This approach has the advantage of utilizing as much data as possible for training. However, leave-one-out can produce quite misleading results in some special scenarios, as illustrated in Exercise 11. Furthermore, leave-one-out can be computationally expensive for large data sets as the cross-validation procedure needs to be repeated N times. For most practical applications, the choice of k between 5 and 10 provides a reasonable approach for estimating the generalization error rate, because each fold is able to make use of 80% to 90% of the labeled data for training.
The k-fold cross-validation method, as described above, produces a single estimate of the generalization error rate, without providing any information about the variance of the estimate. To obtain this information, we can run k- fold cross-validation for every possible partitioning of the data into k partitions, and obtain a distribution of test error rates computed for every such partitioning. The average test error rate across all possible partitionings serves as a more robust estimate of generalization error rate. This approach of estimating the generalization error rate and its variance is known as the complete cross-validation approach. Even though such an estimate is quite robust, it is usually too expensive to consider all possible partitionings of a large data set into k partitions. A more practical solution is to repeat the cross-validation approach multiple times, using a different random partitioning of the data into k partitions at every time, and use the average test error rate as the estimate of generalization error rate. Note that since there is only one possible partitioning for the leave-one-out approach, it is not possible to estimate the variance of generalization error rate, which is another limitation of this method.
The k-fold cross-validation does not guarantee that the fraction of positive and negative instances in every partition of the data is equal to the fraction observed in the overall data. A simple solution to this problem is to perform a stratified sampling of the positive and negative instances into k partitions, an
approach called stratified cross-validation.
In k-fold cross-validation, a different model is learned at every run and the performance of every one of the k models on their respective test folds is then aggregated to compute the overall test error rate, errtest. Hence, errtest does not reflect the generalization error rate of any of the k models. Instead, it reflects the expected generalization error rate of the model selection approach, when applied on a training set of the same size as one of the training folds (N(k−1)/k). This is different than the errtest computed in the holdout method, which exactly corresponds to the specific model learned over D.train. Hence, although effectively utilizing every data instance in D for training and testing, the errtest computed in the cross-validation method does not represent the performance of a single model learned over a specific D.train.
Nonetheless, in practice, errtest is typically used as an estimate of the generalization error of a model built on D. One motivation for this is that when the size of the training folds is closer to the size of the overall data (when k is large), then errtest resembles the expected performance of a model learned over a data set of the same size as D. For example, when k is 10, every training fold is 90% of the overall data. The errtest then should approach the expected performance of a model learned over 90% of the overall data, which will be close to the expected performance of a model learned over D.
3.7 Presence of Hyper-parameters Hyper-parameters are parameters of learning algorithms that need to be determined before learning the classification model. For instance, consider the hyper-parameter α that appeared in Equation 3.11, which is repeated here for convenience. This equation was used for estimating the generalization error for a model selection approach that used an explicit representations of model complexity. (See Section 3.5.2.)
gen.error(m)=train.error(m, D.train)+α×complexity(M)
For other examples of hyper-parameters, see Chapter 4.
Unlike regular model parameters, such as the test conditions in the internal nodes of a decision tree, hyper-parameters such as α do not appear in the final classification model that is used to classify unlabeled instances. However, the values of hyper-parameters need to be determined during model selection—a process known as hyper-parameter selection—and must be taken into account during model evaluation. Fortunately, both tasks can be effectively accomplished via slight modifications of the cross-validation approach described in the previous section.
3.7.1 Hyper-parameter Selection In Section 3.5.2, a validation set was used to select α and this approach is generally applicable for hyper-parameter section. Let p be the hyper- parameter that needs to be selected from a finite range of values, P= {p1, p2, … pn }. Partition D.train into D.tr and D.val. For every choice of hyper-parameter value pi, we can learn a model mi on D.tr, and apply this model on D.val to obtain the validation error rate errval(pi). Let p* be the hyper-parameter value that provides the lowest validation error rate. We can then use the model m* corresponding to p* as the final choice of classification model.
The above approach, although useful, uses only a subset of the data, D.train, for training and a subset, D.val, for validation. The framework of cross- validation, presented in Section 3.6.2, addresses both of those issues, albeit in the context of model evaluation. Here we indicate how to use a cross- validation approach for hyper-parameter selection. To illustrate this approach, let us partition D.train into three folds as shown in Figure 3.34. At every run, one of the folds is used as D.val for validation, and the remaining two folds are used as D.tr for learning a model, for every choice of hyper-parameter value pi. The overall validation error rate corresponding to each pi is computed by summing the errors across all the three folds. We then select the hyper-parameter value p* that provides the lowest validation error rate, and use it to learn a model m* on the entire training set D.train.
Figure 3.34. Example demonstrating the 3-fold cross-validation framework for hyper-parameter selection using D.train.
Figure 3.34. Full Alternative Text
Algorithm 3.2 generalizes the above approach using a k-fold cross-validation framework for hyper-parameter selection. At the ith run of cross-validation, the data in the ith fold is used as D.val(i) for validation (Step 4), while the remainder of the data in D.train is used as D.tr(i) for training (Step 5). Then for every choice of hyper-parameter value pi, a model is learned on D.tr(i) (Step 7), which is applied on D.val(i) to compute its validation error (Step 8).
This is used to compute the validation error rate corresponding to models learning using pi over all the folds (Step 11). The hyper-parameter value p* that provides the lowest validation error rate (Step 12) is now used to learn the final model m* on the entire training set D.train (Step 13). Hence, at the end of this algorithm, we obtain the best choice of the hyper-parameter value as well as the final classification model (Step 14), both of which are obtained by making an effective use of every data instance in D.train.
Algorithm 3.2 Procedure model- select(k, P, D.train)
1: Ntrain = |D.train| {Size of D.train.} 2: Divide D.train into k partitions, D.train1 to D.traink. 3: for each run i = 1 to k do 4: D.val(i) = D.traini. {Partition used for validation.} 5: D.tr(i) = D.train \ D.traini. {Partitions used for training.} 6: for each parameter p ∈ P do 7: m = model-train(p, D.tr(i)). {Train model} 8: errsum(p, i) = model-test(m, D.val(i)). {Sum of validation errors.} 9: end for 10: end for 11: errval(p)=∑ik errsum(p, i)/Ntrain. {Compute validation error rate.} 12: p* = argminp errval(p). {Select best hyper-parameter value.} 13: m* = model-train(p*, D.train). {Learn final model on D.train} 14: return (p*, m*).
3.7.2 Nested Cross-Validation The approach of the previous section provides a way to effectively use all the instances in D.train to learn a classification model when hyper-parameter selection is required. This approach can be applied over the entire data set D to learn the final classification model. However, applying Algorithm 3.2 on D would only return the final classification model m* but not an estimate of its generalization performance, errtest. Recall that the validation error rates used
in Algorithm 3.2 cannot be used as estimates of generalization performance, since they are used to guide the selection of the final model m*. However, to compute errtest, we can again use a cross-validation framework for evaluating the performance on the entire data set D, as described originally in Section 3.6.2. In this approach, D is partitioned into D.train (for training) and D.test (for testing) at every run of cross-validation. When hyper-parameters are involved, we can use Algorithm 3.2 to train a model using D.train at every run, thus “internally” using cross-validation for model selection. This approach is called nested cross-validation or double cross-validation. Algorithm 3.3 describes the complete approach for estimating errtest using nested cross-validation in the presence of hyper-parameters.
As an illustration of this approach, see Figure 3.35 where the labeled set D is partitioned into D.train and D.test, using a 3-fold cross-validation method.
Figure 3.35. Example demonstrating 3-fold nested cross-validation for computing errtest.
Figure 3.35. Full Alternative Text
At the ith run of this method, one of the folds is used as the test set, D.test(i),
while the remaining two folds are used as the training set, D.train(i). This is represented in Figure 3.35 as the ith “outer” run. In order to select a model using D.train(i), we again use an “inner” 3-fold cross-validation framework that partitions D.train(i) into D.tr and D.val at every one of the three inner runs (iterations). As described in Section 3.7, we can use the inner cross- validation framework to select the best hyper-parameter value p*(i) as well as its resulting classification model m*(i) learned over D.train(i). We can then apply m*(i) on D.test(i) to obtain the test error at the ith outer run. By repeating this process for every outer run, we can compute the average test error rate, errtest, over the entire labeled set D. Note that in the above approach, the inner cross-validation framework is being used for model selection while the outer cross-validation framework is being used for model evaluation.
Algorithm 3.3 The nested cross- validation approach for computing errtest. 1: Divide D into k partitions, D1 to Dk. 2: for each outer run i = 1 to k do 3: D.test(i) = Di. {Partition used for testing.} 4: D.train(i) = D \ Di. {Partitions used for model selection.} 5: (p*, m*(i)) = model-select(k, P, D.train(i)). {Inner cross-validation.} 6: errsum(i) = model-test(m*(i), D.test(i)). {Sum of test errors.} 7: end for 8: errtest=∑ikerrtest(i)/N. {Compute test error rate.}
3.8 Pitfalls of Model Selection and Evaluation Model selection and evaluation, when used effectively, serve as excellent tools for learning classification models and assessing their generalization performance. However, when using them effectively in practical settings, there are several pitfalls that can result in improper and often misleading conclusions. Some of these pitfalls are simple to understand and easy to avoid, while others are quite subtle in nature and difficult to catch. In the following, we present two of these pitfalls and discuss best practices to avoid them.
3.8.1 Overlap between Training and Test Sets One of the basic requirements of a clean model selection and evaluation setup is that the data used for model selection (D.train) must be kept separate from the data used for model evaluation (D.test). If there is any overlap between the two, the test error rate errtest computed over D.test cannot be considered representative of the performance on unseen instances. Comparing the effectiveness of classification models using errtest can then be quite misleading, as an overly complex model can show an inaccurately low value of errtest due to model overfitting (see Exercise 12 at the end of this chapter).
To illustrate the importance of ensuring no overlap between D.train and D.test, consider a labeled data set where all the attributes are irrelevant, i.e. they have no relationship with the class labels. Using such attributes, we should expect no classification model to perform better than random guessing. However, if the test set involves even a small number of data instances that were used for training, there is a possibility for an overly
complex model to show better performance than random, even though the attributes are completely irrelevant. As we will see later in Chapter 10, this scenario can actually be used as a criterion to detect overfitting due to improper setup of experiment. If a model shows better performance than a random classifier even when the attributes are irrelevant, it is an indication of a potential feedback between the training and test sets.
3.8.2 Use of Validation Error as Generalization Error The validation error rate errval serves an important role during model selection, as it provides “out-of-sample” error estimates of models on D.val, which is not used for training the models. Hence, errval serves as a better metric than the training error rate for selecting models and hyper-parameter values, as described in Sections 3.5.1 and 3.7, respectively. However, once the validation set has been used for selecting a classification model m*, errval no longer reflects the performance of m* on unseen instances.
To realize the pitfall in using validation error rate as an estimate of generalization performance, consider the problem of selecting a hyper- parameter value p from a range of values P, using a validation set D.val. If the number of possible values in P is quite large and the size of D.val is small, it is possible to select a hyper-parameter value p* that shows favorable performance on D.val just by random chance. Notice the similarity of this problem with the multiple comparisons problem discussed in Section 3.4.1. Even though the classification model m* learned using p* would show a low validation error rate, it would lack generalizability on unseen test instances.
The correct approach for estimating the generalization error rate of a model m* is to use an independently chosen test set D.test that hasn't been used in any way to influence the selection of m*. As a rule of thumb, the test set should never be examined during model selection, to ensure the absence of any form of overfitting. If the insights gained from any portion of a labeled data set help in improving the classification model even in an indirect way, then that portion of data must be discarded during testing.
3.9 Model Comparison*
One difficulty when comparing the performance of different classification models is whether the observed difference in their performance is statistically significant. For example, consider a pair of classification models, MA and MB. Suppose MA achieves 85% accuracy when evaluated on a test set containing 30 instances, while MB achieves 75% accuracy on a different test set containing 5000 instances. Based on this information, is MA a better model than MB? This example raises two key questions regarding the statistical significance of a performance metric:
1. Although MA has a higher accuracy than MB, it was tested on a smaller test set. How much confidence do we have that the accuracy for MA is
actually 85%?
2. Is it possible to explain the difference in accuracies between MA and MB as a result of variations in the composition of their test sets?
The first question relates to the issue of estimating the confidence interval of model accuracy. The second question relates to the issue of testing the statistical significance of the observed deviation. These issues are investigated in the remainder of this section.
3.9.1 Estimating the Confidence Interval for Accuracy To determine its confidence interval, we need to establish the probability distribution for sample accuracy. This section describes an approach for deriving the confidence interval by modeling the classification task as a binomial random experiment. The following describes characteristics of such an experiment:
1. The random experiment consists of N independent trials, where each trial has two possible outcomes: success or failure.
2. The probability of success, p, in each trial is constant.
An example of a binomial experiment is counting the number of heads that turn up when a coin is flipped N times. If X is the number of successes observed in N trials, then the probability that X takes a particular value is given by a binomial distribution with mean Np and variance Np(1−p):
P(X=υ)=(Nυ)pυ(1−p)N−υ.
For example, if the coin is fair (p=0.5) and is flipped fifty times, then the probability that the head shows up 20 times is
P(X=20)=(5020)0.520(1−0.5)30=0.0419.
If the experiment is repeated many times, then the average number of heads expected to show up is 50×0.5=25, while its variance is 50×0.5×0.5=12.5.
The task of predicting the class labels of test instances can also be considered as a binomial experiment. Given a test set that contains N instances, let X be the number of instances correctly predicted by a model and p be the true accuracy of the model. If the prediction task is modeled as a binomial experiment, then X has a binomial distribution with mean Np and variance Np(1−p). It can be shown that the empirical accuracy, acc=X/N, also has a binomial distribution with mean p and variance p(1−p)/N (see Exercise 14). The binomial distribution can be approximated by a normal distribution when N is sufficiently large. Based on the normal distribution, the confidence interval for acc can be derived as follows:
P(−Zα/2≤acc−pp(1−p)/N≤Z1−α/2)=1−α, (3.15)
where Zα/2 and Z1−α/2 are the upper and lower bounds obtained from a standard normal distribution at confidence level (1−α). Since a standard normal distribution is symmetric around Z=0, it follows that Zα/2=Z1−α/2. Rearranging this inequality leads to the following confidence interval for p:
2×N×acc×Zα/22±Zα/2Zα/22+4Nacc−4Nacc22(N+Zα/22). (3.16)
The following table shows the values of Zα/2 at different confidence levels:
1−α 0.99 0.98 0.95 0.9 0.8 0.7 0.5 Zα/2 2.58 2.33 1.96 1.65 1.28 1.04 0.67
3.11. Example Confidence Interval for Accuracy Consider a model that has an accuracy of 80% when evaluated on 100 test instances. What is the confidence interval for its true accuracy at a 95% confidence level? The confidence level of 95% corresponds to Za/2=1.96 according to the table given above. Inserting this term into Equation 3.16 yields a confidence interval between 71.1% and 86.7%. The following table shows the confidence interval when the number of instances, N, increases:
N 20 50 100 500 1000 5000 Confidence 0.584 0.670 0.711 0.763 0.774 0.789
Interval −0.919 −0.888 −0.867 −0.833 −0.824 −0.811
Note that the confidence interval becomes tighter when N increases.
3.9.2 Comparing the Performance of Two Models Consider a pair of models, M1 and M2, which are evaluated on two independent test sets, D1 and D2. Let n1 denote the number of instances in D1 and n2 denote the number of instances in D2. In addition, suppose the error rate for M1 on D1 is e1 and the error rate for M2 on D2 is e2. Our goal is to test whether the observed difference between e1 and e2 is statistically significant.
Assuming that n1 and n2 are sufficiently large, the error rates e1 and e2 can be approximated using normal distributions. If the observed difference in the error rate is denoted as d=e1−e2, then d is also normally distributed with mean dt, its true difference, and variance, σd2. The variance of d can be computed as follows:
σd2≃σ^d2=e1(1−e1)n1+e2(1−e2)n2, (3.17)
where e1(1−e1)/n1 and e2(1−e1)/n2 are the variances of the error rates. Finally, at the (1−α)% confidence level, it can be shown that the confidence interval for the true difference dt is given by the following equation:
dt=d±zα/2σ^d. (3.18)
3.12. Example Significance Testing Consider the problem described at the beginning of this section. Model MA has an error rate of e1=0.15 when applied to N1=30 test instances, while model MB has an error rate of e2=0.25 when applied to N2=5000 test instances. The observed difference in their error rates is d=|0.15−0.25|=0.1. In this example, we are performing a two-sided test to check whether dt=0 or dt≠0. The estimated variance of the observed difference in error rates can be computed as follows:
σ^d2=0.15(1−0.15)30+0.25(1−0.25)5000=0.0043
or σ^d=0.0655. Inserting this value into Equation 3.18, we obtain the following confidence interval for dt at 95% confidence level:
dt=0.1±1.96×0.0655=0.1±0.128.
As the interval spans the value zero, we can conclude that the observed difference is not statistically significant at a 95% confidence level.
At what confidence level can we reject the hypothesis that dt=0? To do this, we need to determine the value of Zα/2 such that the confidence interval for dt does not span the value zero. We can reverse the preceding computation
and look for the value Zα/2 such that d>Zσ/2σ^d. Replacing the values of d and σ^d gives Zσ/2<1.527. This value first occurs when (1−α)<~0.936 (for a two-sided test). The result suggests that the null hypothesis can be rejected at confidence level of 93.6% or lower.
3.10 Bibliographic Notes Early classification systems were developed to organize various collections of objects, from living organisms to inanimate ones. Examples abound, from Aristotle's cataloguing of species to the Dewey Decimal and Library of Congress classification systems for books. Such a task typically requires considerable human efforts, both to identify properties of the objects to be classified and to organize them into well distinguished categories.
With the development of statistics and computing, automated classification has been a subject of intensive research. The study of classification in classical statistics is sometimes known as discriminant analysis, where the objective is to predict the group membership of an object based on its corresponding features. A well-known classical method is Fisher's linear discriminant analysis [142], which seeks to find a linear projection of the data that produces the best separation between objects from different classes.
Many pattern recognition problems also require the discrimination of objects from different classes. Examples include speech recognition, handwritten character identification, and image classification. Readers who are interested in the application of classification techniques for pattern recognition may refer to the survey articles by Jain et al. [150] and Kulkarni et al. [157] or classic pattern recognition books by Bishop [125], Duda et al. [137], and Fukunaga [143]. The subject of classification is also a major research topic in neural networks, statistical learning, and machine learning. An in-depth treatment on the topic of classification from the statistical and machine learning perspectives can be found in the books by Bishop [126], Cherkassky and Mulier [132], Hastie et al. [148], Michie et al. [162], Murphy [167], and Mitchell [165]. Recent years have also seen the release of many publicly available software packages for classification, which can be embedded in programming languages such as Java (Weka [147]) and Python (scikit-learn [174]).
An overview of decision tree induction algorithms can be found in the survey articles by Buntine [129], Moret [166], Murthy [168], and Safavian et al.
[179]. Examples of some well-known decision tree algorithms include CART [127], ID3 [175], C4.5 [177], and CHAID [153]. Both ID3 and C4.5 employ the entropy measure as their splitting function. An in-depth discussion of the C4.5 decision tree algorithm is given by Quinlan [177]. The CART algorithm was developed by Breiman et al. [127] and uses the Gini index as its splitting function. CHAID [153] uses the statistical χ2 test to determine the best split during the tree-growing process.
The decision tree algorithm presented in this chapter assumes that the splitting condition at each internal node contains only one attribute. An oblique decision tree can use multiple attributes to form the attribute test condition in a single node [149, 187]. Breiman et al. [127] provide an option for using linear combinations of attributes in their CART implementation. Other approaches for inducing oblique decision trees were proposed by Heath et al. [149], Murthy et al. [169], Cantú-Paz and Kamath [130], and Utgoff and Brodley [187]. Although an oblique decision tree helps to improve the expressiveness of the model representation, the tree induction process becomes computationally challenging. Another way to improve the expressiveness of a decision tree without using oblique decision trees is to apply a method known as constructive induction [161]. This method simplifies the task of learning complex splitting functions by creating compound features from the original data.
Besides the top-down approach, other strategies for growing a decision tree include the bottom-up approach by Landeweerd et al. [159] and Pattipati and Alexandridis [173], as well as the bidirectional approach by Kim and Landgrebe [154]. Schuermann and Doster [181] and Wang and Suen [193] proposed using a soft splitting criterion to address the data fragmentation problem. In this approach, each instance is assigned to different branches of the decision tree with different probabilities.
Model overfitting is an important issue that must be addressed to ensure that a decision tree classifier performs equally well on previously unlabeled data instances. The model overfitting problem has been investigated by many authors including Breiman et al. [127], Schaffer [180], Mingers [164], and Jensen and Cohen [151]. While the presence of noise is often regarded as one of the primary reasons for overfitting [164, 170], Jensen and Cohen [151]
viewed overfitting as an artifact of failure to compensate for the multiple comparisons problem.
Bishop [126] and Hastie et al. [148] provide an excellent discussion of model overfitting, relating it to a well-known framework of theoretical analysis, known as bias-variance decomposition [146]. In this framework, the prediction of a learning algorithm is considered to be a function of the training set, which varies as the training set is changed. The generalization error of a model is then described in terms of its bias (the error of the average prediction obtained using different training sets), its variance (how different are the predictions obtained using different training sets), and noise (the irreducible error inherent to the problem). An underfit model is considered to have high bias but low variance, while an overfit model is considered to have low bias but high variance. Although the bias-variance decomposition was originally proposed for regression problems (where the target attribute is a continuous variable), a unified analysis that is applicable for classification has been proposed by Domingos [136]. The bias variance decomposition will be discussed in more detail while introducing ensemble learning methods in Chapter 4.
Various learning principles, such as the Probably Approximately Correct (PAC) learning framework [188], have been developed to provide a theoretical framework for explaining the generalization performance of learning algorithms. In the field of statistics, a number of performance estimation methods have been proposed that make a trade-off between the goodness of fit of a model and the model complexity. Most noteworthy among them are the Akaike's Information Criterion [120] and the Bayesian Information Criterion [182]. They both apply corrective terms to the training error rate of a model, so as to penalize more complex models. Another widely-used approach for measuring the complexity of any general model is the VapnikChervonenkis (VC) Dimension [190]. The VC dimension of a class of functions C is defined as the maximum number of points that can be shattered (every point can be distinguished from the rest) by functions belonging to C, for any possible configuration of points. The VC dimension lays the foundation of the structural risk minimization principle [189], which is extensively used in many learning algorithms, e.g., support vector machines, which will be discussed in detail in Chapter 4.
The Occam's razor principle is often attributed to the philosopher William of Occam. Domingos [135] cautioned against the pitfall of misinterpreting Occam's razor as comparing models with similar training errors, instead of generalization errors. A survey on decision tree-pruning methods to avoid overfitting is given by Breslow and Aha [128] and Esposito et al. [141]. Some of the typical pruning methods include reduced error pruning [176], pessimistic error pruning [176], minimum error pruning [171], critical value pruning [163], cost-complexity pruning [127], and error-based pruning [177]. Quinlan and Rivest proposed using the minimum description length principle for decision tree pruning in [178].
The discussions in this chapter on the significance of cross-validation error estimates is inspired from Chapter 7 in Hastie et al. [148]. It is also an excellent resource for understanding “the right and wrong ways to do cross- validation”, which is similar to the discussion on pitfalls in Section 3.8 of this chapter. A comprehensive discussion of some of the common pitfalls in using cross-validation for model selection and evaluation is provided in Krstajic et al. [156].
The original cross-validation method was proposed independently by Allen [121], Stone [184], and Geisser [145] for model assessment (evaluation). Even though cross-validation can be used for model selection [194], its usage for model selection is quite different than when it is used for model evaluation, as emphasized by Stone [184]. Over the years, the distinction between the two usages has often been ignored, resulting in incorrect findings. One of the common mistakes while using cross-validation is to perform pre-processing operations (e.g., hyper-parameter tuning or feature selection) using the entire data set and not “within” the training fold of every cross-validation run. Ambroise et al., using a number of gene expression studies as examples, [124] provide an extensive discussion of the selection bias that arises when feature selection is performed outside cross-validation. Useful guidelines for evaluating models on microarray data have also been provided by Allison et al. [122].
The use of the cross-validation protocol for hyper-parameter tuning has been described in detail by Dudoit and van der Laan [138]. This approach has been called “grid-search cross-validation.” The correct approach in using cross-
validation for both hyper-parameter selection and model evaluation, as discussed in Section 3.7 of this chapter, is extensively covered by Varma and Simon [191]. This combined approach has been referred to as “nested cross- validation” or “double cross-validation” in the existing literature. Recently, Tibshirani and Tibshirani [185] have proposed a new approach for hyper- parameter selection and model evaluation. Tsamardinos et al. [186] compared this approach to nested cross-validation. The experiments they performed found that, on average, both approaches provide conservative estimates of model performance with the Tibshirani and Tibshirani approach being more computationally efficient.
Kohavi [155] has performed an extensive empirical study to compare the performance metrics obtained using different estimation methods such as random subsampling and k-fold cross-validation. Their results suggest that the best estimation method is ten-fold, stratified cross-validation.
An alternative approach for model evaluation is the bootstrap method, which was presented by Efron in 1979 [139]. In this method, training instances are sampled with replacement from the labeled set, i.e., an instance previously selected to be part of the training set is equally likely to be drawn again. If the original data has N instances, it can be shown that, on average, a bootstrap sample of size N contains about 63.2% of the instances in the original data. Instances that are not included in the bootstrap sample become part of the test set. The bootstrap procedure for obtaining training and test sets is repeated b times, resulting in a different error rate on the test set, err(i), at the ith run. To obtain the overall error rate, errboot, the .632 bootstrap approach combines err(i) with the error rate obtained from a training set containing all the labeled examples, errs, as follows:
errboot=1b∑i=1b(0.632)×err(i)+0.386×errs). (3.19)
Efron and Tibshirani [140] provided a theoretical and empirical comparison between cross-validation and a bootstrap method known as the 632+ rule.
While the .632 bootstrap method presented above provides a robust estimate of the generalization performance with low variance in its estimate, it may produce misleading results for highly complex models in certain conditions, as demonstrated by Kohavi [155]. This is because the overall error rate is not
truly an out-of-sample error estimate as it depends on the training error rate, errs, which can be quite small if there is overfitting.
Current techniques such as C4.5 require that the entire training data set fit into main memory. There has been considerable effort to develop parallel and scalable versions of decision tree induction algorithms. Some of the proposed algorithms include SLIQ by Mehta et al. [160], SPRINT by Shafer et al. [183], CMP by Wang and Zaniolo [192], CLOUDS by Alsabti et al. [123], RainForest by Gehrke et al. [144], and ScalParC by Joshi et al. [152]. A survey of parallel algorithms for classification and other data mining tasks is given in [158]. More recently, there has been extensive research to implement large-scale classifiers on the compute unified device architecture (CUDA) [131, 134] and MapReduce [133, 172] platforms.
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3.11 Exercises 1. 1. Draw the full decision tree for the parity function of four Boolean
attributes, A, B, C, and D. Is it possible to simplify the tree?
2. 2. Consider the training examples shown in Table 3.5 for a binary classification problem.
Table 3.5. Data set for Exercise 2. Customer ID Gender Car Type Shirt Size Class
1 M Family Small C0 2 M Sports Medium C0 3 M Sports Medium C0 4 M Sports Large C0 5 M Sports Extra Large C0 6 M Sports Extra Large C0 7 F Sports Small C0 8 F Sports Small C0 9 F Sports Medium C0 10 F Luxury Large C0 11 M Family Large C1 12 M Family Extra Large C1 13 M Family Medium C1 14 M Luxury Extra Large C1 15 F Luxury Small C1 16 F Luxury Small C1 17 F Luxury Medium C1
18 F Luxury Medium C1 19 F Luxury Medium C1 20 F Luxury Large C1
1. Compute the Gini index for the overall collection of training examples.
2. Compute the Gini index for the Customer ID attribute.
3. Compute the Gini index for the Gender attribute.
4. Compute the Gini index for the Car Type attribute using multiway split.
5. Compute the Gini index for the Shirt Size attribute using multiway split.
6. Which attribute is better, Gender, Car Type, or Shirt Size?
7. Explain why Customer ID should not be used as the attribute test condition even though it has the lowest Gini.
3. 3. Consider the training examples shown in Table 3.6 for a binary classification problem.
Table 3.6. Data set for Exercise 3. Instance a1 a2 a3 Target Class
1 T T 1.0 + 2 T T 6.0 + 3 T F 5.0 − 4 F F 4.0 + 5 F T 7.0 −
6 F T 3.0 − 7 F F 8.0 − 8 T F 7.0 + 9 F T 5.0 −
1. What is the entropy of this collection of training examples with respect to the class attribute?
2. What are the information gains of a1 and a2 relative to these training examples?
3. For a3, which is a continuous attribute, compute the information gain for every possible split.
4. What is the best split (among a1, a2, and a3) according to the information gain?
5. What is the best split (between a1 and a2) according to the misclassification error rate?
6. What is the best split (between a1 and a2) according to the Gini index?
4. 4. Show that the entropy of a node never increases after splitting it into smaller successor nodes.
5. 5. Consider the following data set for a binary class problem.
A B Class Label T F + T T + T T + T F − T T + F F − F F −
F F − T T − T F −
1. Calculate the information gain when splitting on A and B. Which attribute would the decision tree induction algorithm choose?
2. Calculate the gain in the Gini index when splitting on A and B. Which attribute would the decision tree induction algorithm choose?
3. Figure 3.11 shows that entropy and the Gini index are both monotonically increasing on the range [0, 0.5] and they are both monotonically decreasing on the range [0.5, 1]. Is it possible that information gain and the gain in the Gini index favor different attributes? Explain.
6. 6. Consider splitting a parent node P into two child nodes, C1 and C2, using some attribute test condition. The composition of labeled training instances at every node is summarized in the Table below.
P C1 C2 Class 0 7 3 4 Class 1 3 0 3
1. Calculate the Gini index and misclassification error rate of the parent node P .
2. Calculate the weighted Gini index of the child nodes. Would you consider this attribute test condition if Gini is used as the impurity measure?
3. Calculate the weighted misclassification rate of the child nodes. Would you consider this attribute test condition if misclassification rate is used as the impurity measure?
7. 7. Consider the following set of training examples.
X Y Z No. of Class C1 Examples No. of Class C2 Examples 0 0 0 5 40 0 0 1 0 15 0 1 0 10 5 0 1 1 45 0 1 0 0 10 5 1 0 1 25 0 1 1 0 5 20 1 1 1 0 15
1. Compute a two-level decision tree using the greedy approach described in this chapter. Use the classification error rate as the criterion for splitting. What is the overall error rate of the induced tree?
2. Repeat part (a) using X as the first splitting attribute and then choose the best remaining attribute for splitting at each of the two successor nodes. What is the error rate of the induced tree?
3. Compare the results of parts (a) and (b). Comment on the suitability of the greedy heuristic used for splitting attribute selection.
8. 8. The following table summarizes a data set with three attributes A, B, C and two class labels +, −. Build a two-level decision tree.
A B C Number of Instances + −
T T T 5 0 F T T 0 20 T F T 20 0 F F T 0 5 T T F 0 0 F T F 25 0 T F F 0 0
F F F 0 25
1. According to the classification error rate, which attribute would be chosen as the first splitting attribute? For each attribute, show the contingency table and the gains in classification error rate.
2. Repeat for the two children of the root node.
3. How many instances are misclassified by the resulting decision tree?
4. Repeat parts (a), (b), and (c) using C as the splitting attribute.
5. Use the results in parts (c) and (d) to conclude about the greedy nature of the decision tree induction algorithm.
9. 9. Consider the decision tree shown in Figure 3.36.
Figure 3.36. Decision tree and data sets for Exercise 9.
Figure 3.36. Full Alternative Text
1. Compute the generalization error rate of the tree using the optimistic approach.
2. Compute the generalization error rate of the tree using the pessimistic approach. (For simplicity, use the strategy of adding a factor of 0.5 to each leaf node.)
3. Compute the generalization error rate of the tree using the validation set shown above. This approach is known as reduced error pruning.
10. 10. Consider the decision trees shown in Figure 3.37. Assume they are generated from a data set that contains 16 binary attributes and 3 classes, C1, C2, and C3.
Compute the total description length of each decision tree according to the following formulation of the minimum description length principle.
The total description length of a tree is given by
Cost(tree,data)=Cost(tree)+Cost(data|tree).
Each internal node of the tree is encoded by the ID of the splitting attribute. If there are m attributes, the cost of encoding each attribute is log2m bits.
Figure 3.37. Decision trees for Exercise 10.
Figure 3.37. Full Alternative Text
Each leaf is encoded using the ID of the class it is associated with. If there are k classes, the cost of encoding a class is log2 k bits.
Cost(tree) is the cost of encoding all the nodes in the tree. To simplify the computation, you can assume that the total cost of the tree is obtained by adding up the costs of encoding each internal node and each leaf node.
Cost(data|tree) is encoded using the classification errors the tree commits on the training set. Each error is encoded by log2 n bits, where n is the total number of training instances.
Which decision tree is better, according to the MDL principle?
11. 11. This exercise, inspired by the discussions in [155], highlights one of
the known limitations of the leave-one-out model evaluation procedure. Let us consider a data set containing 50 positive and 50 negative instances, where the attributes are purely random and contain no information about the class labels. Hence, the generalization error rate of any classification model learned over this data is expected to be 0.5. Let us consider a classifier that assigns the majority class label of training instances (ties resolved by using the positive label as the default class) to any test instance, irrespective of its attribute values. We can call this approach as the majority inducer classifier. Determine the error rate of this classifier using the following methods.
1. Leave-one-out.
2. 2-fold stratified cross-validation, where the proportion of class labels at every fold is kept same as that of the overall data.
3. From the results above, which method provides a more reliable evaluation of the classifier's generalization error rate?
12. 12. Consider a labeled data set containing 100 data instances, which is randomly partitioned into two sets A and B, each containing 50 instances. We use A as the training set to learn two decision trees, T10 with 10 leaf nodes and T100 with 100 leaf nodes. The accuracies of the two decision trees on data sets A and B are shown in Table 3.7.
Table 3.7. Comparing the test accuracy of decision trees T10 and T100.
Accuracy Data Set T10 T100
A 0.86 0.97 B 0.84 0.77
1. Based on the accuracies shown in Table 3.7, which classification model would you expect to have better performance on unseen instances?
2. Now, you tested T10 and T100 on the entire data set (A+B) and found that the classification accuracy of T10 on data set (A+B) is 0.85, whereas the classification accuracy of T100 on the data set (A+B) is 0.87. Based on this new information and your observations from Table 3.7, which classification model would you finally choose for classification?
13. 13. Consider the following approach for testing whether a classifier A beats another classifier B. Let N be the size of a given dataset, pA be the accuracy of classifier A, pB be the accuracy of classifier B, and p= (pA+pB)/2 be the average accuracy for both classifiers. To test whether classifier A is significantly better than B, the following Z-statistic is used:
Z=pA−pB2p(1−p)N.
Classifier A is assumed to be better than classifier B if Z>1.96.
Table 3.8 compares the accuracies of three different classifiers, decision tree classifiers, naïve Bayes classifiers, and support vector machines, on various data sets. (The latter two classifiers are described in Chapter 4.)
Summarize the performance of the classifiers given in Table 3.8 using the following 3×3 table:
win-loss-draw Decisiontree Naïve Bayes
Support vector machine
Decision tree 0 - 0 - 23 Naïve Bayes 0 - 0 - 23 Support vector machine 0 - 0 - 23
Table 3.8. Comparing the accuracy of various classification methods.
Data Set Size(N) DecisionTree (%)
naïve Bayes (%)
Support vector machine (%)
Anneal 898 92.09 79.62 87.19 Australia 690 85.51 76.81 84.78 Auto 205 81.95 58.05 70.73 Breast 699 95.14 95.99 96.42 Cleve 303 76.24 83.50 84.49 Credit 690 85.80 77.54 85.07 Diabetes 768 72.40 75.91 76.82 German 1000 70.90 74.70 74.40 Glass 214 67.29 48.59 59.81 Heart 270 80.00 84.07 83.70 Hepatitis 155 81.94 83.23 87.10 Horse 368 85.33 78.80 82.61 Ionosphere 351 89.17 82.34 88.89 Iris 150 94.67 95.33 96.00 Labor 57 78.95 94.74 92.98 Led7 3200 73.34 73.16 73.56 Lymphography 148 77.03 83.11 86.49 Pima 768 74.35 76.04 76.95 Sonar 208 78.85 69.71 76.92 Tic-tac-toe 958 83.72 70.04 98.33 Vehicle 846 71.04 45.04 74.94 Wine 178 94.38 96.63 98.88
Zoo 101 93.07 93.07 96.04
Each cell in the table contains the number of wins, losses, and draws when comparing the classifier in a given row to the classifier in a given column.
14. 14. Let X be a binomial random variable with mean Np and variance Np(1−p). Show that the ratio X/N also has a binomial distribution with mean p and variance p(1−p)N.
4 Classification: Alternative Techniques The previous chapter introduced the classification problem and presented a technique known as the decision tree classifier. Issues such as model overfitting and model evaluation were also discussed. This chapter presents alternative techniques for building classification models—from simple techniques such as rule-based and nearest neighbor classifiers to more sophisticated techniques such as artificial neural networks, deep learning, support vector machines, and ensemble methods. Other practical issues such as the class imbalance and multiclass problems are also discussed at the end of the chapter.
4.1 Types of Classifiers Before presenting specific techniques, we first categorize the different types of classifiers available. One way to distinguish classifiers is by considering the characteristics of their output.
Binary versus Multiclass Binary classifiers assign each data instance to one of two possible labels, typically denoted as +1 and −1. The positive class usually refers to the category we are more interested in predicting correctly compared to the negative class (e.g., the spam category in email classification problems). If there are more than two possible labels available, then the technique is known as a multiclass classifier. As some classifiers were designed for binary classes only, they must be adapted to deal with multiclass problems. Techniques for transforming binary classifiers into multiclass classifiers are described in Section 4.12.
Deterministic versus Probabilistic A deterministic classifier produces a discrete-valued label to each data instance it classifies whereas a probabilistic classifier assigns a continuous score between 0 and 1 to indicate how likely it is that an instance belongs to a particular class, where the probability scores for all the classes sum up to 1. Some examples of probabilistic classifiers include the naïve Bayes classifier, Bayesian networks, and logistic regression. Probabilistic classifiers provide additional information about the confidence in assigning an instance to a class than deterministic classifiers. A data instance is typically assigned to the class with the highest probability score, except when the cost of misclassifying the class with lower probability is significantly higher. We will discuss the topic of cost-sensitive classification with probabilistic outputs in Section 4.11.2.
Another way to distinguish the different types of classifiers is based on their technique for discriminating instances from different classes.
Linear versus Nonlinear A linear classifier uses a linear separating hyperplane to discriminate instances from different classes whereas a nonlinear classifier enables the construction of more complex, nonlinear decision surfaces. We illustrate an example of a linear classifier (perceptron) and its nonlinear counterpart (multi-layer neural network) in Section 4.7. Although the linearity assumption makes the model less flexible in terms of fitting complex data, linear classifiers are thus less susceptible to model overfitting compared to nonlinear classifiers. Furthermore, one can transform the original set of attributes, x=(x1, x2, ⋯ ,xd), into a more complex feature set, e.g., Φ(x)= (x1, x2, x1x2, x12, x22, ⋯), before applying the linear classifier. Such feature transformation allows the linear classifier to fit data sets with nonlinear decision surfaces (see Section 4.9.4).
Global versus Local A global classifier fits a single model to the entire data set. Unless the model is highly nonlinear, this one-size-fits-all strategy may not be effective when the relationship between the attributes and the class labels varies over the input space. In contrast, a local classifier partitions the input space into smaller regions and fits a distinct model to training instances in each region. The k-nearest neighbor classifier (see Section 4.3) is a classic example of local classifiers. While local classifiers are more flexible in terms of fitting complex decision boundaries, they are also more susceptible to the model overfitting problem, especially when the local regions contain few training examples.
Generative versus Discriminative
Given a data instance x, the primary objective of any classifier is to predict the class label, y, of the data instance. However, apart from predicting the class label, we may also be interested in describing the underlying mechanism that generates the instances belonging to every class label. For example, in the process of classifying spam email messages, it may be useful to understand the typical characteristics of email messages that are labeled as spam, e.g., specific usage of keywords in the subject or the body of the email. Classifiers that learn a generative model of every class in the process of predicting class labels are known as generative classifiers. Some examples of generative classifiers include the naïve Bayes classifier and Bayesian networks. In contrast, discriminative classifiers directly predict the class labels without explicitly describing the distribution of every class label. They solve a simpler problem than generative models since they do not have the onus of deriving insights about the generative mechanism of data instances. They are thus sometimes preferred over generative models, especially when it is not crucial to obtain information about the properties of every class. Some examples of discriminative classifiers include decision trees, rule-based classifier, nearest neighbor classifier, artificial neural networks, and support vector machines.
4.2 Rule-Based Classifier A rule-based classifier uses a collection of “if …then…” rules (also known as a rule set) to classify data instances. Table 4.1 shows an example of a rule set generated for the vertebrate classification problem described in the previous chapter. Each classification rule in the rule set can be expressed in the following way:
ri:(Conditioni)→yi. (4.1)
The left-hand side of the rule is called the rule antecedent or precondition. It contains a conjunction of attribute test conditions:
Conditioni=(A1 op v1)∧(A2 op v2)…(Ak op vk), (4.2)
where (Aj, vj) is an attribute-value pair and op is a comparison operator chosen from the set {=, ≠, <, >, ≤, ≥}. Each attribute test (Aj op vj) is also known as a conjunct. The right-hand side of the rule is called the rule consequent, which contains the predicted class yi.
A rule r covers a data instance x if the precondition of r matches the attributes of x. r is also said to be fired or triggered whenever it covers a given instance. For an illustration, consider the rule r1 given in Table 4.1 and the following attributes for two vertebrates: hawk and grizzly bear.
Table 4.1. Example of a rule set for the vertebrate classification problem. r1:(Gives Birth=no)∧(Aerial Creature=yes)→Birdsr2:(Gives Birth=no)∧ blooded)→Mammalsr4:(Gives Birth=no)∧(Aerial Creature=no)→Reptiles
Name BodyTemperature Skin
Cover Gives Birth
Aquatic Creature
Aerial Creature
Has Legs Hibernates
hawk warm-blooded feather no no yes yes no
grizzly bear
warm- blooded fur yes no no yes yes
r1 covers the first vertebrate because its precondition is satisfied by the hawk's attributes. The rule does not cover the second vertebrate because grizzly bears give birth to their young and cannot fly, thus violating the precondition of r1.
The quality of a classification rule can be evaluated using measures such as coverage and accuracy. Given a data set D and a classification rule r : A→y, the coverage of the rule is the fraction of instances in D that trigger the rule r. On the other hand, its accuracy or confidence factor is the fraction of instances triggered by r whose class labels are equal to y. The formal definitions of these measures are
Coverage(r)=| A || D |Coverage(r)=|A∩y || A |, (4.3)
where |A| is the number of instances that satisfy the rule antecedent, |A∩y| is the number of instances that satisfy both the antecedent and consequent, and |D| is the total number of instances.
Example 4.1. Consider the data set shown in Table 4.2. The rule
(Gives Birth=yes)∧(Body Temperature=warm-blooded)→Mammals
has a coverage of 33% since five of the fifteen instances support the rule antecedent. The rule accuracy is 100% because all five vertebrates covered by the rule are mammals.
Table 4.2. The vertebrate data set.
Name BodyTemperature Skin
Cover Gives Birth
Aquatic Creature
Aerial Creature
Has Legs Hibernates
human warm-blooded hair yes no no yes no
python cold-blooded scales no no no no yes salmon cold-blooded scales no yes no no no
whale warm-blooded hair yes yes no no no
frog cold-blooded none no semi no yes yes komodo dragon cold-blooded scales no no no yes no
bat warm-blooded hair yes no yes yes yes
pigeon warm-blooded feathers no no yes yes no
cat warm-blooded fur yes no no yes no
guppy cold-blooded scales yes yes no no no alligator cold-blooded scales no semi no yes no
penguin warm-blooded feathers no semi no yes no
porcupine warm-blooded quills yes no no yes yes
eel cold-blooded scales no yes no no no salamander cold-blooded none no semi no yes yes
4.2.1 How a Rule-Based Classifier
Works A rule-based classifier classifies a test instance based on the rule triggered by the instance. To illustrate how a rule-based classifier works, consider the rule set shown in Table 4.1 and the following vertebrates:
Name BodyTemperature Skin
Cover Gives Birth
Aquatic Creature
Aerial Creature
Has Legs Hibernates
lemur warm-blooded fur yes no no yes yes
turtle cold-blooded scales no semi no yes no dogfish shark cold-blooded scales yes yes no no no
The first vertebrate, which is a lemur, is warm-blooded and gives birth to its young. It triggers the rule r3, and thus, is classified as a mammal.
The second vertebrate, which is a turtle, triggers the rules r4 and r5. Since the classes predicted by the rules are contradictory (reptiles versus amphibians), their conflicting classes must be resolved.
None of the rules are applicable to a dogfish shark. In this case, we need to determine what class to assign to such a test instance.
4.2.2 Properties of a Rule Set The rule set generated by a rule-based classifier can be characterized by the following two properties.
Definition 4.1 (Mutually Exclusive Rule Set).
The rules in a rule set R are mutually exclusive if no two rules in R are triggered by the same instance. This property ensures that every instance is covered by at most one rule in R.
Definition 4.2 (Exhaustive Rule Set). A rule set R has exhaustive coverage if there is a rule for each combination of attribute values. This property ensures that every instance is covered by at least one rule in R.
Table 4.3. Example of a mutually exclusive and exhaustive rule set. r1: (Body Temperature=cold-blooded)→Non- mammalsr2: (Body Temperature=warm- blooded)∧(Gives Birth=yes)→Mammalsr3: (Body Temperature=warm- blooded)∧(Gives Birth=no)→Non-mammals
Together, these two properties ensure that every instance is covered by exactly one rule. An example of a mutually exclusive and exhaustive rule set is shown in Table 4.3. Unfortunately, many rule-based classifiers, including the one shown in Table 4.1, do not have such properties. If the rule set is not exhaustive, then a default rule, rd: ()→yd, must be added to cover the remaining cases. A default rule has an empty antecedent and is triggered when all other rules have failed. yd is known as the default class and is typically assigned to the majority class of training instances not covered by the existing rules. If the rule set is not mutually exclusive, then an instance can be covered by more than one rule, some of which may predict conflicting classes.
Definition 4.3 (Ordered Rule Set). The rules in an ordered rule set R are ranked in decreasing order of their priority. An ordered rule set is also known as a decision list.
The rank of a rule can be defined in many ways, e.g., based on its accuracy or total description length. When a test instance is presented, it will be classified by the highest-ranked rule that covers the instance. This avoids the problem of having conflicting classes predicted by multiple classification rules if the rule set is not mutually exclusive.
An alternative way to handle a non-mutually exclusive rule set without ordering the rules is to consider the consequent of each rule triggered by a test instance as a vote for a particular class. The votes are then tallied to determine the class label of the test instance. The instance is usually assigned to the class that receives the highest number of votes. The vote may also be weighted by the rule's accuracy. Using unordered rules to build a rule-based classifier has both advantages and disadvantages. Unordered rules are less susceptible to errors caused by the wrong rule being selected to classify a test instance unlike classifiers based on ordered rules, which are sensitive to the choice of rule-ordering criteria. Model building is also less expensive because the rules do not need to be kept in sorted order. Nevertheless, classifying a test instance can be quite expensive because the attributes of the test instance must be compared against the precondition of every rule in the rule set.
In the next two sections, we present techniques for extracting an ordered rule set from data. A rule-based classifier can be constructed using (1) direct methods, which extract classification rules directly from data, and (2) indirect methods, which extract classification rules from more complex classification models, such as decision trees and neural networks. Detailed discussions of these methods are presented in Sections 4.2.3 and 4.2.4, respectively.
4.2.3 Direct Methods for Rule Extraction
To illustrate the direct method, we consider a widely-used rule induction algorithm called RIPPER. This algorithm scales almost linearly with the number of training instances and is particularly suited for building models from data sets with imbalanced class distributions. RIPPER also works well with noisy data because it uses a validation set to prevent model overfitting.
RIPPER uses the sequential covering algorithm to extract rules directly from data. Rules are grown in a greedy fashion one class at a time. For binary class problems, RIPPER chooses the majority class as its default class and learns the rules to detect instances from the minority class. For multiclass problems, the classes are ordered according to their prevalence in the training set. Let (y1, y2, … ,yc) be the ordered list of classes, where y1 is the least prevalent class and yc is the most prevalent class. All training instances that belong to y1 are initially labeled as positive examples, while those that belong to other classes are labeled as negative examples. The sequential covering algorithm learns a set of rules to discriminate the positive from negative examples. Next, all training instances from y2 are labeled as positive, while those from classes y3, y4, ⋯, yc are labeled as negative. The sequential covering algorithm would learn the next set of rules to distinguish y2 from other remaining classes. This process is repeated until we are left with only one class, yc, which is designated as the default class.
Example 4.1. Sequential covering algorithm. 1: Let E be the training instances and A be the set of attribute-value pairs, {(A 2: Let Yo be an ordered set of classes {y1, y2, … ,yk. 3: Let R = { } be the initial rule list. 4: for each class y ∈ Yo −{yk} do 5: while stopping condition is not met do 6: r← Learn-One-Rule (E, A, y). 7: Remove training instances from E that are covered by r. 8: Add r to the bottom of the rule list: R ← R ∨ r. 9: end while 10: end for 11: Insert the default rule, {} → yk, to the bottom of the rule list
A summary of the sequential covering algorithm is shown in Algorithm 4.1. The algorithm starts with an empty decision list, R, and extracts rules for each class based on the ordering specified by the class prevalence. It iteratively extracts the rules for a given class y using the Learn-One-Rule function. Once such a rule is found, all the training instances covered by the rule are eliminated. The new rule is added to the bottom of the decision list R. This procedure is repeated until the stopping criterion is met. The algorithm then proceeds to generate rules for the next class.
Figure 4.1 demonstrates how the sequential covering algorithm works for a data set that contains a collection of positive and negative examples. The rule R1, whose coverage is shown in Figure 4.1(b), is extracted first because it covers the largest fraction of positive examples. All the training instances covered by R1 are subsequently removed and the algorithm proceeds to look for the next best rule, which is R2.
Learn-One-Rule Function Finding an optimal rule is computationally expensive due to the exponential search space to explore. The Learn-One-Rule function addresses this problem by growing the rules in a greedy fashion. It generates an initial rule r: {}→+, where the left-hand side is an empty set and the right-hand side corresponds to the positive class. It then refines the rule until a certain stopping criterion is met. The accuracy of the initial rule may be poor because some of the training instances covered by the rule belong to the negative class. A new conjunct must be added to the rule antecedent to improve its accuracy.
Figure 4.1. An example of the sequential covering algorithm.
Figure 4.1. Full Alternative Text
RIPPER uses the FOIL's information gain measure to choose the best conjunct to be added into the rule antecedent. The measure takes into consideration both the gain in accuracy and support of a candidate rule, where support is defined as the number of positive examples covered by the rule. For example, suppose the rule r: A→+ initially covers p0 positive examples and n0 negative examples. After adding a new conjunct B, the
extended rule r′: A∧B→+ covers p1 positive examples and n1 negative examples. The FOIL's information gain of the extended rule is computed as follows:
FOIL's information gain=p1×(log2p1p1+n1−log2p0p0+n0). (4.4)
RIPPER chooses the conjunct with highest FOIL's information gain to extend the rule, as illustrated in the next example.
Example 4.2. [Foil's Information Gain] Consider the training set for the vertebrate classification problem shown in Table 4.2. Suppose the target class for the Learn-One-Rule function is mammals. Initially, the antecedent of the rule {}→Mammals covers 5 positive and 10 negative examples. Thus, the accuracy of the rule is only 0.333. Next, consider the following three candidate conjuncts to be added to the left-hand side of the rule: Skin cover=hair, Body temperature=warm, and Has legs=No. The number of positive and negative examples covered by the rule after adding each conjunct along with their respective accuracy and FOIL's information gain are shown in the following table.
Candidate rule p1 n1 Accuracy Info Gain {Skin Cover=hair}→mammals 3 0 1.000 4.755 {Body temperature=wam}→mammals 5 1 0.714 5.498 {Has legs=No}→mammals 2 4 0.200 −0.737
Although Skin cover=hair has the highest accuracy among the three candidates, the conjunct Body temperature=warm has the highest FOIL's information gain. Thus, it is chosen to extend the rule (see Figure 4.2). This process continues until adding new conjuncts no longer improves the information gain measure.
Rule Pruning The rules generated by the Learn-One-Rule function can be pruned to improve their generalization errors. RIPPER prunes the rules based on their performance on the validation set. The following metric is computed to determine whether pruning is needed: (p−n)/(p+n), where p(n) is the number of positive (negative) examples in the validation set covered by the rule. This metric is monotonically related to the rule's accuracy on the validation set. If the metric improves after pruning, then the conjunct is removed. Pruning is done starting from the last conjunct added to the rule. For example, given a rule ABCD→y, RIPPER checks whether D should be pruned first, followed by CD, BCD, etc. While the original rule covers only positive examples, the pruned rule may cover some of the negative examples in the training set.
Building the Rule Set After generating a rule, all the positive and negative examples covered by the rule are eliminated. The rule is then added into the rule set as long as it does not violate the stopping condition, which is based on the minimum description length principle. If the new rule increases the total description length of the rule set by at least d bits, then RIPPER stops adding rules into its rule set (by default, d is chosen to be 64 bits). Another stopping condition used by RIPPER is that the error rate of the rule on the validation set must not exceed 50%.
Figure 4.2. General-to-specific and specific-to-general rule-growing strategies.
Figure 4.2. Full Alternative Text
RIPPER also performs additional optimization steps to determine whether some of the existing rules in the rule set can be replaced by better alternative rules. Readers who are interested in the details of the optimization method may refer to the reference cited at the end of this chapter.
Instance Elimination After a rule is extracted, RIPPER eliminates the positive and negative examples covered by the rule. The rationale for doing this is illustrated in the next example.
Figure 4.3 shows three possible rules, R1, R2, and R3, extracted from a training set that contains 29 positive examples and 21 negative examples. The accuracies of R1, R2, and R3 are 12/15 (80%), 7/10 (70%), and 8/12 (66.7%), respectively. R1 is generated first because it has the highest accuracy. After generating R1, the algorithm must remove the examples covered by the rule so that the next rule generated by the algorithm is different than R1. The question is, should it remove the positive examples only, negative examples only, or both? To answer this, suppose the algorithm must choose between generating R2 or R3 after R1. Even though R2 has a higher accuracy than R3 (70% versus 66.7%), observe that the region covered by R2 is disjoint from R1, while the region covered by R3 overlaps with R1. As a result, R1 and R3 together cover 18 positive and 5 negative examples (resulting in an overall accuracy of 78.3%), whereas R1 and R2 together cover 19 positive and 6 negative examples (resulting in a lower overall accuracy of 76%). If the positive examples covered by R1 are not removed, then we may overestimate the effective accuracy of R3. If the negative examples covered by R1 are not removed, then we may underestimate the accuracy of R3. In the latter case, we might end up preferring R2 over R3 even though half of the false positive errors committed by R3 have already been accounted for by the preceding rule, R1. This example shows that the effective accuracy after adding R2 or R3 to the rule set becomes evident only when both positive and negative examples covered by R1 are removed.
Figure 4.3. Elimination of training instances by the sequential covering algorithm. R1, R2, and R3 represent regions covered by three different rules.
Figure 4.3. Full Alternative Text
4.2.4 Indirect Methods for Rule Extraction This section presents a method for generating a rule set from a decision tree. In principle, every path from the root node to the leaf node of a decision tree can be expressed as a classification rule. The test conditions encountered along the path form the conjuncts of the rule antecedent, while the class label at the leaf node is assigned to the rule consequent. Figure 4.4 shows an example of a rule set generated from a decision tree. Notice that the rule set is exhaustive and contains mutually exclusive rules. However, some of the rules can be simplified as shown in the next example.
Figure 4.4. Converting a decision tree into classification rules.
Figure 4.4. Full Alternative Text
Example 4.3. Consider the following three rules from Figure 4.4:
r2:(P=No)∧(Q=Yes)→+r3:(P=Yes)∧(R=No)→+r5: (P=Yes)∧(R=Yes)∧(Q=Yes)→+.
Observe that the rule set always predicts a positive class when the value of Q is Yes. Therefore, we may simplify the rules as follows:
r2′:(Q=Yes)→+r3:(P=Yes)∧(R=No)→+.
r3 is retained to cover the remaining instances of the positive class. Although the rules obtained after simplification are no longer mutually exclusive, they are less complex and are easier to interpret.
In the following, we describe an approach used by the C4.5rules algorithm to generate a rule set from a decision tree. Figure 4.5 shows the decision tree and resulting classification rules obtained for the data set given in Table 4.2.
Rule Generation Classification rules are extracted for every path from the root to one of the leaf nodes in the decision tree. Given a classification rule r:A→y, we consider a simplified rule, r′:A′→y where A′ is obtained by removing one of the conjuncts in A. The simplified rule with the lowest pessimistic error rate is retained provided its error rate is less than that of the original rule. The rule-pruning step is repeated until the pessimistic error of the rule cannot be improved further. Because some of the rules may become identical after pruning, the duplicate rules are discarded.
Figure 4.5. Classification rules extracted from a decision tree for the vertebrate classification problem.
Figure 4.5. Full Alternative Text
Rule Ordering After generating the rule set, C4.5rules uses the class-based ordering scheme to order the extracted rules. Rules that predict the same class are grouped together into the same subset. The total description length for each subset is computed, and the classes are arranged in increasing order of their total description length. The class that has the smallest description length is given the highest priority because it is expected to contain the best set of rules. The total description length for a class is given by Lexception+g×Lmodel, where Lexception is the number of bits needed to encode the misclassified examples, Lmodel is the number of bits needed to encode the model, and g is a tuning parameter whose default value is 0.5. The tuning parameter depends on the number of redundant attributes present in the model. The value of the tuning parameter is small if the model contains many redundant attributes.
4.2.5 Characteristics of Rule-Based Classifiers
1. Rule-based classifiers have very similar characteristics as decision trees. The expressiveness of a rule set is almost equivalent to that of a decision tree because a decision tree can be represented by a set of mutually exclusive and exhaustive rules. Both rule-based and decision tree classifiers create rectilinear partitions of the attribute space and assign a class to each partition. However, a rule-based classifier can allow multiple rules to be triggered for a given instance, thus enabling the
learning of more complex models than decision trees.
2. Like decision trees, rule-based classifiers can handle varying types of categorical and continuous attributes and can easily work in multiclass classification scenarios. Rule-based classifiers are generally used to produce descriptive models that are easier to interpret but give comparable performance to the decision tree classifier.
3. Rule-based classifiers can easily handle the presence of redundant attributes that are highly correlated with one other. This is because once an attribute has been used as a conjunct in a rule antecedent, the remaining redundant attributes would show little to no FOIL's information gain and would thus be ignored.
4. Since irrelevant attributes show poor information gain, rule-based classifiers can avoid selecting irrelevant attributes if there are other relevant attributes that show better information gain. However, if the problem is complex and there are interacting attributes that can collectively distinguish between the classes but individually show poor information gain, it is likely for an irrelevant attribute to be accidentally favored over a relevant attribute just by random chance. Hence, rule- based classifiers can show poor performance in the presence of interacting attributes, when the number of irrelevant attributes is large.
5. The class-based ordering strategy adopted by RIPPER, which emphasizes giving higher priority to rare classes, is well suited for handling training data sets with imbalanced class distributions.
6. Rule-based classifiers are not well-suited for handling missing values in the test set. This is because the position of rules in a rule set follows a certain ordering strategy and even if a test instance is covered by multiple rules, they can assign different class labels depending on their position in the rule set. Hence, if a certain rule involves an attribute that is missing in a test instance, it is difficult to ignore the rule and proceed to the subsequent rules in the rule set, as such a strategy can result in incorrect class assignments.
4.3 Nearest Neighbor Classifiers The classification framework shown in Figure 3.3 involves a two-step process:
(1) an inductive step for constructing a classification model from data, and
(2) a deductive step for applying the model to test examples. Decision tree and rule-based classifiers are examples of eager learners because they are designed to learn a model that maps the input attributes to the class label as soon as the training data becomes available. An opposite strategy would be to delay the process of modeling the training data until it is needed to classify the test instances. Techniques that employ this strategy are known as lazy learners. An example of a lazy learner is the Rote classifier, which memorizes the entire training data and performs classification only if the attributes of a test instance match one of the training examples exactly. An obvious drawback of this approach is that some test instances may not be classified because they do not match any training example.
One way to make this approach more flexible is to find all the training examples that are relatively similar to the attributes of the test instances. These examples, which are known as nearest neighbors, can be used to determine the class label of the test instance. The justification for using nearest neighbors is best exemplified by the following saying: “If it walks like a duck, quacks like a duck, and looks like a duck, then it's probably a duck.” A nearest neighbor classifier represents each example as a data point in a d-dimensional space, where d is the number of attributes. Given a test instance, we compute its proximity to the training instances according to one of the proximity measures described in Section 2.4 on page 71. The k-nearest neighbors of a given test instance z refer to the k training examples that are closest to z.
Figure 4.6 illustrates the 1-, 2-, and 3-nearest neighbors of a test instance located at the center of each circle. The instance is classified based on the class labels of its neighbors. In the case where the neighbors have more than
one label, the test instance is assigned to the majority class of its nearest neighbors. In Figure 4.6(a), the 1-nearest neighbor of the instance is a negative example. Therefore the instance is assigned to the negative class. If the number of nearest neighbors is three, as shown in Figure 4.6(c), then the neighborhood contains two positive examples and one negative example. Using the majority voting scheme, the instance is assigned to the positive class. In the case where there is a tie between the classes (see Figure 4.6(b)), we may randomly choose one of them to classify the data point.
Figure 4.6. The 1-, 2-, and 3-nearest neighbors of an instance.
Figure 4.6. Full Alternative Text
The preceding discussion underscores the importance of choosing the right value for k. If k is too small, then the nearest neighbor classifier may be susceptible to overfitting due to noise, i.e., mislabeled examples in the training data. On the other hand, if k is too large, the nearest neighbor classifier may misclassify the test instance because its list of nearest neighbors includes training examples that are located far away from its neighborhood (see Figure 4.7).
Figure 4.7. k-nearest neighbor classification with large k.
4.3.1 Algorithm A high-level summary of the nearest neighbor classification method is given in Algorithm 4.2. The algorithm computes the distance (or similarity) between each test instance z=(x′, y′) and all the training examples (x, y)∈D to determine its nearest neighbor list, Dz. Such computation can be costly if the number of training examples is large. However, efficient indexing techniques are available to reduce the computation needed to find the nearest neighbors of a test instance.
Algorithm 4.2 The k-nearest neighbor classifier. 1: Let k be the number of nearest neighbors and D be the set of training examples.
2: for each test instance z=(x′, y′) do 3: Compute z=(x′, x), the distance between z and every example, 4: Select Dz⊆D, the set of k closest training examples to z. 5: y′=argmaxv∑(xi, yi)∈DzI(v=yi) 6: end for
Once the nearest neighbor list is obtained, the test instance is classified based on the majority class of its nearest neighbors:
Majority Voting: y′=argmaxv∑(xi, yi)∈DzI(v=yi), (4.5)
where v is a class label, yi is the class label for one of the nearest neighbors, and I(⋅) is an indicator function that returns the value 1 if its argument is true and 0 otherwise.
In the majority voting approach, every neighbor has the same impact on the classification. This makes the algorithm sensitive to the choice of k, as shown in Figure 4.6. One way to reduce the impact of k is to weight the influence of each nearest neighbor xi according to its distance: wi=1/d(x′, xi)2. As a result, training examples that are located far away from z have a weaker impact on the classification compared to those that are located close to z. Using the distance-weighted voting scheme, the class label can be determined as follows:
Distance-Weighted Voting: y′=argmaxv∑(xi, yi)∈Dzwi×I(v=yi). (4.6)
4.3.2 Characteristics of Nearest Neighbor Classifiers
1. Nearest neighbor classification is part of a more general technique known as instance-based learning, which does not build a global model, but rather uses the training examples to make predictions for a test instance. (Thus, such classifiers are often said to be “model free.”) Such algorithms require a proximity measure to determine the similarity or distance between instances and a classification function that returns the predicted class of a test instance based on its proximity to other
instances.
2. Although lazy learners, such as nearest neighbor classifiers, do not require model building, classifying a test instance can be quite expensive because we need to compute the proximity values individually between the test and training examples. In contrast, eager learners often spend the bulk of their computing resources for model building. Once a model has been built, classifying a test instance is extremely fast.
3. Nearest neighbor classifiers make their predictions based on local information. (This is equivalent to building a local model for each test instance.) By contrast, decision tree and rule-based classifiers attempt to find a global model that fits the entire input space. Because the classification decisions are made locally, nearest neighbor classifiers (with small values of k) are quite susceptible to noise.
4. Nearest neighbor classifiers can produce decision boundaries of arbitrary shape. Such boundaries provide a more flexible model representation compared to decision tree and rule-based classifiers that are often constrained to rectilinear decision boundaries. The decision boundaries of nearest neighbor classifiers also have high variability because they depend on the composition of training examples in the local neighborhood. Increasing the number of nearest neighbors may reduce such variability.
5. Nearest neighbor classifiers have difficulty handling missing values in both the training and test sets since proximity computations normally require the presence of all attributes. Although, the subset of attributes present in two instances can be used to compute a proximity, such an approach may not produce good results since the proximity measures may be different for each pair of instances and thus hard to compare.
6. Nearest neighbor classifiers can handle the presence of interacting attributes, i.e., attributes that have more predictive power taken in combination then by themselves, by using appropriate proximity measures that can incorporate the effects of multiple attributes together.
7. The presence of irrelevant attributes can distort commonly used
proximity measures, especially when the number of irrelevant attributes is large. Furthermore, if there are a large number of redundant attributes that are highly correlated with each other, then the proximity measure can be overly biased toward such attributes, resulting in improper estimates of distance. Hence, the presence of irrelevant and redundant attributes can adversely affect the performance of nearest neighbor classifiers.
8. Nearest neighbor classifiers can produce wrong predictions unless the appropriate proximity measure and data preprocessing steps are taken. For example, suppose we want to classify a group of people based on attributes such as height (measured in meters) and weight (measured in pounds). The height attribute has a low variability, ranging from 1.5 m to 1.85 m, whereas the weight attribute may vary from 90 lb. to 250 lb. If the scale of the attributes are not taken into consideration, the proximity measure may be dominated by differences in the weights of a person.
4.4 Naïve Bayes Classifier Many classification problems involve uncertainty. First, the observed attributes and class labels may be unreliable due to imperfections in the measurement process, e.g., due to the limited preciseness of sensor devices. Second, the set of attributes chosen for classification may not be fully representative of the target class, resulting in uncertain predictions. To illustrate this, consider the problem of predicting a person's risk for heart disease based on a model that uses their diet and workout frequency as attributes. Although most people who eat healthily and exercise regularly have less chance of developing heart disease, they may still be at risk due to other latent factors, such as heredity, excessive smoking, and alcohol abuse, that are not captured in the model. Third, a classification model learned over a finite training set may not be able to fully capture the true relationships in the overall data, as discussed in the context of model overfitting in the previous chapter. Finally, uncertainty in predictions may arise due to the inherent random nature of real-world systems, such as those encountered in weather forecasting problems.
In the presence of uncertainty, there is a need to not only make predictions of class labels but also provide a measure of confidence associated with every prediction. Probability theory offers a systematic way for quantifying and manipulating uncertainty in data, and thus, is an appealing framework for assessing the confidence of predictions. Classification models that make use of probability theory to represent the relationship between attributes and class labels are known as probabilistic classification models. In this section, we present the naïve Bayes classifier, which is one of the simplest and most widely-used probabilistic classification models.
4.4.1 Basics of Probability Theory Before we discuss how the naïve Bayes classifier works, we first introduce some basics of probability theory that will be useful in understanding the
probabilistic classification models presented in this chapter. This involves defining the notion of probability and introducing some common approaches for manipulating probability values.
Consider a variable X, which can take any discrete value from the set {x1, …, xk}. When we have multiple observations of that variable, such as in a data set where the variable describes some characteristic of data objects, then we can compute the relative frequency with which each value occurs. Specifically, suppose that X has the value xi for ni data objects. The relative frequency with which we observe the event X=xi is then ni/N, where N denotes the total number of occurrences (N=∑i=1kni). These relative frequencies characterize the uncertainty that we have with respect to what value X may take for an unseen observation and motivates the notion of probability.
More formally, the probability of an event e, e.g., P(X=xi), measures how likely it is for the event e to occur. The most traditional view of probability is based on relative frequency of events (frequentist), while the Bayesian viewpoint (described later) takes a more flexible view of probabilities. In either case, a probability is always a number between 0 and 1. Further, the sum of probability values of all possible events, e.g., outcomes of a variable X is equal to 1. Variables that have probabilities associated with each possible outcome (values) are known as random variables.
Now, let us consider two random variables, X and Y , that can each take k discrete values. Let nij be the number of times we observe X=xi and Y=yj, out of a total number of N occurrences. The joint probability of observing X=xi and Y=yj together can be estimated as
P(X=xi, Y=yi)=nijN. (4.7)
(This is an estimate since we typically have only a finite subset of all possible observations.) Joint probabilities can be used to answer questions such as “what is the probability that there will be a surprise quiz today AND I will be late for the class.” Joint probabilities are symmetric, i.e., P(X=x, Y=y)=P(Y=y, X=x). For joint probabilities, it is to useful to consider their sum with respect to one of the random variables, as described in the following equation:
∑j=1kP(X=xi,Y=yj)=∑j=1knijN=niN=P(X=xi), (4.8)
where ni is the total number of times we observe X=xi irrespective of the value of Y. Notice that ni/N is essentially the probability of observing X=xi. Hence, by summing out the joint probabilities with respect to a random variable Y , we obtain the probability of observing the remaining variable X. This operation is called marginalization and the probability value P(X=xi) obtained by marginalizing out Y is sometimes called the marginal probability of X. As we will see later, joint probability and marginal probability form the basic building blocks of a number of probabilistic classification models discussed in this chapter.
Notice that in the previous discussions, we used P(X=xi) to denote the probability of a particular outcome of a random variable X. This notation can easily become cumbersome when a number of random variables are involved. Hence, in the remainder of this section, we will use P(X) to denote the probability of any generic outcome of the random variable X, while P(xi) will be used to represent the probability of the specific outcome xi.
Bayes Theorem Suppose you have invited two of your friends Alex and Martha to a dinner party. You know that Alex attends 40% of the parties he is invited to. Further, if Alex is going to a party, there is an 80% chance of Martha coming along. On the other hand, if Alex is not going to the party, the chance of Martha coming to the party is reduced to 30%. If Martha has responded that she will be coming to your party, what is the probability that Alex will also be coming?
Bayes theorem presents the statistical principle for answering questions like the previous one, where evidence from multiple sources has to be combined with prior beliefs to arrive at predictions. Bayes theorem can be briefly described as follows.
Let P(Y|X) denotethe conditional probability of observing the random variable Y whenever the random variable X takes a particular value. P(Y|X) is
often read as the probability of observing Y conditioned on the outcome of X. Conditional probabilities can be used for answering questions such as “given that it is going to rain today, what will be the probability that I will go to the class.” Conditional probabilities of X and Y are related to their joint
probability in the following way:
P(Y|X)=P(X, Y)P(X), which implies (4.9) P(X, Y)=P(Y|X)×P(X)=P(X|Y)×P(Y). (4.10)
Rearranging the last two expressions in Equation 4.10 leads to Equation 4.11, which is known as Bayes theorem:
P(Y|X)=P(X|Y)P(Y)P(X). (4.11)
Bayes theorem provides a relationship between the conditional probabilities P(Y|X) and P(X|Y). Note that the denominator in Equation 4.11 involves the marginal probability of X, which can also be represented as
P(X)=∑i=1kP(X, yi)=∑i=1kP(X|yi)×P(yi).
Using the previous expression for P(X), we can obtain the following equation for P(Y|X) solely in terms of P(X|Y) and P(Y):
P(Y|X)=P(X|Y)P(Y)∑i−1kP(X|yi)P(yi). (4.12)
Example 4.4. [Bayes Theorem] Bayes theorem can be used to solve a number of inferential questions about random variables. For example, consider the problem stated at the beginning on inferring whether Alex will come to the party. Let P(A=1) denote the probability of Alex going to a party, while P(A=0) denotes the probability of him not going to a party. We know that
P(A=1)=0.4,andP(A=0)=1−P(A=1)=0.6.
Further, let P(M=1|A) denote the conditional probability of Martha going to a
party conditioned on whether Alex is going to the party. P(M=1|A) takes the following values:
P(M=1|A=1)=0.8,andP(M=1|A=0)=0.3.
We can use the above values of P(M|A) and P(A) to compute the probability of Alex going to the party given Martha is going to the party, P(A=1|M=1), as follows:
P(A=1|M=1)=P(M=1|A=1)P(A=1)P(M=1|A=0)P(A=0)+P(M=1|A=1)P(A=1),= (4.13)
Notice that even though the prior probability P(A) of Alex going to the party is low, the observation that Martha is going, M=1, affects the conditional probability P(A=1|M=1). This shows the value of Bayes theorem in combining prior assumptions with observed outcomes to make predictions. Since P(A=1|M=1)>0.5, it is more likely for Alex to join if Martha is going to the party.
Using Bayes Theorem for Classification For the purpose of classification, we are interested in computing the probability of observing a class label y for a data instance given its set of attribute values x. This can be represented as P(y|x), which is known as the posterior probability of the target class. Using the Bayes Theorem, we can represent the posterior probability as
P(y|x)=P(x|y)P(y)P(x) (4.14)
Note that the numerator of the previous equation involves two terms, P(x|y) and P(y), both of which contribute to the posterior probability P(y|x). We describe both of these terms in the following.
The first term P(x|y) is known as the class-conditional probability of the
attributes given the class label. P(x|y) measures the likelihood of observing x from the distribution of instances belonging to y. If x indeed belongs to class y, then we should expect P(x|y) to be high. From this point of view, the use of class-conditional probabilities attempts to capture the process from which the data instances were generated. Because of this interpretation, probabilistic classification models that involve computing class-conditional probabilities are known as generative classification models. Apart from their use in computing posterior probabilities and making predictions, class-conditional probabilities also provide insights about the underlying mechanism behind the generation of attribute values.
The second term in the numerator of Equation 4.14 is the prior probability P(y). The prior probability captures our prior beliefs about the distribution of class labels, independent of the observed attribute values. (This is the Bayesian viewpoint.) For example, we may have a prior belief that the likelihood of any person to suffer from a heart disease is α, irrespective of their diagnosis reports. The prior probability can either be obtained using expert knowledge, or inferred from historical distribution of class labels.
The denominator in Equation 4.14 involves the probability of evidence, P (x). Note that this term does not depend on the class label and thus can be treated as a normalization constant in the computation of posterior probabilities. Further, the value of P(x) can be calculated as P(x)=∑iP(x|yi)P(yi).
Bayes theorem provides a convenient way to combine our prior beliefs with the likelihood of obtaining the observed attribute values. During the training phase, we are required to learn the parameters for P(y) and P(x|y). The prior probability P(y) can be easily estimated from the training set by computing the fraction of training instances that belong to each class. To compute the class-conditional probabilities, one approach is to consider the fraction of training instances of a given class for every possible combination of attribute values. For example, suppose that there are two attributes X1 and X2 that can each take a discrete value from c1 to ck. Let n0 denote the number of training instances belonging to class 0, out of which nij0 number of training instances have X1=ci and X2=cj. The class-conditional probability can then be given as
P(X1=ci, X2=cj|Y=0)=nij0n0.
This approach can easily become computationally prohibitive as the number of attributes increase, due to the exponential growth in the number of attribute value combinations. For example, if every attribute can take k discrete values, then the number of attribute value combinations is equal to kd, where d is the number of attributes. The large number of attribute value combinations can also result in poor estimates of class-conditional probabilities, since every combination will have fewer training instances when the size of training set is small.
In the following, we present the naïve Bayes classifier, which makes a simplifying assumption about the class-conditional probabilities, known as the naïve Bayes assumption. The use of this assumption significantly helps in obtaining reliable estimates of class-conditional probabilities, even when the number of attributes are large.
4.4.2 Naïve Bayes Assumption The naïve Bayes classifier assumes that the class-conditional probability of all attributes x can be factored as a product of class-conditional probabilities of every attribute xi, as described in the following equation:
P(x|y)=∏i=1dP(xi|y), (4.15)
where every data instance x consists of d attributes, {x1, x2, …, xd}. The basic assumption behind the previous equation is that the attribute values xi are conditionally independent of each other, given the class label y. This means that the attributes are influenced only by the target class and if we know the class label, then we can consider the attributes to be independent of each other. The concept of conditional independence can be formally stated as follows.
Conditional Independence Let X1, X2,, and Y denote three sets of random variables. The variables in X1 are said to be conditionally independent of X2, given Y, if the following
condition holds:
P(X1|X2, Y)=P(X1|Y). (4.16)
This means that conditioned on Y, the distribution of X1 is not influenced by the outcomes of X2, and hence is conditionally independent of X2. To illustrate the notion of conditional independence, consider the relationship between a person's arm length (X1) and his or her reading skills (X2). One might observe that people with longer arms tend to have higher levels of reading skills, and thus consider X1 and X2 to be related to each other. However, this relationship can be explained by another factor, which is the age of the person (Y). A young child tends to have short arms and lacks the reading skills of an adult. If the age of a person is fixed, then the observed relationship between arm length and reading skills disappears. Thus, we can conclude that arm length and reading skills are not directly related to each other and are conditionally independent when the age variable is fixed.
Another way of describing conditional independence is to consider the joint conditional probability, P(X1, X2|Y), as follows:
P(X1, X2|Y)=P(X1, X2, Y)P(Y)=P(X1, X2, Y)P(X2, Y)×P(X2, Y)P(Y)=P(X1 (4.17)
where Equation 4.16 was used to obtain the last line of Equation 4.17. The previous description of conditional independence is quite useful from an operational perspective. It states that the joint conditional probability of X1 and X2 given Y can be factored as the product of conditional probabilities of X1 and X2 considered separately. This forms the basis of the naïve Bayes assumption stated in Equation 4.15.
How a Naïve Bayes Classifier Works Using the naïve Bayes assumption, we only need to estimate the conditional probability of each xi given Y separately, instead of computing the class-
conditional probability for every combination of attribute values. For example, if ni0 and nj0 denote the number of training instances belonging to class 0 with X1=ci and X2=cj, respectively, then the class-conditional probability can be estimated as
P(X1=ci, X2=xj|Y=0)=ni0n0×nj0n0.
In the previous equation, we only need to count the number of training instances for every one of the k values of an attribute X, irrespective of the values of other attributes. Hence, the number of parameters needed to learn class-conditional probabilities is reduced from dk to dk. This greatly simplifies the expression for the class-conditional probability and makes it more amenable to learning parameters and making predictions, even in high- dimensional settings.
The naïve Bayes classifier computes the posterior probability for a test instance x by using the following equation:
P(y|x)=P(y)∏i=1dP(xi|y)P(x) (4.18)
Since P (x) is fixed for every y and only acts as a normalizing constant to ensure that P(y|x)∈[0, 1], we can write
P(y|x)∝P(y)∏i=1dP(xi|y).
Hence, it is sufficient to choose the class that maximizes P(y)∏i=1dP(xi|y).
One of the useful properties of the naïve Bayes classifier is that it can easily work with incomplete information about data instances, when only a subset of attributes are observed at every instance. For example, if we only observe p out of d attributes at a data instance, then we can still compute P(y)∏i=1pP(xi|y) using those p attributes and choose the class with the maximum value. The naïve Bayes classifier can thus naturally handle missing values in test instances. In fact, in the extreme case where no attributes are observed, we can still use the prior probability P(y) as an estimate of the posterior probability. As we observe more attributes, we can keep refining the posterior probability to better reflect the likelihood of observing the data instance.
In the next two subsections, we describe several approaches for estimating the conditional probabilities P(xi|y) for categorical and continuous attributes from the training set.
Estimating Conditional Probabilities for Categorical Attributes For a categorical attribute Xi, the conditional probability P(Xi=c|y) is estimated according to the fraction of training instances in class y where Xi takes on a particular categorical value c.
P(Xi=c|y)=ncn,
where n is the number of training instances belonging to class y, out of which nc number of instances have Xi=c. For example, in the training set given in Figure 4.8, seven people have the class label Defaulted Borrower=No, out of which three people have Home Owner=Yes while the remaining four have Home Owner=No. As a result, the conditional probability for P(Home Owner=Yes|Defaulted Borrower=No) is equal to 3/7. Similarly, the conditional probability for defaulted borrowers with Marital Status=Single is given by P(Marital Status=Single|Defaulted Borrower=Yes)=2/3. Note that the sum of conditional probabilities over all possible outcomes of Xi is equal to one, i.e., ∑cP(Xi=c|y)=1,.
Figure 4.8. Training set for predicting the loan default problem.
Estimating Conditional Probabilities for Continuous Attributes There are two ways to estimate the class-conditional probabilities for continuous attributes:
1. We can discretize each continuous attribute and then replace the continuous values with their corresponding discrete intervals. This
approach transforms the continuous attributes into ordinal attributes, and the simple method described previously for computing the conditional probabilities of categorical attributes can be employed. Note that the estimation error of this method depends on the discretization strategy (as described in Section 2.3.6 on page 63), as well as the number of discrete intervals. If the number of intervals is too large, every interval may have an insufficient number of training instances to provide a reliable estimate of P(Xi|Y). On the other hand, if the number of intervals is too small, then the discretization process may loose information about the true distribution of continuous values, and thus result in poor predictions.
2. We can assume a certain form of probability distribution for the continuous variable and estimate the parameters of the distribution using the training data. For example, we can use a Gaussian distribution to represent the conditional probability of continuous attributes. The Gaussian distribution is characterized by two parameters, the mean, μ, and the variance, σ2. For each class yj, the class-conditional probability
for attribute Xi is
P(Xi=xi|Y=yj)=12πσijexp[−(xi−μij)22σij2 ]. (4.19)
The parameter μij can be estimated using the sample mean of Xi(x¯) for all training instances that belong to yj. Similarly, σij2 can be estimated from the sample variance (s2) of such training instances. For example, consider the annual income attribute shown in Figure 4.8. The sample mean and variance for this attribute with respect to the class No are
x¯=125+100+70+…+757=100s2=(125−110)2+(100−110)2+… (75−110)26=2975s=2975=54.54.
Given a test instance with taxable income equal to $120K, we can use the following value as its conditional probability given class No:
P(Income=120|No)=12π(54.54)exp−(120−110)22×2975=0.0072.
Example 4.5. [Naïve Bayes Classifier] Consider the data set shown in Figure 4.9(a), where the target class is Defaulted Borrower, which can take two values Yes and No. We can compute the class-conditional probability for each categorical attribute and the sample mean and variance for the continuous attribute, as summarized in Figure 4.9(b).
We are interested in predicting the class label of a test instance x= (Home Owner=No, Marital Status=Married, Annual Income=$120K). To do this, we first compute the prior probabilities by counting the number of training instances belonging to every class. We thus obtain P(yes)=0.3 and P(No)=0.7. Next, we can compute the class-conditional probability as follows:
Figure 4.9. The naïve Bayes classifier for the loan classification problem.
Figure 4.9. Full Alternative Text P(x|NO)=P(Home Owner=No|No)×P(Status=Married|No)×P(Annual Income=
Notice that the class-conditional probability for class Yes has become 0 because there are no instances belonging to class Yes with Status=Married in the training set. Using these class-conditional probabilities, we can estimate the posterior probabilities as
P(No|x)=0.7×0.0024P(x).=0.0016α.P(Yes|x)=0.3×0P(x)=0.
where α=1/P(x) is a normalizing constant. Since P(No|x)>P(Yes|x), the instance is classified as No.
Handling Zero Conditional Probabilities The preceding example illustrates a potential problem with using the naïve Bayes assumption in estimating class-conditional probabilities. If the conditional probability for any of the attributes is zero, then the entire expression for the class-conditional probability becomes zero. Note that zero conditional probabilities arise when the number of training instances is small and the number of possible values of an attribute is large. In such cases, it may happen that a combination of attribute values and class labels are never observed, resulting in a zero conditional probability.
In a more extreme case, if the training instances do not cover some combinations of attribute values and class labels, then we may not be able to even classify some of the test instances. For example, if P(Marital Status=Divorced|No) is zero instead of 1/7, then a data instance with attribute set x=
(Home Owner=Yes, Marital Status=Divorced, Income=$120K) has the following class-conditional probabilities:
P(x|No)=3/7×0×0.0072=0.P(x|Yes)=0×1/3×1.2×10−9=0.
Since both the class-conditional probabilities are 0, the naïve Bayes classifier will not be able to classify the instance. To address this problem, it is important to adjust the conditional probability estimates so that they are not as brittle as simply using fractions of training instances. This can be achieved by using the following alternate estimates of conditional probability:
Laplace estimate:P(Xi=c|y)=nc+1n+v, (4.20) m-estimate:P(Xi=c|y)=nc+mpn+m, (4.21)
where n is the number of training instances belonging to class y, nc is the number of training instances with Xi=c and Y=y, v is the total number of attribute values that Xi can take, p is some initial estimate of P(Xi=c|y) that is known a priori, and m is a hyper-parameter that indicates our confidence in using p when the fraction of training instances is too brittle. Note that even if nc=0, both Laplace and m-estimate provide non-zero values of conditional probabilities. Hence, they avoid the problem of vanishing class-conditional probabilities and thus generally provide more robust estimates of posterior probabilities.
Characteristics of Naïve Bayes Classifiers
1. Naïve Bayes classifiers are probabilistic classification models that are able to quantify the uncertainty in predictions by providing posterior probability estimates. They are also generative classification models as they treat the target class as the causative factor for generating the data instances. Hence, apart from computing posterior probabilities, naïve Bayes classifiers also attempt to capture the underlying mechanism behind the generation of data instances belonging to every class. They are thus useful for gaining predictive as well as descriptive insights.
2. By using the naïve Bayes assumption, they can easily compute class- conditional probabilities even in high-dimensional settings, provided that the attributes are conditionally independent of each other given the class labels. This property makes naïve Bayes classifier a simple and effective classification technique that is commonly used in diverse application problems, such as text classification.
3. Naïve Bayes classifiers are robust to isolated noise points because such points are not able to significantly impact the conditional probability estimates, as they are often averaged out during training.
4. Naïve Bayes classifiers can handle missing values in the training set by ignoring the missing values of every attribute while computing its conditional probability estimates. Further, naïve Bayes classifiers can effectively handle missing values in a test instance, by using only the non-missing attribute values while computing posterior probabilities. If the frequency of missing values for a particular attribute value depends on class label, then this approach will not accurately estimate posterior probabilities.
5. Naïve Bayes classifiers are robust to irrelevant attributes. If Xi is an irrelevant attribute, then P(Xi|Y) becomes almost uniformly distributed for every class y. The class-conditional probabilities for every class thus receive similar contributions of P(Xi|Y), resulting in negligible impact on the posterior probability estimates.
6. Correlated attributes can degrade the performance of naïve Bayes classifiers because the naïve Bayes assumption of conditional independence no longer holds for such attributes. For example, consider the following probabilities:
P(A=0|Y=0)=0.4,P(A=1|Y=0)=0.6,P(A=0|Y=1)=0.6,P(A=1|Y=1)=0.4,
where A is a binary attribute and Y is a binary class variable. Suppose there is another binary attribute B that is perfectly correlated with A when Y=0, but is independent of A when Y=1. For simplicity, assume that the conditional probabilities for B are the same as for A. Given an instance with attributes A=0, B=0, and assuming conditional
independence, we can compute its posterior probabilities as follows:
P(Y=0|A=0, B=0)=P(A=0|Y=0)P(B=0|Y=0)P(Y=0)P(A=0, B=0)=0.16×P
If P(Y=0)=P(Y=1), then the naïve Bayes classifier would assign the instance to class 1. However, the truth is,
P(A=0, B=0|Y=0)=P(A=0|Y=0)=0.4,
because A and B are perfectly correlated when Y=0. As a result, the posterior probability for Y=0 is
P(Y=0|A=0, B=0)=P(A=0, B=0|Y=0)P(Y=0)P(A=0, B=0)=0.4×P(Y=0)P(A=0
which is larger than that for Y=1. The instance should have been classified as class 0. Hence, the naïve Bayes classifier can produce incorrect results when the attributes are not conditionally independent given the class labels. Naïve Bayes classifiers are thus not well-suited for handling redundant or interacting attributes.
4.5 Bayesian Networks The conditional independence assumption made by naïve Bayes classifiers may seem too rigid, especially for classification problems where the attributes are dependent on each other even after conditioning on the class labels. We thus need an approach to relax the naïve Bayes assumption so that we can capture more generic representations of conditional independence among attributes.
In this section, we present a flexible framework for modeling probabilistic relationships between attributes and class labels, known as Bayesian Networks. By building on concepts from probability theory and graph theory, Bayesian networks are able to capture more generic forms of conditional independence using simple schematic representations. They also provide the necessary computational structure to perform inferences over random variables in an efficient way. In the following, we first describe the basic representation of a Bayesian network, and then discuss methods for performing inference and learning model parameters in the context of classification.
4.5.1 Graphical Representation Bayesian networks belong to a broader family of models for capturing probabilistic relationships among random variables, known as probabilistic graphical models. The basic concept behind these models is to use graphical representations where the nodes of the graph correspond to random variables and the edges between the nodes express probabilistic relationships. Figures 4.10(a) and 4.10(b) show examples of probabilistic graphical models using directed edges (with arrows) and undirected edges (without arrows), respectively. Directed graphical models are also known as Bayesian networks while undirected graphical models are known as Markov random fields. The two approaches use different semantics for expressing relationships among random variables and are thus useful in different contexts. In the
following, we briefly describe Bayesian networks that are useful in the context of classification.
A Bayesian network (also referred to as a belief network) involves directed edges between nodes, where every edge represents a direction of influence among random variables. For example, Figure 4.10(a) shows a Bayesian network where variable C depends upon the values of variables A and B, as indicated by the arrows pointing toward C from A and B. Consequently, the variable C influences the values of variables D and E. Every edge in a Bayesian network thus encodes a dependence relationship between random variables with a particular directionality.
Figure 4.10. Illustrations of two basic types of graphical models.
Figure 4.10. Full Alternative Text
Bayesian networks are directed acyclic graphs (DAG) because they do not contain any directed cycles such that the influence of a node loops back to the same node. Figure 4.11 shows some examples of Bayesian networks that capture different types of dependence structures among random variables. In a directed acyclic graph, if there is a directed edge from X to Y ,then X is called the parent of Y and Y is called the child of X. Note that a node can have multiple parents in a Bayesian network, e.g., node D has two parent nodes, B and C, in Figure 4.11(a). Furthermore, if there is a directed path in the network from X to Z, then X is an ancestor of Z, while Z is a descendant of X. For example, in the diagram shown in Figure 4.11(b), A is a descendant of D and D is an ancestor of B. Note that there can be multiple directed paths between two nodes of a directed acyclic graph, as is the case for nodes A and D in Figure 4.11(a).
Figure 4.11. Examples of Bayesian networks.
Conditional Independence An important property of a Bayesian network is its ability to represent
varying forms of conditional independence among random variables. There are several ways of describing the conditional independence assumptions captured by Bayesian networks. One of the most generic ways of expressing conditional independence is the concept of d-separation, which can be used to determine if any two sets of nodes A and B are conditionally independent given another set of nodes C. Another useful concept is that of the Markov blanket of a node Y , which denotes the minimal set of nodes X that makes Y independent of the other nodes in the graph, when conditioned on X. (See Bibliographic Notes for more details on d-separation and Markov blanket.) However, for the purpose of classification, it is sufficient to describe a simpler expression of conditional independence in Bayesian networks, known as the local Markov property.
Property 1 (Local Markov Property). A node in a Bayesian network is conditionally independent of its non- descendants, if its parents are known.
To illustrate the local Markov property, consider the Bayes network shown in Figure 4.11(b). We can state that A is conditionally independent of both B and D given C, because C is the parent of A and nodes B and D are non- descendants of A. The local Markov property helps in interpreting parent- child relationships in Bayesian networks as representations of conditional probabilities. Since a node is conditionally independent of its non- descendants given it parents, the conditional independence assumptions imposed by a Bayesian network is often sparse in structure. Nonetheless, Bayesian networks are able to express a richer class of conditional independence statements among attributes and class labels than the naïve Bayes classifier. In fact, the naïve Bayes classifier can be viewed as a special type of Bayesian network, where the target class Y is at the root of a tree and every attribute Xi is connected to the root node by a directed edge, as shown in Figure 4.12(a).
Figure 4.12. Comparing the graphical representation of a naïve Bayes classifier with that of a generic Bayesian network.
Figure 4.12. Full Alternative Text
Note that in a naïve Bayes classifier, every directed edge points from the target class to the observed attributes, suggesting that the class label is a factor behind the generation of attributes. Inferring the class label can thus be viewed as diagnosing the root cause behind the observed attributes. On the other hand, Bayesian networks provide a more generic structure of probabilistic relationships, since the target class is not required to be at the root of a tree but can appear anywhere in the graph, as shown in Figure 4.12(b). In this diagram, inferring Y not only helps in diagnosing the factors influencing X3 and X4, but also helps in predicting the influence of X1 and X2.
Joint Probability The local Markov property can be used to succinctly express the joint
probability of the set of random variables involved in a Bayesian network. To realize this, let us first consider a Bayesian network consisting of d nodes, X1 to Xd, where the nodes have been numbered in such a way that Xi is an ancestor of Xj only if i<j. The joint probability of X={X1, …, Xd} can be generically factorized using the chain rule of probability as
P(X)=P(X1)P(X2|X1)P(X3|X1, X2) … P(Xd|X1, … Xd −1)=∏i=1dP(Xi|X1, … Xi−1) (4.22)
By the way we have constructed the graph, note that the set {X1, … Xi−1 } contains only non-descendants of Xi. Hence, by using the local Markov property, we can write P(Xi|X1, … Xi−1) as P(Xi|pa(Xi)), where pa(Xi) denotes the parents of Xi. The joint probability can then be represented as
P(X)=∏i=1dP(Xi|pa(Xi)) (4.23)
It is thus sufficient to represent the probability of every node Xi in terms of its parent nodes, pa(Xi), for computing P(x). This is achieved with the help of probability tables that associate every node to its parent nodes as follows:
1. The probability table for node Xi contains the conditional probability values P(Xi|pa(Xi)) for every combination of values in Xi and pa(Xi).
2. If Xi has no parents (pa(Xi)=ϕ), then the table contains only the prior probability P(Xi).
Example 4.6. [Probability Tables] Figure 4.13 shows an example of a Bayesian network for modeling the relationships between a patient's symptoms and risk factors. The probability tables are shown at the side of every node in the figure. The probability tables associated with the risk factors (Exercise and Diet) contain only the prior probabilities, whereas the tables for heart disease, heartburn, blood pressure, and chest pain, contain the conditional probabilities.
Figure 4.13. A Bayesian network for detecting heart disease and heartburn in patients.
Figure 4.13. Full Alternative Text
Use of Hidden Variables A Bayesian network typically involves two types of variables: observed variables that are clamped to specific observed values, and unobserved variables, whose values are not known and need to be inferred from the
network. To distinguish between these two types of variables, observed variables are generally represented using shaded nodes while unobserved variables are represented using empty nodes. Figure 4.14 shows an example of a Bayesian network with observed variables (A, B, and E ) and unobserved variables (C and D).
Figure 4.14. Observed and unobserved variables are represented using unshaded and shaded circles, respectively.
In the context of classification, the observed variables correspond to the set of attributes X, while the target class is represented using an unobserved variable Y that needs to be inferred during testing. However, note that a generic Bayesian network may contain many other unobserved variables apart from the target class, as represented in Figure 4.15 as the set of variables H. These unobserved variables represent hidden or confounding factors that affect the probabilities of attributes and class labels, although they are never directly observed. The use of hidden variables enhances the expressive power of Bayesian networks in representing complex probabilistic relationships between attributes and class labels. This is one of the key distinguishing properties of Bayesian networks as compared to naïve Bayes classifiers.
4.5.2 Inference and Learning Given the probability tables corresponding to every node in a Bayesian network, the problem of inference corresponds to computing the probabilities of different sets of random variables. In the context of classification, one of the key inference problems is to compute the probability of a target class Y taking on a specific value y, given the set of observed attributes at a data instance, x. This can be represented using the following conditional probability:
P(Y=y|x)=(y, x)P(x)=(y, x)∑y′P(y′, x) (4.24)
The previous equation involves marginal probabilities of the form P(y, x). They can be computed by marginalizing out the hidden variables H from the joint probability as follows:
P(y, x)=∑HP(y, x, H), (4.25)
where the joint probability P(y, x, H) can be obtained by using the factorization described in Equation 4.23. To understand the nature of computations involved in estimating P(y, x), consider the example Bayesian network shown in Figure 4.15, which involves a target class, Y , three observed attributes, X1 to X3, and four hidden variables, H1 to H4. For this network, we can express P(y, x) as
Figure 4.15. An example of a Bayesian network with four hidden variables, H1 to H4, three observed attributes, X1 to X3, and one target class Y .
P(y, x)=∑h1∑h2∑h3∑h4P(y, x1, x2, h1, h2, h3, h4),=∑h1∑h2∑h3∑h4 [P(h1)P(h2)P(x2)P(h4)P(x1|h1, h2) ×P(h3|x2, h2)P(y|x1, h3)P(x3|h3, h4) ], (4.26) =∑h1∑h2∑h3∑h4f(h1, h2, h3, h4), (4.27)
where f is a factor that depends on the values of h1 to h4. In the previous simplistic expression of P(y, x), a different summand is considered for every combination of values, h1 to h4, in the hidden variables, H1 to H4. If we assume that every variable in the network can take k discrete values, then the summation has to be carried out for a total number of k4 times. The computational complexity of this approach is thus O(k4). Moreover, the number of computations grows exponentially with the number of hidden variables, making it difficult to use this approach with networks that have a large number of hidden variables. In the following, we present different computational techniques for efficiently performing inferences in Bayesian networks.
Variable Elimination To reduce the number of computations involved in estimating P(y, x), let us closely examine the expressions in Equations 4.26 and 4.27. Notice that although f(h1, h2, h3, h4) depends on the values of all four hidden variables, it can be decomposed as a product of several smaller factors, where every factor involves only a small number of hidden variables. For example, the factor P(h4) depends only on the value of h4, and thus acts as a constant multiplicative term when summations are performed over h1, h2, or h3. Hence, if we place P(h4) outside the summations of h1 to h3, we can save some repeated multiplications occurring inside every summand.
In general, we can push every summation as far inside as possible, so that the
factors that do not depend on the summing variable are placed outside the summation. This will help reduce the number of wasteful computations by using smaller factors at every summation. To illustrate this process, consider the following sequence of steps for computing P(y, x), by rearranging the order
of summations in Equation 4.26.
P(y, x)=P(x2)∑h4P(h4)∑h3P(y|x1, h3)P(x3|h3, h4)×∑h2P(h2)P(h3|x2, h2)∑h1 (4.28) =P(x2)∑h4P(h4)∑h3P(y|x1, h3)P(x3|h3, h4)×∑h2P(h2)P(h3|x2, h2)f1(h2) (4.29) =P(x2)∑h4P(h4)∑h3P(y|x1, h3)P(x3|h3, h4)f2(h3) (4.30) =P(x2)∑h4P(h4)f3(h4) (4.31)
where fi represents the intermediate factor term obtained by summing out hi. To check if the previous rearrangements provide any improvements in computational efficiency, let us count the number of computations occurring at every step of the process. At the first step (Equation 4.28), we perform a summation over h1 using factors that depend on h1 and h2. This requires considering every pair of values in h1 and h2, resulting in O(k2) computations. Similarly, the second step (Equation 4.29) involves summing out h2 using factors of h2 and h3, leading to O(k2) computations. The third step (Equation 4.30) again requires O(k2) computations as it involves summing out h3 over factors depending on h3 and h4. Finally, the fourth step (Equation 4.31) involves summing out h4 using factors depending on h4, resulting in O(k) computations.
The overall complexity of the previous approach is thus O(k2), which is considerably smaller than the O(k4) complexity of the basic approach. Hence, by merely rearranging summations and using algebraic manipulations, we are able to improve the computational efficiency in computing P(y, x). This procedure is known as variable elimination.
The basic concept that variable elimination exploits to reduce the number of computations is the distributive nature of multiplication over addition operations. For example, consider the following multiplication and addition operations:
a.(b+c+d)=a.b+a.c+a.d
Notice that the right-hand side of the previous equation involves three multiplications and three additions, while the left-hand side involves only one multiplication and three additions, thus saving on two arithmetic operations. This property is utilized by variable elimination in pushing out constant terms outside the summation, such that they are multiplied only once.
Note that the efficiency of variable elimination depends on the order of hidden variables used for performing summations. Hence, we would ideally like to find the optimal order of variables that result in the smallest number of computations. Unfortunately, finding the optimal order of summations for a generic Bayesian network is an NP-Hard problem, i.e., there does not exist an efficient algorithm for finding the optimal ordering that can run in polynomial time. However, there exists efficient techniques for handling special types of Bayesian networks, e.g., those involving tree-like graphs, as described in the following.
Sum-Product Algorithm for Trees Note that in Equations 4.28 and 4.29, whenever a variable hi is eliminated during marginalization, it results in the creation of a factor fi that depends on the neighboring nodes of hi. fi is then absorbed in the factors of neighboring variables and the process is repeated until all unobserved variables have marginalized. This phenomena of variable elimination can be viewed as transmitting a local message from the variable being marginalized to its neighboring nodes. This idea of message passing utilizes the structure of the graph for performing computations, thus making it possible to use graph- theoretic approaches for making effective inferences. The sum-product algorithm builds on the concept of message passing for computing marginal and conditional probabilities on tree-based graphs.
Figure 4.16 shows an example of a tree involving five variables, X1 to X5. A key characteristic of a tree is that every node in the tree has exactly one parent, and there is only one directed edge between any two nodes in the tree. For the purpose of illustration, let us consider the problem of estimating the
marginal probability of X2, P(X2). This can be obtained by marginalizing out every variable in the graph except X2 and rearranging the summations to obtain the following expression:
Figure 4.16. An example of a Bayesian network with a tree structure.
P(x2)=∑x1∑x3∑x4∑x5P(x1)P(x2|x1)P(x3|x2)P(x4|x3)P(x5|x3),= (∑x1P(x1)P(x2|x1))︸m12(x2)(∑x3P(x3|x2)(∑x4P(x4|x3))︸m43(x3) (∑x5P(x5|x3))︸m53(x3)),︸m32(x2)
where mij(xj) has been conveniently chosen to represent the factor of xj that is obtained by summing out xi. We can view mij(xj) as a local message passed from node xi to node xj, as shown using arrows in Figure 4.17(a). These local messages capture the influence of eliminating nodes on the marginal probabilities of neighboring nodes.
Before we formally describe the formula for computing mij(xj) and P(xj), we first define a potential function ψ(⋅) that is associated every node and edge of
the graph. We can define the potential of a node Xi as
ψ(Xi)={P(Xi),if Xi is the root node.1,otherwise. (4.32)
Figure 4.17. Illustration of message passing in the sum-product algorithm.
Figure 4.17. Full Alternative Text
Similarly, we can define the potential of an edge between nodes Xi and Xj (where Xi is the parent of Xj) as
ψ(Xi, Xj)=P(Xj|Xi).
Using ψ(Xi) and ψ(Xi, Xj), we can represent mij(xj) using the following
equation:
mij(xj)=∑xi(ψ(xi)ψ(xi, xj)∏k∈N(i)\imki(xi)), (4.33)
where N(i) represents the set of neighbors of node Xi. The message mij that is transmitted from Xi to Xj can thus be recursively computed using the messages incident on Xi from its neighboring nodes excluding Xi. Note that the formula for mij involves taking a sum over all possible values of Xj, after multiplying the factors obtained from the neighbors of Xj. This approach of message passing is thus called the “sum-product” algorithm. Further, since mij represents a notion of “belief” propagated from Xi to Xj, this algorithm is also known as belief propagation. The marginal probability of a node Xi
is then given as
P(xi)=ψ(xi)∏j∈N(i)mji(xi). (4.34)
A useful property of the sum-product algorithm is that it allows the messages to be reused for computing a different marginal probability in the future. For example, if we had to compute the marginal probability for node X3, we would require the following messages from its neighboring nodes: m23(x3), m43(x3), and m53(x3). However, note that m43(x3), and m53(x3) have already been computed in the process of computing the marginal probability of X2 and thus can be reused.
Notice that the basic operations of the sum-product algorithm resemble a message passing protocol over the edges of the network. A node sends out a message to all its neighboring nodes only after it has received incoming messages from all its neighbors. Hence, we can initialize the message passing protocol from the leaf nodes, and transmit messages till we reach the root node. We can then run a second pass of messages from the root node back to the leaf nodes. In this way, we can compute the messages for every edge in both directions, using just O(2|E|) operations, where |E| is the number of edges. Once we have transmitted all possible messages as shown in Figure 4.17(b), we can easily compute the marginal probability of every node in the graph using Equation 4.34.
In the context of classification, the sum-product algorithm can be easily
modified for computing the conditional probability of the class label y given the set of observed attributes x^, i.e., P(y|x^). This basically amounts to computing P(y, X=x^) in Equation 4.24, where X is clamped to the observed values x^. To handle the scenario where some of the random variables are fixed and do not need to be normalized, we consider the following modification.
If Xi is a random variable that is fixed to a specific value x^i, then we can simply modify ψ(Xi) and ψ(Xi, Xj) as follows:
ψ(Xi)={1,if Xi=x^i.0,otherwise. (4.35) ψ(Xi, Xj)={P(Xi|x^i),if Xi=x^i.0,otherwise. (4.36)
We can run the sum-product algorithm using these modified values for every observed variable and thus compute P(y, X=x^).
Figure 4.18.
Example of a poly-tree and its corresponding factor graph.
Figure 4.18. Full Alternative Text
Generalizations for Non-Tree Graphs The sum-product algorithm is guaranteed to optimally converge in the case of trees using a single run of message passing in both directions of every edge. This is because any two nodes in a tree have a unique path for the transmission of messages. Furthermore, since every node in a tree has a single parent, the joint probability involves only factors of at most two variables. Hence, it is sufficient to consider potentials over edges and not other generic substructures in the graph.
Both of the previous properties are violated in graphs that are not trees, thus making it difficult to directly apply the sum-product algorithm for making inferences. However, a number of variants of the sum-product algorithm have been devised to perform inferences on a broader family of graphs than trees. Many of these variants transform the original graph into an alternative tree- based representation, and then apply the sum-product algorithm on the transformed tree. In this section, we briefly discuss one such transformations known as factor graphs.
Factor graphs are useful for making inferences over graphs that violate the condition that every node has a single parent. Nonetheless, they still require the absence of multiple paths between any two nodes, to guarantee convergence. Such graphs are known as poly-trees. An example of a poly- tree is shown in Figure 4.18(a).
A poly-tree can be transformed into a tree-based representation with the help of factor graphs. These graphs consist of two types of nodes, variables nodes (that are represented using circles) and factor nodes (that are represented using squares). The factor nodes represent conditional independence relationships among the variables of the poly-tree. In particular, every
probability table can be represented as a factor node. The edges in a factor graph are undirected in nature and relate a variable node to a factor node if the variable is involved in the probability table corresponding to the factor node. Figure 4.18(b) presents the factor graph representation of the poly-tree shown in Figure 4.18(a).
Note that the factor graph of a poly-tree always forms a tree-like structure, where there is a unique path of influence between any two nodes in the factor graph. Hence, we can apply a modified form of sum-product algorithm to transmit messages between variable nodes and factor nodes, which is guaranteed to converge to optimal values.
Learning Model Parameters In all our previous discussions on Bayesian networks, we had assumed that the topology of the Bayesian network and the values in the probability tables of every node were already known. In this section, we discuss approaches for learning both the topology and the probability table values of a Bayesian network from the training data.
Let us first consider the case where the topology of the network is known and we are only required to compute the probability tables. If there are no unobserved variables in the training data, then we can easily compute the probability table for P(Xi|pa(Xi)), by counting the fraction of training instances for every value of Xi and every combination of values in pa(Xi). However, if there are unobserved variables in Xi or pa(Xi), then computing the fraction of training instances for such variables is non-trivial and requires the use of advances techniques such as the Expectation-Maximization algorithm (described later in Chapter 8).
Learning the structure of the Bayesian network is a much more challenging task than learning the probability tables. Although there are some scoring approaches that attempt to find a graph structure that maximizes the training likelihood, they are often computationally infeasible when the graph is large. Hence, a common approach for constructing Bayesian networks is to use the subjective knowledge of domain experts.
4.5.3 Characteristics of Bayesian Networks
1. Bayesian networks provide a powerful approach for representing probabilistic relationships between attributes and class labels with the help of graphical models. They are able to capture complex forms of dependencies among variables. Apart from encoding prior beliefs, they are also able to model the presence of latent (unobserved) factors as hidden variables in the graph. Bayesian networks are thus quite expressive and provide predictive as well as descriptive insights about the behavior of attributes and class labels.
2. Bayesian networks can easily handle the presence of correlated or redundant attributes, as opposed to the naïve Bayes classifier. This is because Bayesian networks do not use the naïve Bayes assumption about conditional independence, but instead are able to express richer forms of conditional independence.
3. Similar to the naïve Bayes classifier, Bayesian networks are also quite robust to the presence of noise in the training data. Further, they can handle missing values during training as well as testing. If a test instance contains an attribute Xi with a missing value, then a Bayesian network can perform inference by treating Xi as an unobserved node and marginalizing out its effect on the target class. Hence, Bayesian networks are well-suited for handling incompleteness in the data, and can work with partial information. However, unless the pattern with which missing values occurs is completely random, then their presence will likely introduce some degree of error and/or bias into the analysis.
4. Bayesian networks are robust to irrelevant attributes that contain no discriminatory information about the class labels. Such attributes show no impact on the conditional probability of the target class, and are thus rightfully ignored.
5. Learning the structure of a Bayesian network is a cumbersome task that
often requires assistance from expert knowledge. However, once the structure has been decided, learning the parameters of the network can be quite straightforward, especially if all the variables in the network are observed.
6. Due to its additional ability of representing complex forms of relationships, Bayesian networks are more susceptible to overfitting as compared to the naïve Bayes classifier. Furthermore, Bayesian networks typically require more training instances for effectively learning the probability tables than the naïve Bayes classifier.
7. Although the sum-product algorithm provides computationally efficient techniques for performing inference over tree-like graphs, the complexity of the approach increase significantly when dealing with generic graphs of large sizes. In situations where exact inference is computationally infeasible, it is quite common to use approximate inference techniques.
4.6 Logistic Regression The naïve Bayes and the Bayesian network classifiers described in the previous sections provide different ways of estimating the conditional probability of an instance x given class y, P(x|y). Such models are known as probabilistic generative models. Note that the conditional probability P(x|y) essentially describes the behavior of instances in the attribute space that are generated from class y. However, for the purpose of making predictions, we are finally interested in computing the posterior probability P(y|x). For example, computing the following ratio of posterior probabilities is sufficient for inferring class labels in a binary classification problem:
P(y=1|x)P(y=0|x)
This ratio is known as the odds. If this ratio is greater than 1, then x is classified as y=1. Otherwise, it is assigned to class y=0. Hence, one may simply learn a model of the odds based on the attribute values of training instances, without having to compute P(x|y) as an intermediate quantity in the Bayes theorem.
Classification models that directly assign class labels without computing class-conditional probabilities are called discriminative models. In this section, we present a probabilistic discriminative model known as logistic regression, which directly estimates the odds of a data instance x using its attribute values. The basic idea of logistic regression is to use a linear predictor, z=wTx+b, for representing the odds of x as follows:
P(y=1|x)P(y=0|x)=ez=ewTx+b, (4.37)
where w and b are the parameters of the model and aT denotes the transpose of a vector a. Note that if wTx+b>0, then x belongs to class 1 since its odds is greater than 1. Otherwise, x belongs to class 0.
Figure 4.19. Plot of sigmoid (logistic) function, σ(z).
Since P(y=0|x)+P(y=1|x)=1, we can re-write Equation 4.37 as
P(y=1|x)1−P(y=1|x)=ez.
This can be further simplified to express P(y=1|x) as a function of z.
P(y=1|x)=11+e−z=σ(z), (4.38)
where the function σ(⋅) is known as the logistic or sigmoid function. Figure 4.19 shows the behavior of the sigmoid function as we vary z. We can see that σ(z)≥0.5 only when z≥0. We can also derive P(y=0|x) using σ(z) as follows:
P(y=0|x)=1−σ(z)=11+e−z (4.39)
Hence, if we have learned a suitable value of parameters w and b, we can use Equations 4.38 and 4.39 to estimate the posterior probabilities of any data instance x and determine its class label.
4.6.1 Logistic Regression as a Generalized Linear Model Since the posterior probabilities are real-valued, their estimation using the previous equations can be viewed as solving a regression problem. In fact, logistic regression belongs to a broader family of statistical regression models, known as generalized linear models (GLM). In these models, the target variable y is considered to be generated from a probability distribution P(y|x), whose mean μ can be estimated using a link function g(⋅) as follows:
g(μ)=z=wT x + b. (4.40)
For binary classification using logistic regression, y follows a Bernoulli distribution (y can either be 0 or 1) and μ is equal to P(y=1|x). The link function g(⋅) of logistic regression, called the logit function, can thus be represented as
g(μ)=log(μ1−μ).
Depending on the choice of link function g(⋅) and the form of probability distribution P(y|x), GLMs are able to represent a broad family of regression models, such as linear regression and Poisson regression. They require different approaches for estimating their model parameters, (w, b). In this chapter, we will only discuss approaches for estimating the model parameters of logistic regression, although methods for estimating parameters of other types of GLMs are often similar (and sometimes even simpler). (See Bibliographic Notes for more details on GLMs.)
Note that even though logistic regression has relationships with regression models, it is a classification model since the computed posterior probabilities are eventually used to determine the class label of a data instance.
4.6.2 Learning Model Parameters
The parameters of logistic regression, (w, b), are estimated during training using a statistical approach known as the maximum likelihood estimation (MLE) method. This method involves computing the likelihood of observing the training data given (w, b), and then determining the model parameters (w*, b*) that yield maximum likelihood.
Let D.train={(x1, y1), (x2, y2), … , (xn, yn)} denote a set of n training instances, where yi is a binary variable (0 or 1). For a given training instance xi, we can compute its posterior probabilities using Equations 4.38 and 4.39. We can then express the likelihood of observing yi given xi, w, and b as
P(yi|xi, w, b)=P(y=1|xi)yi×P(y=0|xi)1−yi,=(σ(zi))yi×(1−σ(zi))1−yi,= (σ(wTxi+b))yi×(1−σ(wTxi+b))1−yi, (4.41)
where σ(⋅) is the sigmoid function as described above, Equation 4.41 basically means that the likelihood P(yi|xi, w, b) is equal to P(y=1|xi) when yi=1, and equal to P(y=0|xi) when yi=0. The likelihood of all training instances, L(w, b), can then be computed by taking the product of individual likelihoods (assuming independence among training instances) as follows:
L(w, b)=∏i=1nP(yi|xi, w, b)=∏i=1nP(y=1|xi)yi×P(y=0|xi)1−yi. (4.42)
The previous equation involves multiplying a large number of probability values, each of which are smaller than or equal to 1. Since this naïve computation can easily become numerically unstable when n is large, a more practical approach is to consider the negative logarithm (to base e) of the likelihood function, also known as the cross entropy function:
−logL(w, b)=−∑i=1nyilog(P(y=1|xi))+(1−yi)log(P(y=0|xi)).= −∑i=1nyilog(σ(wTxi+b))+(1−yi)log(1−σ(wTxi+b)).
The cross entropy is a loss function that measures how unlikely it is for the training data to be generated from the logistic regression model with parameters (w, b). Intuitively, we would like to find model parameters (w*, b*) that result in the lowest cross entropy, −logL(w*, b*).
(w*, b*)=argmin(w, b)E(w, b)=argmin(w, b)−logL(w, b) (4.43)
where E(w, b)=−logL(w, b) is the loss function. It is worth emphasizing that E(w, b) is a convex function, i.e., any minima of E(w, b) will be a global minima. Hence, we can use any of the standard convex optimization techniques to solve Equation 4.43, which are mentioned in Appendix E. Here, we briefly describe the Newton-Raphson method that is commonly used for estimating the parameters of logistic regression. For ease of representation, we will use a single vector to describe w˜=(wT b)T, which is of size one greater than w. Similarly, we will consider the concatenated feature vector x˜= (xT 1)T, such that the linear predictor z=wTx+b can be succinctly written as z=w˜Tx˜. Also, the concatenation of all training labels, y1 to yn, will be represented as y, the set consisting of σ(z1) to σ(zn) will be represented as σ, and the concatenation of x˜1 to x˜n will be represented as X˜.
The Newton-Raphson is an iterative method for finding w˜* that uses the following equation to update the model parameters at every iteration:
w˜(new)=w˜(old)−H−1∇E(w˜), (4.44)
where ∇E(w˜) and H are the first- and second-order derivatives of the loss function E( w˜) with respect to w˜, respectively. The key intuition behind Equation 4.44 is to move the model parameters in the direction of maximum gradient, such that w˜ takes larger steps when ∇E(w˜) is large. When w˜ arrives at a minima after some number of iterations, then ∇E(w˜) would become equal to 0 and thus result in convergence. Hence, we start with some initial values of w˜ (either randomly assigned or set to 0) and use Equation 4.44 to iteratively update w˜ till there are no significant changes in its value (beyond a certain threshold).
The first-order derivative of E(w˜) is given by
∇E(w˜)=−∑i=1nyix˜i(1−σ(w˜Tx˜i))−(1−yi)x˜iσ(w˜Tx˜i),=−∑i=1n(σ(w˜Tx˜i) −yi)x˜i,=X˜(σ−y), (4.45)
where we have used the fact that dσ(z)/dz=σ(z)(1−σ(z)). Using ∇E(w˜), we can compute the second-order derivative of E(w˜) as
H=∇∇E(w˜)=∑i=1nσ(w˜Tx˜i)(1−σ(w˜Tx˜i)x˜ix˜iT)=X˜TRX˜, (4.46)
where R is a diagonal matrix whose ith diagonal element Rii=σi(1−σi). We can now use the first- and second-order derivatives of E(w˜) in Equation 4.44 to obtain the following update equation at the kth iteration:
w˜(k+1)=w˜(k)−(X˜TRkX˜)−1X˜T(σk−y) (4.47)
where the subscript k under Rk and σk refers to using w˜(k) to compute both terms.
4.6.3 Characteristics of Logistic Regression
1. Logistic Regression is a discriminative model for classification that directly computes the poster probabilities without making any assumption about the class conditional probabilities. Hence, it is quite generic and can be applied in diverse applications. It can also be easily extended to multiclass classification, where it is known as multinomial logistic regression. However, its expressive power is limited to learning only linear decision boundaries.
2. Because there are different weights (parameters) for every attribute, the learned parameters of logistic regression can be analyzed to understand the relationships between attributes and class labels.
3. Because logistic regression does not involve computing densities and distances in the attribute space, it can work more robustly even in high- dimensional settings than distance-based methods such as nearest neighbor classifiers. However, the objective function of logistic regression does not involve any term relating to the complexity of the model. Hence, logistic regression does not provide a way to make a trade-off between model complexity and training performance, as compared to other classification models such as support vector machines. Nevertheless, variants of logistic regression can easily be developed to account for model complexity, by including appropriate terms in the objective function along with the cross entropy function.
4. Logistic regression can handle irrelevant attributes by learning weight parameters close to 0 for attributes that do not provide any gain in performance during training. It can also handle interacting attributes since the learning of model parameters is achieved in a joint fashion by considering the effects of all attributes together. Furthermore, if there are redundant attributes that are duplicates of each other, then logistic regression can learn equal weights for every redundant attribute, without degrading classification performance. However, the presence of a large number of irrelevant or redundant attributes in high-dimensional settings can make logistic regression susceptible to model overfitting.
5. Logistic regression cannot handle data instances with missing values, since the posterior probabilities are only computed by taking a weighted sum of all the attributes. If there are missing values in a training instance, it can be discarded from the training set. However, if there are missing values in a test instance, then logistic regression would fail to predict its class label.
4.7 Artificial Neural Network (ANN) Artificial neural networks (ANN) are powerful classification models that are able to learn highly complex and nonlinear decision boundaries purely from the data. They have gained widespread acceptance in several applications such as vision, speech, and language processing, where they have been repeatedly shown to outperform other classification models (and in some cases even human performance). Historically, the study of artificial neural networks was inspired by attempts to emulate biological neural systems. The human brain consists primarily of nerve cells called neurons, linked together with other neurons via strands of fiber called axons. Whenever a neuron is stimulated (e.g., in response to a stimuli), it transmits nerve activations via axons to other neurons. The receptor neurons collect these nerve activations using structures called dendrites, which are extensions from the cell body of the neuron. The strength of the contact point between a dendrite and an axon, known as a synapse, determines the connectivity between neurons. Neuroscientists have discovered that the human brain learns by changing the strength of the synaptic connection between neurons upon repeated stimulation by the same impulse.
The human brain consists of approximately 100 billion neurons that are inter- connected in complex ways, making it possible for us to learn new tasks and perform regular activities. Note that a single neuron only performs a simple modular function, which is to respond to the nerve activations coming from sender neurons connected at its dendrite, and transmit its activation to receptor neurons via axons. However, it is the composition of these simple functions that together is able to express complex functions. This idea is at the basis of constructing artificial neural networks.
Analogous to the structure of a human brain, an artificial neural network is composed of a number of processing units, called nodes, that are connected with each other via directed links. The nodes correspond to neurons that perform the basic units of computation, while the directed links correspond to
connections between neurons, consisting of axons and dendrites. Further, the weight of a directed link between two neurons represents the strength of the synaptic connection between neurons. As in biological neural systems, the primary objective of ANN is to adapt the weights of the links until they fit the input-output relationships of the underlying data.
The basic motivation behind using an ANN model is to extract useful features from the original attributes that are most relevant for classification. Traditionally, feature extraction has been achieved by using dimensionality reduction techniques such as PCA (introduced in Chapter 2), which show limited success in extracting nonlinear features, or by using hand-crafted features provided by domain experts. By using a complex combination of inter-connected nodes, ANN models are able to extract much richer sets of features, resulting in good classification performance. Moreover, ANN models provide a natural way of representing features at multiple levels of abstraction, where complex features are seen as compositions of simpler features. In many classification problems, modeling such a hierarchy of features turns out to be very useful. For example, in order to detect a human face in an image, we can first identify low-level features such as sharp edges with different gradients and orientations. These features can then be combined to identify facial parts such as eyes, nose, ears, and lips. Finally, an appropriate arrangement of facial parts can be used to correctly identify a human face. ANN models provide a powerful architecture to represent a hierarchical abstraction of features, from lower levels of abstraction (e.g., edges) to higher levels (e.g., facial parts).
Artificial neural networks have had a long history of developments spanning over five decades of research. Although classical models of ANN suffered from several challenges that hindered progress for a long time, they have re- emerged with widespread popularity because of a number of recent developments in the last decade, collectively known as deep learning. In this section, we examine classical approaches for learning ANN models, starting from the simplest model called perceptrons to more complex architectures called multi-layer neural networks. In the next section, we discuss some of the recent advancements in the area of ANN that have made it possible to effectively learn modern ANN models with deep architectures.
4.7.1 Perceptron A perceptron is a basic type of ANN model that involves two types of nodes: input nodes, which are used to represent the input attributes, and an output node, which is used to represent the model output. Figure 4.20 illustrates the basic architecture of a perceptron that takes three input attributes, x1, x2, and x3, and produces a binary output y. The input node corresponding to an attribute xi is connected via a weighted link wi to the output node. The weighted link is used to emulate the strength of a synaptic connection between neurons.
Figure 4.20. Basic architecture of a perceptron.
The output node is a mathematical device that computes a weighted sum of its inputs, adds a bias factor b to the sum, and then examines the sign of the result to produce the output y^ as follows:
3^y={1,if wTx+b>0.−1,otherwise. (4.48)
To simplify notations, w and b can be concatenated to form w˜=(wT b)T, while x can be appended with 1 at the end to form x˜=(xT 1)T. The output of the perceptron y^ can then be written:
y^=sign(w˜Tx˜),
where the sign function acts as an activation function by providing an output value of +1 if the argument is positive and −1 if its argument is negative.
Learning the Perceptron Given a training set, we are interested in learning parameters w˜ such that y^ closely resembles the true y of training instances. This is achieved by using the perceptron learning algorithm given in Algorithm 4.3. The key computation for this algorithm is the iterative weight update formula given in Step 8 of the algorithm:
wj(k+1)=wj(k)+λ(yi−yi^(k))xij, (4.49)
where w(k) is the weight parameter associated with the ith input link after the kth iteration, λ is a parameter known as the learning rate, and xij is the value of the jth attribute of the training example xi. The justification for Equation 4.49 is rather intuitive. Note that (yi−y^i) captures the discrepancy between yi and y^i, such that its value is 0 only when the true label and the predicted output match. Assume xij is positive. If y^=0 and y=1, then wj is increased at the next iteration so that w˜Txi can become positive. On the other hand, if y^=1 and y=0, then wj is decreased so that w˜Txi can become negative. Hence, the weights are modified at every iteration to reduce the discrepancies between y^ and y across all training instances. The learning rate λ, a parameter whose value is between 0 and 1, can be used to control the amount of adjustments made in each iteration. The algorithm halts when the average number of discrepancies are smaller than a threshold γ.
Algorithm 4.3 Perceptron learning algorithm. 1: Let D.train={(x˜i, yi)|i=1, 2, …, n} be the set of training instances. 2: Set k ← 0. 3: Initialize the weight vector w˜(0) with random values. 4: repeat
5: for each training instance (x˜i, yi)∈D.train do 6: Compute the predicted output y^i(k) using w˜(k). 7: for each weight component wj do 8: Update the weight, wj(k+1)=wj(k)+λ(yi−y^i(k))xij. 9: end for 10: Update k ← k + 1. 11: end for 12: until ∑i=1n|yi−y^i(k)|/n is less than a threshold γ
The perceptron is a simple classification model that is designed to learn linear decision boundaries in the attribute space. Figure 4.21 shows the decision boundary obtained by applying the perceptron learning algorithm to the data set provided on the left of the figure. However, note that there can be multiple decision boundaries that can separate the two classes, and the perceptron arbitrarily learns one of these boundaries depending on the random initial values of parameters. (The selection of the optimal decision boundary is a problem that will be revisited in the context of support vector machines in Section 4.9.) Further, the perceptron learning algorithm is only guaranteed to converge when the classes are linearly separable. However, if the classes are not linearly separable, the algorithm fails to converge. Figure 4.22 shows an example of a nonlinearly separable data given by the XOR function. The perceptron cannot find the right solution for this data because there is no linear decision boundary that can perfectly separate the training instances. Thus, the stopping condition at line 12 of Algorithm 4.3 would never be met and hence, the perceptron learning algorithm would fail to converge. This is a major limitation of perceptrons since real-world classification problems often involve nonlinearly separable classes.
Figure 4.21. Perceptron decision boundary for the data given on the left (+ represents a positively labeled instance while o represents a negatively labeled instance.
Figure 4.21. Full Alternative Text
Figure 4.22. XOR classification problem. No linear hyperplane can separate the two classes.
Figure 4.22. Full Alternative Text
4.7.2 Multi-layer Neural Network A multi-layer neural network generalizes the basic concept of a perceptron to more complex architectures of nodes that are capable of learning nonlinear decision boundaries. A generic architecture of a multi-layer neural network is shown in Figure 4.23 where the nodes are arranged in groups called layers. These layers are commonly organized in the form of a chain such that every layer operates on the outputs of its preceding layer. In this way, the layers represent different levels of abstraction that are applied on the input features in a sequential manner. The composition of these abstractions generates the final output at the last layer, which is used for making predictions. In the following, we briefly describe the three types of layers used in multi-layer neural networks.
Figure 4.23. Example of a multi-layer artificial neural network (ANN).
The first layer of the network, called the input layer, is used for representing inputs from attributes. Every numerical or binary attribute is typically represented using a single node on this layer, while a categorical attribute is either represented using a different node for each categorical value, or by encoding the k-ary attribute using ⌈log2k ⌉ input nodes. These inputs are fed into intermediary layers known as hidden layers, which are made up of processing units known as hidden nodes. Every hidden node operates on signals received from the input nodes or hidden nodes at the preceding layer, and produces an activation value that is transmitted to the next layer. The final layer is called the output layer and processes the activation values from its preceding layer to produce predictions of output variables. For binary classification, the output layer contains a single node representing the binary class label. In this architecture, since the signals are propagated only in the forward direction from the input layer to the output layer, they are also called feedforward neural networks.
A major difference between multi-layer neural networks and perceptrons is the inclusion of hidden layers, which dramatically improves their ability to represent arbitrarily complex decision boundaries. For example, consider the XOR problem described in the previous section. The instances can be classified using two hyperplanes that partition the input space into their respective classes, as shown in Figure 4.24(a). Because a perceptron can create only one hyperplane, it cannot find the optimal solution. However, this problem can be addressed by using a hidden layer consisting of two nodes, as shown in Figure 4.24(b). Intuitively, we can think of each hidden node as a perceptron that tries to construct one of the two hyperplanes, while the output node simply combines the results of the perceptrons to yield the decision boundary shown in Figure 4.24(a).
Figure 4.24. A two-layer neural network for the XOR problem.
Figure 4.24. Full Alternative Text
The hidden nodes can be viewed as learning latent representations or features that are useful for distinguishing between the classes. While the first hidden
layer directly operates on the input attributes and thus captures simpler features, the subsequent hidden layers are able to combine them and construct more complex features. From this perspective, multi-layer neural networks learn a hierarchy of features at different levels of abstraction that are finally combined at the output nodes to make predictions. Further, there are combinatorially many ways we can combine the features learned at the hidden layers of ANN, making them highly expressive. This property chiefly distinguishes ANN from other classification models such as decision trees, which can learn partitions in the attribute space but are unable to combine them in exponential ways.
Figure 4.25. Schematic illustration of the parameters of an ANN model with (L −1) hidden layers.
Figure 4.25. Full Alternative Text
To understand the nature of computations happening at the hidden and output nodes of ANN, consider the ith node at the lth layer of the network (l>0), where the layers are numbered from 0 (input layer) to L (output layer), as
shown in Figure 4.25. The activation value generated at this node, ail, can be represented as a function of the inputs received from nodes at the preceding layer. Let wijl represent the weight of the connection from the jth node at layer (l−1) to the ith node at layer l. Similarly, let us denote the bias term at this node as bjl. The activation value ail can then be expressed as
ail=f(zil)=f(∑jwijlajl−1+bil),
where z is called the linear predictor and f(⋅) is the activation function that converts z to a. Further, note that, by definition, aj0=xj at the input layer and aL=y^ at the output node.
There are a number of alternate activation functions apart from the sign function that can be used in multi-layer neural networks. Some examples include linear, sigmoid (logistic), and hyperbolic tangent functions, as shown in Figure 4.26. These functions are able to produce real-valued and nonlinear activation values. Among these activation functions, the sigmoid σ(⋅) has been widely used in many ANN models, although the use of other types of activation functions in the context of deep learning will be discussed in Section 4.8. We can thus represent ail as
Figure 4.26. Types of activation functions used in multi-layer neural networks.
Figure 4.26. Full Alternative Text ail=σ(zil)=11+e−zil. (4.50)
Learning Model Parameters The weights and bias terms (w, b) of the ANN model are learned during training so that the predictions on training instances match the true labels.
This is achieved by using a loss function
E(w, b)=∑k=1nLoss (yk, y^k) (4.51)
where yk is the true label of the kth training instance and y^k is equal to aL, produced by using xk. A typical choice of the loss function is the squared loss function:.
Loss (yk, y^k)=(yk, y^k)2. (4.52)
Note that E(w, b) is a function of the model parameters (w, b) because the output activation value aL depends on the weights and bias terms. We are interested in choosing (w, b) that minimizes the training loss E(w, b). Unfortunately, because of the use of hidden nodes with nonlinear activation functions, E(w, b) is not a convex function of w and b, which means that E(w, b) can have local minima that are not globally optimal. However, we can still apply standard optimization techniques such as the gradient descent method to arrive at a locally optimal solution. In particular, the weight parameter wijl and the bias term bil can be iteratively updated using the following equations:
wijl←wijl−λ∂E∂wijl, (4.53) bil←bil−λ∂E∂bil, (4.54)
where λ is a hyper-parameter known as the learning rate. The intuition behind this equation is to move the weights in a direction that reduces the training loss. If we arrive at a minima using this procedure, the gradient of the training loss will be close to 0, eliminating the second term and resulting in the convergence of weights. The weights are commonly initialized with values drawn randomly from a Gaussian or a uniform distribution.
A necessary tool for updating weights in Equation 4.53 is to compute the partial derivative of E with respect to wijl. This computation is nontrivial especially at hidden layers (l<L), since wijl does not directly affect y^=aL (and hence the training loss), but has complex chains of influences via activation values at subsequent layers. To address this problem, a technique known as backpropagation was developed, which propagates the derivatives backward from the output layer to the hidden layers. This technique can be described as follows.
Recall that the training loss E is simply the sum of individual losses at training instances. Hence the partial derivative of E can be decomposed as a sum of partial derivatives of individual losses.
∂E∂wjl=∑k=1n∂ Loss (yk, y^k)∂wjl.
To simplify discussions, we will consider only the derivatives of the loss at the kth training instance, which will be generically represented as Loss(y, aL). By using the chain rule of differentiation, we can represent the partial derivatives of the loss with respect to wijl as
∂ Loss∂wijl=∂ Loss∂ail×∂ail∂zil×∂zil∂wijl. (4.55)
The last term of the previous equation can be written as
∂zil∂wijl=∂(∑jwijlajl−1+bil)∂wijl=ajl−1.
Also, if we use the sigmoid activation function, then
∂ail∂zil=∂ σ(zil)∂zil=ail(1−ai1).
Equation 4.55 can thus be simplified as
∂ Loss∂wijl=δil×ail(1−ai1)×ajl−1,where δil=∂ Loss∂ail. (4.56)
A similar formula for the partial derivatives with respect to the bias terms bli is given by
∂ Loss∂bil=δil×ail(1−ai1). (4.57)
Hence, to compute the partial derivatives, we only need to determine δil. Using a squared loss function, we can easily write δL at the output node as
δL=∂ Loss∂aL=∂ (y−aL)2∂aL=2(aL−y). (4.58)
However, the approach for computing δjl at hidden nodes (l<L) is more involved. Notice that ajl affects the activation values ail+1 of all nodes at the next layer, which in turn influences the loss. Hence, again using the chain rule of differentiation, δjl can be represented as
δjl=∂ Loss∂ajl=∑i(∂ Loss∂ail+1×∂ail+1∂ajl).=∑i(∂ Loss∂ail+1×∂ail+1∂zil+1× (4.59)
The previous equation provides a concise representation of the δjl values at layer l in terms of the δjl+1 values computed at layer l+1. Hence, proceeding backward from the output layer L to the hidden layers, we can recursively apply Equation 4.59 to compute δil at every hidden node. δil can then be used in Equations 4.56 and 4.57 to compute the partial derivatives of the loss with respect to wijl and bil, respectively. Algorithm 4.4 summarizes the complete approach for learning the model parameters of ANN using backpropagation and gradient descent method.
Algorithm 4.4 Learning ANN using backpropagation and gradient descent.
1: Let D.train = {(xk, yk) | k = 1, 2, …, n} be the set of training instances. 2: Set counter c ← 0. 3: Initialize the weight and bias terms (w(0), b(0)) with random values. 4: repeat 5: for each training instance (xk, yk) ∈ D.train do 6: Compute the set of activations (ail)k by making a forward pass using 7: Compute the set (δil)k by backpropagation using Equations 8: Compute (∂ Loss/∂wijl, ∂ Loss/∂bil)k using Equations 4.56 9: end for 10: Compute ∂E/∂wijl←∑k=1n(∂ Loss/∂wijl)k. 11: Compute ∂E/∂bil←∑k=1n(∂ Loss/∂bil)k. 12: Update (w(c + 1), b(c + 1)) by gradient descent using Equations 13: Update c ← c + 1. 14: until (w(c + 1), b(c + 1)) and (w(c), b(c)) converge to the same value
4.7.3 Characteristics of ANN 1. Multi-layer neural networks with at least one hidden layer are universal
approximators; i.e., they can be used to approximate any target function. They are thus highly expressive and can be used to learn complex decision boundaries in diverse applications. ANN can also be used for multiclass classification and regression problems, by appropriately modifying the output layer. However, the high model complexity of classical ANN models makes it susceptible to overfitting, which can be overcome to some extent by using deep learning techniques discussed in Section 4.8.3.
2. ANN provides a natural way to represent a hierarchy of features at multiple levels of abstraction. The outputs at the final hidden layer of the ANN model thus represent features at the highest level of abstraction that are most useful for classification. These features can also be used as inputs in other supervised classification models, e.g., by replacing the output node of the ANN by any generic classifier.
3. ANN represents complex high-level features as compositions of simpler lower-level features that are easier to learn. This provides ANN the ability to gradually increase the complexity of representations, by adding more hidden layers to the architecture. Further, since simpler features can be combined in combinatorial ways, the number of complex features learned by ANN is much larger than traditional classification models. This is one of the main reasons behind the high expressive power of deep neural networks.
4. ANN can easily handle irrelevant attributes, by using zero weights for attributes that do not help in improving the training loss. Also, redundant attributes receive similar weights and do not degrade the quality of the classifier. However, if the number of irrelevant or redundant attributes is large, the learning of the ANN model may suffer from overfitting, leading to poor generalization performance.
5. Since the learning of ANN model involves minimizing a non-convex function, the solutions obtained by gradient descent are not guaranteed to be globally optimal. For this reason, ANN has a tendency to get stuck in local minima, a challenge that can be addressed by using deep learning techniques discussed in Section 4.8.4.
6. Training an ANN is a time consuming process, especially when the number of hidden nodes is large. Nevertheless, test examples can be classified rapidly.
7. Just like logistic regression, ANN can learn in the presence of interacting variables, since the model parameters are jointly learned over all variables together. In addition, ANN cannot handle instances with missing values in the training or testing phase.
4.8 Deep Learning As described above, the use of hidden layers in ANN is based on the general belief that complex high-level features can be constructed by combining simpler lower-level features. Typically, the greater the number of hidden layers, the deeper the hierarchy of features learned by the network. This motivates the learning of ANN models with long chains of hidden layers, known as deep neural networks. In contrast to “shallow” neural networks that involve only a small number of hidden layers, deep neural networks are able to represent features at multiple levels of abstraction and often require far fewer nodes per layer to achieve generalization performance similar to shallow networks.
Despite the huge potential in learning deep neural networks, it has remained challenging to learn ANN models with a large number of hidden layers using classical approaches. Apart from reasons related to limited computational resources and hardware architectures, there have been a number of algorithmic challenges in learning deep neural networks. First, learning a deep neural network with low training error has been a daunting task because of the saturation of sigmoid activation functions, resulting in slow convergence of gradient descent. This problem becomes even more serious as we move away from the output node to the hidden layers, because of the compounded effects of saturation at multiple layers, known as the vanishing gradient problem. Because of this reason, classical ANN models have suffered from slow and ineffective learning, leading to poor training and test performance. Second, the learning of deep neural networks is quite sensitive to the initial values of model parameters, chiefly because of the non-convex nature of the optimization function and the slow convergence of gradient descent. Third, deep neural networks with a large number of hidden layers have high model complexity, making them susceptible to overfitting. Hence, even if a deep neural network has been trained to show low training error, it can still suffer from poor generalization performance.
These challenges have deterred progress in building deep neural networks for several decades and it is only recently that we have started to unlock their
immense potential with the help of a number of advances being made in the area of deep learning. Although some of these advances have been around for some time, they have only gained mainstream attention in the last decade, with deep neural networks continually beating records in various competitions and solving problems that were too difficult for other classification approaches.
There are two factors that have played a major role in the emergence of deep learning techniques. First, the availability of larger labeled data sets, e.g., the ImageNet data set contains more than 10 million labeled images, has made it possible to learn more complex ANN models than ever before, without falling easily into the traps of model overfitting. Second, advances in computational abilities and hardware infrastructures, such as the use of graphical processing units (GPU) for distributed computing, have greatly helped in experimenting with deep neural networks with larger architectures that would not have been feasible with traditional resources.
In addition to the previous two factors, there have been a number of algorithmic advancements to overcome the challenges faced by classical methods in learning deep neural networks. Some examples include the use of more responsive combinations of loss functions and activation functions, better initialization of model parameters, novel regularization techniques, more agile architecture designs, and better techniques for model learning and hyper-parameter selection. In the following, we describe some of the deep learning advances made to address the challenges in learning deep neural networks. Further details on recent developments in deep learning can be obtained from the Bibliographic Notes.
4.8.1 Using Synergistic Loss Functions One of the major realizations leading to deep learning has been the importance of choosing appropriate combinations of activation and loss functions. Classical ANN models commonly made use of the sigmoid activation function at the output layer, because of its ability to produce real-
valued outputs between 0 and 1, which was combined with a squared loss objective to perform gradient descent. It was soon noticed that this particular combination of activation and loss function resulted in the saturation of output activation values, which can be described as follows.
Saturation of Outputs Although the sigmoid has been widely-used as an activation function, it easily saturates at high and low values of inputs that are far away from 0. Observe from Figure 4.27(a) that σ(z) shows variance in its values only when z is close to 0. For this reason, ∂σ(z)/∂z is non-zero for only a small range of z around 0, as shown in Figure 4.27(b). Since ∂σ(z)/∂z is one of the components in the gradient of loss (see Equation 4.55), we get a diminishing gradient value when the activation values are far from 0.
Figure 4.27. Plots of sigmoid function and its derivative.
Figure 4.27. Full Alternative Text
To illustrate the effect of saturation on the learning of model parameters at the output node, consider the partial derivative of loss with respect to the weight wjL at the output node. Using the squared loss function, we can write this as
∂ Loss∂wjL=2(aL−y)×σ(zL)(1−σ(zL))×ajL−1. (4.60)
In the previous equation, notice that when zL is highly negative, σ(zL) (and hence the gradient) is close to 0. On the other hand, when zL is highly positive, (1−σ(zL)) becomes close to 0, nullifying the value of the gradient. Hence, irrespective of whether the prediction aL matches the true label y or not, the gradient of the loss with respect to the weights is close to 0 whenever zL is highly positive or negative. This causes an unnecessarily slow convergence of the model parameters of the ANN model, often resulting in poor learning.
Note that it is the combination of the squared loss function and the sigmoid activation function at the output node that together results in diminishing gradients (and thus poor learning) upon saturation of outputs. It is thus important to choose a synergistic combination of loss function and activation function that does not suffer from the saturation of outputs.
Cross entropy loss function The cross entropy loss function, which was described in the context of logistic regression in Section 4.6.2, can significantly avoid the problem of saturating outputs when used in combination with the sigmoid activation function. The cross entropy loss function of a real-valued prediction y^∈(0, 1) on a data instance with binary label y∈{0, 1} can be defined as
Loss(y, y^)=−ylog(y^)−(1−y)log(1−y^), (4.61)
where log represents the natural logarithm (to base e) and 0 log(0)=0 for convenience. The cross entropy function has foundations in information theory and measures the amount of disagreement between y and y^. The partial derivative of this loss function with respect to y^=aL can be given as
δL=∂ Loss∂aL=−yaL+(1−y)(1−aL).=(aL−y)aL(1−aL). (4.62)
Using this value of δL in Equation 4.56, we can obtain the partial derivative of the loss with respect to the weight wjl at the output node as
∂ Loss∂wjL=(aL−y)aL(1−aL)×aL(1−aL)×ajL−1.=(aL−y)×ajL−1. (4.63)
Notice the simplicity of the previous formula using the cross entropy loss function. The partial derivatives of the loss with respect to the weights at the output node depend only on the difference between the prediction aL and the true label y. In contrast to Equation 4.60, it does not involve terms such as σ(zL)(1−σ(zL)) that can be impacted by saturation of zL. Hence, the gradients are high whenever (aL−y) is large, promoting effective learning of the model parameters at the output node. This has been a major breakthrough in the learning of modern ANN models and it is now a common practice to use the cross entropy loss function with sigmoid activations at the output node.
4.8.2 Using Responsive Activation Functions Even though the cross entropy loss function helps in overcoming the problem of saturating outputs, it still does not solve the problem of saturation at hidden layers, arising due to the use of sigmoid activation functions at hidden nodes. In fact, the effect of saturation on the learning of model parameters is even more aggravated at hidden layers, a problem known as the vanishing gradient problem. In the following, we describe the vanishing gradient problem and the use of a more responsive activation function, called the rectified linear output unit (ReLU), to overcome this problem.
Vanishing Gradient Problem The impact of saturating activation values on the learning of model parameters increases at deeper hidden layers that are farther away from the
output node. Even if the activation in the output layer does not saturate, the repeated multiplications performed as we backpropagate the gradients from the output layer to the hidden layers may lead to decreasing gradients in the hidden layers. This is called the vanishing gradient problem, which has been one of the major hindrances in learning deep neural networks.
To illustrate the vanishing gradient problem, consider an ANN model that consists of a single node at every hidden layer of the network, as shown in Figure 4.28. This simplified architecture involves a single chain of hidden nodes where a single weighted link wl connects the node at layer l−1 to the node at layer l. Using Equations 4.56 and 4.59, we can represent the partial derivative of the loss with respect to wl as
∂ Loss∂wl=δl×al(1−al)×al−1,where δl=2(aL−y)×∏r=lL −1(ar+1(1−ar+1)×wr+1). (4.64)
Notice that if any of the linear predictors zr+1 saturates at subsequent layers, then the term ar+1(1−ar+1) becomes close to 0, thus diminishing the overall gradient. The saturation of activations thus gets compounded and has multiplicative effects on the gradients at hidden layers, making them highly unstable and thus, unsuitable for use with gradient descent. Even though the previous discussion only pertains to the simplified architecture involving a single chain of hidden nodes, a similar argument can be made for any generic ANN architecture involving multiple chains of hidden nodes. Note that the vanishing gradient problem primarily arises because of the use of sigmoid activation function at hidden nodes, which is known to easily saturate especially after repeated multiplications.
Figure 4.28. An example of an ANN model with only one node at every hidden layer.
Figure 4.29. Plot of the rectified linear unit (ReLU) activation function.
Rectified Linear Units (ReLU) To overcome the vanishing gradient problem, it is important to use an activation function f(z)at the hidden nodes that provides a stable and significant value of the gradient whenever a hidden node is active, i.e., z>0. This is achieved by using rectified linear units (ReLU) as activation functions at hidden nodes, which can be defined as
a=f(z)={z,if z>0.0,otherwise. (4.65)
The idea of ReLU has been inspired from biological neurons, which are either in an inactive state (f(z)=0) or show an activation value proportional to the input. Figure 4.29 shows a plot of the ReLU function. We can see that it is linear with respect to z when z>0. Hence, the gradient of the activation value with respect to z can be written as
∂a∂z={1,if z>0.0,if z<0. (4.66)
Although f(z)is not differentiable at 0, it is common practice to use ∂a/∂z=0 when z=0. Since the gradient of the ReLU activation function is equal to 1 whenever z>0, it avoids the problem of saturation at hidden nodes, even after repeated multiplications. Using ReLU, the partial derivatives of the loss with respect to the weight and bias parameters can be given by
∂ Loss∂wijl=δil×I(zil)×ajl−1, (4.67) ∂ Loss∂bil=δil×I(zil),where δil=∑i=1n(δil+1×I(zil+1)×wijl+1),and I(z)= {1,if z>0.0,otherwise. (4.68)
Notice that ReLU shows a linear behavior in the activation values whenever a node is active, as compared to the nonlinear properties of the sigmoid function. This linearity promotes better flows of gradients during backpropagation, and thus simplifies the learning of ANN model parameters. The ReLU is also highly responsive at large values of z away from 0, as opposed to the sigmoid activation function, making it more suitable for gradient descent. These differences give ReLU a major advantage over the sigmoid function. Indeed, ReLU is used as the preferred choice of activation function at hidden layers in most modern ANN models.
4.8.3 Regularization A major challenge in learning deep neural networks is the high model complexity of ANN models, which grows with the addition of hidden layers in the network. This can become a serious concern, especially when the training set is small, due to the phenomena of model overfitting. To overcome this challenge, it is important to use techniques that can help in reducing the complexity of the learned model, known as regularization techniques. Classical approaches for learning ANN models did not have an effective way to promote regularization of the learned model parameters. Hence, they had often been sidelined by other classification methods, such as support vector machines (SVM), which have in-built regularization mechanisms. (SVMs will be discussed in more detail in Section 4.9).
One of the major advancements in deep learning has been the development of novel regularization techniques for ANN models that are able to offer significant improvements in generalization performance. In the following, we discuss one of the regularization techniques for ANN, known as the dropout method, that have gained a lot of attention in several applications.
Dropout The main objective of dropout is to avoid the learning of spurious features at hidden nodes, occurring due to model overfitting. It uses the basic intuition that spurious features often “co-adapt” themselves such that they show good training performance only when used in highly selective combinations. On the other hand, relevant features can be used in a diversity of feature combinations and hence are quite resilient to the removal or modification of other features. The dropout method uses this intuition to break complex “co- adaptations” in the learned features by randomly dropping input and hidden nodes in the network during training.
Dropout belongs to a family of regularization techniques that uses the criteria of resilience to random perturbations as a measure of the robustness (and hence, simplicity) of a model. For example, one approach to regularization is to inject noise in the input attributes of the training set and learn a model with the noisy training instances. If a feature learned from the training data is indeed generalizable, it should not be affected by the addition of noise. Dropout can be viewed as a similar regularization approach that perturbs the information content of the training set not only at the level of attributes but also at multiple levels of abstractions, by dropping input and hidden nodes.
The dropout method draws inspiration from the biological process of gene swapping in sexual reproduction, where half of the genes from both parents are combined together to create the genes of the offspring. This favors the selection of parent genes that are not only useful but can also inter-mingle with diverse combinations of genes coming from the other parent. On the other hand, co-adapted genes that function only in highly selective combinations are soon eliminated in the process of evolution. This idea is used in the dropout method for eliminating spurious co-adapted features. A
simplified description of the dropout method is provided in the rest of this section.
Figure 4.30. Examples of sub-networks generated in the dropout method using γ=0.5.
Figure 4.30. Full Alternative Text
Let (wk, bk) represent the model parameters of the ANN model at the kth iteration of the gradient descent method. At every iteration, we randomly select a fraction γ of input and hidden nodes to be dropped from the network,
where γ∈(0, 1) is a hyper-parameter that is typically chosen to be 0.5. The weighted links and bias terms involving the dropped nodes are then eliminated, resulting in a “thinned” sub-network of smaller size. The model parameters of the sub-network (wsk, bsk) are then updated by computing activation values and performing backpropagation on this smaller sub- network. These updated values are then added back in the original network to obtain the updated model parameters, (wk+1, bk+1), to be used in the next iteration.
Figure 4.30 shows some examples of sub-networks that can be generated at different iterations of the dropout method, by randomly dropping input and hidden nodes. Since every sub-network has a different architecture, it is difficult to learn complex co-adaptations in the features that can result in overfitting. Instead, the features at the hidden nodes are learned to be more agile to random modifications in the network structure, thus improving their generalization ability. The model parameters are updated using a different random sub-network at every iteration, till the gradient descent method converges.
Let (wkmax, bkmax) denote the model parameters at the last iteration kmax of the gradient descent method. These parameters are finally scaled down by a factor of (1−γ), to produce the weights and bias terms of the final ANN model, as follows:
(w*, b*)=((1−γ)×wkmax, (1−γ)×bkmax)
We can now use the complete neural network with model parameters (w*, b*) for testing. The dropout method has been shown to provide significant improvements in the generalization performance of ANN models in a number of applications. It is computationally cheap and can be applied in combination with any of the other deep learning techniques. It also has a number of similarities with a widely-used ensemble learning method known as bagging, which learns multiple models using random subsets of the training set, and then uses the average output of all the models to make predictions. (Bagging will be presented in more detail later in Section 4.10.4). In a similar vein, it can be shown that the predictions of the final network learned using dropout approximates the average output of all possible 2n sub- networks that can be formed using n nodes. This is one of the reasons behind
the superior regularization abilities of dropout.
4.8.4 Initialization of Model Parameters Because of the non-convex nature of the loss function used by ANN models, it is possible to get stuck in locally optimal but globally inferior solutions. Hence, the initial choice of model parameter values plays a significant role in the learning of ANN by gradient descent. The impact of poor initialization is even more aggravated when the model is complex, the network architecture is deep, or the classification task is difficult. In such cases, it is often advisable to first learn a simpler model for the problem, e.g., using a single hidden layer, and then incrementally increase the complexity of the model, e.g., by adding more hidden layers. An alternate approach is to train the model for a simpler task and then use the learned model parameters as initial parameter choices in the learning of the original task. The process of initializing ANN model parameters before the actual training process is known as pretraining.
Pretraining helps in initializing the model to a suitable region in the parameter space that would otherwise be inaccessible by random initialization. Pretraining also reduces the variance in the model parameters by fixing the starting point of gradient descent, thus reducing the chances of overfitting due to multiple comparisons. The models learned by pretraining are thus more consistent and provide better generalization performance.
Supervised Pretraining A common approach for pretraining is to incrementally train the ANN model in a layer-wise manner, by adding one hidden layer at a time. This approach, known as supervised pretraining, ensures that the parameters learned at every layer are obtained by solving a simpler problem, rather than learning all model parameters together. These parameter values thus provide a good choice for initializing the ANN model. The approach for supervised
pretraining can be briefly described as follows.
We start the supervised pretraining process by considering a reduced ANN model with only a single hidden layer. By applying gradient descent on this simple model, we are able to learn the model parameters of the first hidden layer. At the next run, we add another hidden layer to the model and apply gradient descent to learn the parameters of the newly added hidden layer, while keeping the parameters of the first layer fixed. This procedure is recursively applied such that while learning the parameters of the lth hidden layer, we consider a reduced model with only l hidden layers, whose first (l −1) hidden layers are not updated on the lth run but are instead fixed using pretrained values from previous runs. In this way, we are able to learn the model parameters of all (L−1) hidden layers. These pretrained values are used to initialize the hidden layers of the final ANN model, which is fine- tuned by applying a final round of gradient descent over all the layers.
Unsupervised Pretraining Supervised pretraining provides a powerful way to initialize model parameters, by gradually growing the model complexity from shallower to deeper networks. However, supervised pretraining requires a sufficient number of labeled training instances for effective initialization of the ANN model. An alternate pretraining approach is unsupervised pretraining, which initializes model parameters by using unlabeled instances that are often abundantly available. The basic idea of unsupervised pretraining is to initialize the ANN model in such a way that the learned features capture the latent structure in the unlabeled data.
Figure 4.31. The basic architecture of a single-layer autoencoder.
Unsupervised pretraining relies on the assumption that learning the distribution of the input data can indirectly help in learning the classification model. It is most helpful when the number of labeled examples is small and the features for the supervised problem bear resemblance to the factors generating the input data. Unsupervised pretraining can be viewed as a different form of regularization, where the focus is not explicitly toward finding simpler features but instead toward finding features that can best explain the input data. Historically, unsupervised pretraining has played an important role in reviving the area of deep learning, by making it possible to train any generic deep neural network without requiring specialized architectures.
Use of Autoencoders One simple and commonly used approach for unsupervised pretraining is to
use an unsupervised ANN model known as an autoencoder. The basic architecture of an autoencoder is shown in Figure 4.31. An autoencoder attempts to learn a reconstruction of the input data by mapping the attributes x to latent features c, and then re-projecting c back to the original attribute space to create the reconstruction x^. The latent features are represented using a hidden layer of nodes, while the input and output layers represent the attributes and contain the same number of nodes. During training, the goal is to learn an autoencoder model that provides the lowest reconstruction error, RE(x, x^), on all input data instances. A typical choice of the reconstruction error is the squared loss function:
RE(x, x^)=ǁx−x^ ǁ2.
The model parameters of the autoencoder can be learned by using a similar gradient descent method as the one used for learning supervised ANN models for classification. The key difference is the use of the reconstruction error on all training instances as the training loss. Autoencoders that have multiple layers of hidden layers are known as stacked autoencoders.
Autoencoders are able to capture complex representations of the input data by the use of hidden nodes. However, if the number of hidden nodes is large, it is possible for an autoencoder to learn the identity relationship, where the input x is just copied and returned as the output x^, resulting in a trivial solution. For example, if we use as many hidden nodes as the number of attributes, then it is possible for every hidden node to copy an attribute and simply pass it along to an output node, without extracting any useful information. To avoid this problem, it is common practice to keep the number of hidden nodes smaller than the number of input attributes. This forces the autoencoder to learn a compact and useful encoding of the input data, similar to a dimensionality reduction technique. An alternate approach is to corrupt the input instances by adding random noise, and then learn the autoencoder to reconstruct the original instance from the noisy input. This approach is known as the denoising autoencoder, which offers strong regularization capabilities and is often used to learn complex features even in the presence of a large number of hidden nodes.
To use an autoencoder for unsupervised pretraining, we can follow a similar layer-wise approach like supervised pretraining. In particular, to pretrain the
model parameters of the lth hidden layer, we can construct a reduced ANN model with only l hidden layers and an output layer containing the same number of nodes as the attributes and is used for reconstruction. The parameters of the lth hidden layer of this network are then learned using a gradient descent method to minimize the reconstruction error. The use of unlabeled data can be viewed as providing hints to the learning of parameters at every layer that aid in generalization. The final model parameters of the ANN model are then learned by applying gradient descent over all the layers, using the initial values of parameters obtained from pretraining.
Hybrid Pretraining Unsupervised pretraining can also be combined with supervised pretraining by using two output layers at every run of pretraining, one for reconstruction and the other for supervised classification. The parameters of the lth hidden layer are then learned by jointly minimizing the losses on both output layers, usually weighted by a trade-off hyper-parameter α. Such a combined approach often shows better generalization performance than either of the approaches, since it provides a way to balance between the competing objectives of representing the input data and improving classification performance.
4.8.5 Characteristics of Deep Learning Apart from the basic characteristics of ANN discussed in Section 4.7.3, the use of deep learning techniques provides the following additional characteristics:
1. An ANN model trained for some task can be easily re-used for a different task that involves the same attributes, by using pretraining strategies. For example, we can use the learned parameters of the original task as initial parameter choices for the target task. In this way,
ANN promotes re-usability of learning, which can be quite useful when the target application has a smaller number of labeled training instances.
2. Deep learning techniques for regularization, such as the dropout method, help in reducing the model complexity of ANN and thus promoting good generalization performance. The use of regularization techniques is especially useful in high-dimensional settings, where the number of training labels is small but the classification problem is inherently difficult.
3. The use of an autoencoder for pretraining can help eliminate irrelevant attributes that are not related to other attributes. Further, it can help reduce the impact of redundant attributes by representing them as copies of the same attribute.
4. Although the learning of an ANN model can succumb to finding inferior and locally optimal solutions, there are a number of deep learning techniques that have been proposed to ensure adequate learning of an ANN. Apart from the methods discussed in this section, some other techniques involve novel architecture designs such as skip connections between the output layer and lower layers, which aids the easy flow of gradients during backpropagation.
5. A number of specialized ANN architectures have been designed to handle a variety of input data sets. Some examples include convolutional neural networks (CNN) for two-dimensional gridded objects such as images, and recurrent neural network (RNN) for sequences. While CNNs have been extensively used in the area of computer vision, RNNs have found applications in processing speech and language.
4.9 Support Vector Machine (SVM) A support vector machine (SVM) is a discriminative classification model that learns linear or nonlinear decision boundaries in the attribute space to separate the classes. Apart from maximizing the separability of the two classes, SVM offers strong regularization capabilities, i.e., it is able to control the complexity of the model in order to ensure good generalization performance. Due to its unique ability to innately regularize its learning, SVM is able to learn highly expressive models without suffering from overfitting. It has thus received considerable attention in the machine learning community and is commonly used in several practical applications, ranging from handwritten digit recognition to text categorization. SVM has strong roots in statistical learning theory and is based on the principle of structural risk minimization. Another unique aspect of SVM is that it represents the decision boundary using only a subset of the training examples that are most difficult to classify, known as the support vectors. Hence, it is a discriminative model that is impacted only by training instances near the boundary of the two classes, in contrast to learning the generative distribution of every class.
To illustrate the basic idea behind SVM, we first introduce the concept of the margin of a separating hyperplane and the rationale for choosing such a hyperplane with maximum margin. We then describe how a linear SVM can be trained to explicitly look for this type of hyperplane. We conclude by showing how the SVM methodology can be extended to learn nonlinear decision boundaries by using kernel functions.
4.9.1 Margin of a Separating Hyperplane The generic equation of a separating hyperplane can be written as
wTx+b=0,
where x represents the attributes and (w, b) represent the parameters of the hyperplane. A data instance xi can belong to either side of the hyperplane depending on the sign of (wTxi+b). For the purpose of binary classification, we are interested in finding a hyperplane that places instances of both classes on opposite sides of the hyperplane, thus resulting in a separation of the two classes. If there exists a hyperplane that can perfectly separate the classes in the data set, we say that the data set is linearly separable. Figure 4.32 shows an example of linearly separable data involving two classes, squares and circles. Note that there can be infinitely many hyperplanes that can separate the classes, two of which are shown in Figure 4.32 as lines B1 and B2. Even though every such hyperplane will have zero training error, they can provide different results on previously unseen instances. Which separating hyperplane should we thus finally choose to obtain the best generalization performance? Ideally, we would like to choose a simple hyperplane that is robust to small perturbations. This can be achieved by using the concept of the margin of a separating hyperplane, which can be briefly described as follows.
Figure 4.32. Margin of a hyperplane in a two-dimensional data set.
Figure 4.32. Full Alternative Text
For every separating hyperplane Bi, let us associate a pair of parallel hyperplanes, bi1 and bi2, such that they touch the closest instances of both classes, respectively. For example, if we move B1 parallel to its direction, we can touch the first square using b11 and the first circle using b12. bi1 and bi2 are known as the margin hyperplanes of Bi and the distance between them is known as the margin of the separating hyperplane Bi. From the diagram shown in Figure 4.32, notice that the margin for B1 is considerably larger than that for B2. In this example, b1 turns out to be the separating hyperplane with the maximum margin, known as the maximum margin hyperplane.
Rationale for Maximum Margin Hyperplanes with large margins tend to have better generalization performance than those with small margins. Intuitively, if the margin is small, then any slight perturbation in the hyperplane or the training instances located at the boundary can have quite an impact on the classification performance. Small margin hyperplanes are thus more susceptible to overfitting, as they are barely able to separate the classes with a very narrow room to allow perturbations. On the other hand, a hyperplane that is farther away from training instances of both classes has sufficient leeway to be robust to minor modifications in the data, and thus shows superior generalization performance.
The idea of choosing the maximum margin separating hyperplane also has strong foundations in statistical learning theory. It can be shown that the margin of such a hyperplane is inversely related to the VC-dimension of the classifier, which is a commonly used measure of the complexity of a model. As discussed in Section 3.4 of the last chapter, a simpler model should be preferred over a more complex model if they both show similar training performance. Hence, maximizing the margin results in the selection of a separating hyperplane with the lowest model complexity, which is expected to show better generalization performance.
4.9.2 Linear SVM A linear SVM is a classifier that searches for a separating hyperplane with the largest margin, which is why it is often known as a maximal margin classifier. The basic idea of SVM can be described as follows.
Consider a binary classification problem consisting of n training instances, where every training instance xi is associated with a binary label yi∈{−1, 1}. Let wTx+b=0 be the equation of a separating hyperplane that separates the two classes by placing them on opposite sides. This means that
wTxi+b>0if yi=1,wTxi+b<0if yi=−1.
The distance of any point x from the hyperplane is then given by
D(x)=|wTx+b |ǁ w ǁ
where |⋅| denotes the absolute value and ǁ ⋅ ǁ denotes the length of a vector. Let the distance of the closest point from the hyperplane with y=1 be k+>0. Similarly, let k−>0 denote the distance of the closest point from class −1.
This can be represented using the following constraints:
wTxi+bǁ w ǁ≥k+if yi=1,wTxi+bǁ w ǁ≤−k−if yi=−1, (4.69)
The previous equations can be succinctly represented by using the product of yi and (wTxi+b) as
yi(wTxi+b)≥Mǁwǁ (4.70)
where M is a parameter related to the margin of the hyperplane, i.e., if k+=k −=M, then margin =k+−k−=2M. In order to find the maximum margin hyperplane that adheres to the previous constraints, we can consider the following optimization problem:
maxw, bMsubject toyi(wTxi+b)≥Mǁ w ǁ. (4.71)
To find the solution to the previous problem, note that if w and b satisfy the constraints of the previous problem, then any scaled version of w and b would satisfy them too. Hence, we can conveniently choose ǁwǁ=1/M to simplify the right-hand side of the inequalities. Furthermore, maximizing M amounts to minimizing ǁwǁ2. Hence, the optimization problem of SVM is commonly represented in the following form:
minw, bǁ w ǁ22subject toyi(wTxi+b)≥1. (4.72)
Learning Model Parameters Equation 4.72 represents a constrained optimization problem with linear inequalities. Since the objective function is convex and quadratic with respect
to w, it is known as a quadratic programming problem (QPP), which can be solved using standard optimization techniques, as described in Appendix E. In the following, we present a brief sketch of the main ideas for learning the model parameters of SVM.
First, we rewrite the objective function in a form that takes into account the constraints imposed on its solutions. The new objective function is known as the Lagrangian primal problem, which can be represented as follows,
LP=12ǁ w ǁ2−∑i=1nλi(yi(wTxi+b)−1), (4.73)
where the parameters λi≥0 correspond to the constraints and are called the Lagrange multipliers. Next, to minimize the Lagrangian, we take the derivative of LP with respect to w and b and set them equal to zero:
∂LP∂w=0⇒w=∑i=1nλiyixi, (4.74) ∂LP∂b=0⇒∑i=1nλiyi=0. (4.75)
Note that using Equation 4.74, we can represent w completely in terms of the Lagrange multipliers. There is another relationship between (w, b) and λi that is derived from the Karush-Kuhn-Tucker (KKT) conditions, a commonly used technique for solving QPP. This relationship can be described as
λi[yi(wTxi+b)−1]=0. (4.76)
Equation 4.76 is known as the complementary slackness condition, which sheds light on a valuable property of SVM. It states that the Lagrange multiplier λi is strictly greater than 0 only when xi satisfies the equation yi(w⋅xi+b)=1, which means that xi lies exactly on a margin hyperplane. However, if xi is farther away from the margin hyperplanes such that yi(w⋅xi+b)>1, then λi is necessarily 0. Hence, λi>0 for only a small number of instances that are closest to the separating hyperplane, which are known as support vectors. Figure 4.33 shows the support vectors of a hyperplane as filled circles and squares. Further, if we look at Equation 4.74, we will observe that training instances with λi=0 do not contribute to the weight parameter w. This suggests that w can be concisely represented only in terms of the support vectors in the training data, which are quite fewer than the overall number of training instances. This ability to represent the decision
function only in terms of the support vectors is what gives this classifier the name support vector machines.
Figure 4.33. Support vectors of a hyperplane shown as filled circles and squares.
Figure 4.33. Full Alternative Text
Using equations 4.74, 4.75, and 4.76 in Equation 4.73, we obtain the following optimization problem in terms of the Lagrange multipliers λi:
maxλi∑i=1nλi−12∑i=1n∑j=1nλiλjyiyjxiTxjsubject to∑i=1nλiyi=0,λi≥0. (4.77)
The previous optimization problem is called the dual optimization problem.
Maximizing the dual problem with respect to λi is equivalent to minimizing the primal problem with respect to w and b.
The key differences between the dual and primal problems are as follows:
1. Solving the dual problem helps us identify the support vectors in the data that have non-zero values of λi. Further, the solution of the dual problem is influenced only by the support vectors that are closest to the decision boundary of SVM. This helps in summarizing the learning of SVM solely in terms of its support vectors, which are easier to manage computationally. Further, it represents a unique ability of SVM to be dependent only on the instances closest to the boundary, which are harder to classify, rather than the distribution of instances farther away from the boundary.
2. The objective of the dual problem involves only terms of the form xiTxj, which are basically inner products in the attribute space. As we will see later in Section 4.9.4, this property will prove to be quite useful in learning nonlinear decision boundaries using SVM.
Because of these differences, it is useful to solve the dual optimization problem using any of the standard solvers for QPP. Having found an optimal solution for λi, we can use Equation 4.74 to solve for w. We can then use Equation 4.76 on the support vectors to solve for b as follows:
b=1nS∑i∈S1−yiwTxiyi (4.78)
where S represents the set of support vectors (S={i|λi>0}) and nS is the number of support vectors. The maximum margin hyperplane can then be expressed as
f(x)=(∑i=1nλiyixiTx)+b=0. (4.79)
Using this separating hyperplane, a test instance x can be assigned a class label using the sign of f(x).
Example 4.7.
Consider the two-dimensional data set shown in Figure 4.34, which contains eight training instances. Using quadratic programming, we can solve the optimization problem stated in Equation 4.77 to obtain the Lagrange multiplier λi for each training instance. The Lagrange multipliers are depicted in the last column of the table. Notice that only the first two instances have non-zero Lagrange multipliers. These instances correspond to the support vectors for this data set.
Let w=(w1, w2) and b denote the parameters of the decision boundary. Using Equation 4.74, we can solve for w1 and w2 in the following way:
w1=∑iλiyixi1=65.5261×1×0.3858+65.5261×−1×0.4871= −6.64.w2=∑iλiyixi2=65.5261×1×0.4687+65.5261×−1×0.611=−9.32.
Figure 4.34. Example of a linearly separable data set.
Figure 4.34. Full Alternative Text
The bias term b can be computed using Equation 4.76 for each support vector:
b(1)=1−w⋅x1=1−(−6.64)(0.3858)−(−9.32)(0.4687)=7.9300.b(2)=1−w⋅x2= −1−(−6.64)(0.4871)−(−9.32)(0.611)=7.9289.
Averaging these values, we obtain b=7.93. The decision boundary corresponding to these parameters is shown in Figure 4.34.
4.9.3 Soft-margin SVM Figure 4.35 shows a data set that is similar to Figure 4.32, except it has two new examples, P and Q. Although the decision boundary B1 misclassifies the new examples, while B2 classifies them correctly, this does not mean that B2 is a better decision boundary than B1 because the new examples may correspond to noise in the training data. B1 should still be preferred over B2 because it has a wider margin, and thus, is less susceptible to overfitting. However, the SVM formulation presented in the previous section only constructs decision boundaries that are mistake-free.
Figure 4.35. Decision boundary of SVM for the non-separable case.
Figure 4.35. Full Alternative Text
This section examines how the formulation of SVM can be modified to learn a separating hyperplane that is tolerable to small number of training errors using a method known as the soft-margin approach. More importantly, the method presented in this section allows SVM to learn linear hyperplanes even in situations where the classes are not linearly separable. To do this, the learning algorithm in SVM must consider the trade-off between the width of the margin and the number of training errors committed by the linear hyperplane.
To introduce the concept of training errors in the SVM formulation, let us relax the inequality constraints to accommodate for some violations on a small number of training instances. This can be done by introducing a slack variable ξ≥0 for every training instance xi as follows:
yi(wTxi+b)≥1−ξi (4.80)
The variable ξi allows for some slack in the inequalities of the SVM such that every instance xi does not need to strictly satisfy yi(wTxi+b)≥1. Further, ξi is non-zero only if the margin hyperplanes are not able to place xi on the same side as the rest of the instances belonging to yi. To illustrate this, Figure 4.36 shows a circle P that falls on the opposite side of the separating hyperplane as the rest of the circles, and thus satisfies wTx+b=−1+ξ. The distance between P and the margin hyperplane wTx+b=−1 is equal to ξ/ǁ w ǁ. Hence, ξi provides a measure of the error of SVM in representing xi using soft inequality constraints.
Figure 4.36. Slack variables used in soft-margin SVM.
Figure 4.36. Full Alternative Text
In the presence of slack variables, it is important to learn a separating hyperplane that jointly maximizes the margin (ensuring good generalization performance) and minimizes the values of slack variables (ensuring low training error). This can be achieved by modifying the optimization problem of SVM as follows:
minw, b, ξiǁ w ǁ22+c∑i=1nξisubject toyi(wTxi+b)≥1−ξi,ξi≥0. (4.81)
where C is a hyper-parameter that makes a trade-off between maximizing the margin and minimizing the training error. A large value of C pays more emphasis on minimizing the training error than maximizing the margin. Notice the similarity of the previous equation with the generic formula of generalization error rate introduced in Section 3.4 of the previous chapter. Indeed, SVM provides a natural way to balance between model complexity and training error in order to maximize generalization performance.
To solve Equation 4.81 we apply the Lagrange multiplier method and convert the primal problem to its corresponding dual problem, similar to the approach described in the previous section. The Lagrangian primal problem of Equation 4.81 can be written as follows:
LP=12ǁ w ǁ2+C∑i=1nξi−∑i=1nλi(yi(wTxi+b)−1+ξi)−∑i=1nμi(ξi), (4.82)
where λi≥0 and μi≥0 are the Lagrange multipliers corresponding to the inequality constraints of Equation 4.81. Setting the derivative of LP with respect to w, b, and ξi equal to 0, we obtain the following equations:
∂LP∂w=0⇒w=∑i=1nλiyixi. (4.83) ∂L∂b=0⇒∑i=1nλiyi=0. (4.84) ∂L∂ξi=0⇒λi+μi=C. (4.85)
We can also obtain the complementary slackness conditions by using the following KKT conditions:
λi(yi(wTxi+b)−1+ξi)=0, (4.86) μiξi=0. (4.87)
Equation 4.86 suggests that λi is zero for all training instances except those that reside on the margin hyperplanes wTxi+b=±1, or have ξi>0. These instances with λi>0 are known as support vectors. On the other hand, μi given in Equation 4.87 is zero for any training instance that is misclassified, i.e., ξi>0. Further, λi and μi are related with each other by Equation 4.85. This results in the following three configurations of (λi, μi):
1. If λi=0 and μi=C, then xi does not reside on the margin hyperplanes and is correctly classified on the same side as other instances belonging to yi.
2. If λi=C and μi=0, then xi is misclassified and has a non-zero slack variable ξi.
3. If 0<λi<C and 0<μi<C, then xi resides on one of the margin hyperplanes.
Substituting Equations 4.83 to 4.87 into Equation 4.82, we obtain the following dual optimization problem:
maxλi∑i=1nλi−12∑i=1n∑j=1nλiλjyiyjxiTxjsubject to∑i=1nλiyi=0,0≤λi≤C. (4.88)
Notice that the previous problem looks almost identical to the dual problem of SVM for the linearly separable case (Equation 4.77), except that λi is required to not only be greater than 0 but also smaller than a constant value C. Clearly, when C reaches infinity, the previous optimization problem becomes equivalent to Equation 4.77, where the learned hyperplane perfectly separates the classes (with no training errors). However, by capping the values of λi to C, the learned hyperplane is able to tolerate a few training errors that have ξi>0.
Figure 4.37. Hinge loss as a function of yy^.
As before, Equation 4.88 can be solved by using any of the standard solvers for QPP, and the optimal value of w can be obtained by using Equation 4.83. To solve for b, we can use Equation 4.86 on the support vectors that reside on the margin hyperplanes as follows:
b=1nS∑i∈S1−yiwTxiyi (4.89)
where S represents the set of support vectors residing on the margin hyperplanes (S={i|0<λi<C}) and nS is the number of elements in S.
SVM as a Regularizer of Hinge Loss
SVM belongs to a broad class of regularization techniques that use a loss function to represent the training errors and a norm of the model parameters to represent the model complexity. To realize this, notice that the slack variable ξ, used for measuring the training errors in SVM, is equivalent to the hinge loss function, which can be defined as follows:
Loss (y, y^) =max(0, 1−yy^),
where y∈{+1, −1}. In the case of SVM, y^ corresponds to wTx+b. Figure 4.37 shows a plot of the hinge loss function as we vary yy^. We can see that the hinge loss is equal to 0 as long as y and y^ have the same sign and |y^|≥1. However, the hinge loss grows linearly with |y^| whenever y and y^ are of the opposite sign or |y^|<1. This is similar to the notion of the slack variable ξ, which is used to measure the distance of a point from its margin hyperplane. Hence, the optimization problem of SVM can be represented in the following equivalent form:
minw, bǁ w ǁ22+C∑i=1nLoss (yi, wTxi+b) (4.90)
Note that using the hinge loss ensures that the optimization problem is convex and can be solved using standard optimization techniques. However, if we use a different loss function, such as the squared loss function that was introduced in Section 4.7 on ANN, it will result in a different optimization problem that may or may not remain convex. Nevertheless, different loss functions can be explored to capture varying notions of training error, depending on the characteristics of the problem.
Another interesting property of SVM that relates it to a broader class of regularization techniques is the concept of a margin. Although minimizing ǁ w ǁ2 has the geometric interpretation of maximizing the margin of a separating hyperplane, it is essentially the squared L2 norm of the model parameters, ǁ w ǁ22. In general, the Lq norm of w, ǁ w ǁq, is equal to the Minkowski distance of order q from w to the origin, i.e.,
ǁ w ǁq=(∑ipwiq)1/q
Minimizing the Lq norm of w to achieve lower model complexity is a generic regularization concept that has several interpretations. For example,
minimizing the L2 norm amounts to finding a solution on a hypersphere of smallest radius that shows suitable training performance. To visualize this in two-dimensions, Figure 4.38(a) shows the plot of a circle with constant radius r, where every point has the same L2 norm. On the other hand, using the L1 norm ensures that the solution lies on the surface of a hypercube with smallest size, with vertices along the axes. This is illustrated in Figure 4.38(b) as a square with vertices on the axes at a distance of r from the origin. The L1 norm is commonly used as a regularizer to obtain sparse model parameters with only a small number of non-zero parameter values, such as the use of Lasso in regression problems (see Bibliographic Notes).
Figure 4.38. Plots showing the behavior of two-dimensional solutions with constant L2 and L1 norms.
Figure 4.38. Full Alternative Text
In general, depending on the characteristics of the problem, different combinations of Lq norms and training loss functions can be used for learning the model parameters, each requiring a different optimization solver. This forms the backbone of a wide range of modeling techniques that attempt to improve the generalization performance by jointly minimizing training error and model complexity. However, in this section, we focus only on the squared L2 norm and the hinge loss function, resulting in the classical formulation of SVM.
4.9.4 Nonlinear SVM The SVM formulations described in the previous sections construct a linear decision boundary to separate the training examples into their respective classes. This section presents a methodology for applying SVM to data sets that have nonlinear decision boundaries. The basic idea is to transform the data from its original attribute space in x into a new space φ(x) so that a linear hyperplane can be used to separate the instances in the transformed space, using the SVM approach. The learned hyperplane can then be projected back to the original attribute space, resulting in a nonlinear decision boundary.
Figure 4.39. Classifying data with a nonlinear decision boundary.
Figure 4.39. Full Alternative Text
Attribute Transformation To illustrate how attribute transformation can lead to a linear decision boundary, Figure 4.39(a) shows an example of a two-dimensional data set consisting of squares (classified as y=1) and circles (classified as y=−1). The data set is generated in such a way that all the circles are clustered near the center of the diagram and all the squares are distributed farther away from the center. Instances of the data set can be classified using the following equation:
y={1if (x1−0.5)2+(x2−0.5)2>0.2,−1otherwise. (4.91)
The decision boundary for the data can therefore be written as follows:
(x1−0.5)2+(x2−0.5)2>0.2,
which can be further simplified into the following quadratic equation:
x12−x1+x22−x2=−0.46.
A nonlinear transformation φ is needed to map the data from its original attribute space into a new space such that a linear hyperplane can separate the classes. This can be achieved by using the following simple transformation:
φ:(x1, x2)→(x12−x1, x22−x2). (4.92)
Figure 4.39(b) shows the points in the transformed space, where we can see that all the circles are located in the lower left-hand side of the diagram. A linear hyperplane with parameters w and b can therefore be constructed in the transformed space, to separate the instances into their respective classes.
One may think that because the nonlinear transformation possibly increases the dimensionality of the input space, this approach can suffer from the curse of dimensionality that is often associated with high-dimensional data. However, as we will see in the following section, nonlinear SVM is able to avoid this problem by using kernel functions.
Learning a Nonlinear SVM Model Using a suitable function, φ(⋅), we can transform any data instance x to φ(x). (The details on how to choose φ(⋅) will become clear later.) The linear hyperplane in the transformed space can be expressed as wTφ(x)+b=0. To learn the optimal separating hyperplane, we can substitute φ(x) for x in the formulation of SVM to obtain the following optimization problem:
minw, b, ξiǁ w ǁ22+C∑i=1nξisubject toyi(wTφ(xi)+b)≥1−ξi,ξi≥0. (4.93)
Using Lagrange multipliers λi, this can be converted into a dual optimization problem: max
maxλi∑i=1nλi−12∑i=1n∑j=1nλiλjyiyj⟨φ(xi), φ(xj) ⟩subject to∑i=1nλjyi=0,0≤λi≤C, (4.94)
where ⟨ a, b ⟩ denotes the inner product between vectors a and b. Also, the equation of the hyperplane in the transformed space can be represented using
λi as follows:
∑i=1nλiyi⟨φ(xi), φ(x) ⟩+b=0. (4.95)
Further, b is given by
b=1nS(∑i∈S1yi−∑i∈S∑j=1nλjyiyj⟨φ(xi), φ(xj) ⟩yi), (4.96)
where S={i|0>λi<C} is the set of support vectors residing on the margin hyperplanes and nS is the number of elements in S.
Note that in order to solve the dual optimization problem in Equation 4.94, or
to use the learned model parameters to make predictions using Equations 4.95 and 4.96, we need only inner products of φ(x). Hence, even though φ(x) may be nonlinear and high-dimensional, it suffices to use a function of the inner products of φ(x) in the transformed space. This can be achieved by using a kernel trick, which can be described as follows.
The inner product between two vectors is often regarded as a measure of similarity between the vectors. For example, the cosine similarity described in Section 2.4.5 on page 79 can be defined as the dot product between two vectors that are normalized to unit length. Analogously, the inner product φ(xi), φ(xj) can also be regarded as a measure of similarity between two instances, xi and xj, in the transformed space. The kernel trick is a method for computing this similarity as a function of the original attributes. Specifically, the kernel function K(u, v) between two instances u and v can be defined as follows:
K(u, v)=⟨φ(u), φ(v) ⟩=f(u, v) (4.97)
where f(⋅) is a function that follows certain conditions as stated by the Mercer's Theorem. Although the details of this theorem are outside the scope of the book, we provide a list of some of the commonly used kernel functions:
Polynomial kernelK(u, v)=(uTv+1)p (4.98) Radial Basis Function kernelK(u, v)=e−ǁu−v ǁ2/(2σ2) (4.99) Sigmoid kernelK(u, v)=tanh(kuTv−δ) (4.100)
By using a kernel function, we can directly work with inner products in the transformed space without dealing with the exact forms of the nonlinear transformation function φ. Specifically, this allows us to use high- dimensional transformations (sometimes even involving infinitely many dimensions), while performing calculations only in the original attribute space. Computing the inner products using kernel functions is also considerably cheaper than using the transformed attribute set φ(x). Hence, the use of kernel functions provides a significant advantage in representing nonlinear decision boundaries, without suffering from the curse of dimensionality. This has been one of the major reasons behind the widespread usage of SVM in highly complex and nonlinear problems.
Figure 4.40. Decision boundary produced by a nonlinear SVM with polynomial kernel.
Figure 4.40 shows the nonlinear decision boundary obtained by SVM using the polynomial kernel function given in Equation 4.98. We can see that the learned decision boundary is quite close to the true decision boundary shown in Figure 4.39(a). Although the choice of kernel function depends on the characteristics of the input data, a commonly used kernel function is the radial basis function (RBF) kernel, which involves a single hyper-parameter σ, known as the standard deviation of the RBF kernel.
4.9.5 Characteristics of SVM
1. The SVM learning problem can be formulated as a convex optimization problem, in which efficient algorithms are available to find the global minimum of the objective function. Other classification methods, such as rule-based classifiers and artificial neural networks, employ a greedy strategy to search the hypothesis space. Such methods tend to find only locally optimum solutions.
2. SVM provides an effective way of regularizing the model parameters by maximizing the margin of the decision boundary. Furthermore, it is able to create a balance between model complexity and training errors by using a hyper-parameter C. This trade-off is generic to a broader class of model learning techniques that capture the model complexity and the training loss using different formulations.
3. Linear SVM can handle irrelevant attributes by learning zero weights corresponding to such attributes. It can also handle redundant attributes by learning similar weights for the duplicate attributes. Furthermore, the ability of SVM to regularize its learning makes it more robust to the presence of a large number of irrelevant and redundant attributes than other classifiers, even in high-dimensional settings. For this reason, nonlinear SVMs are less impacted by irrelevant and redundant attributes than other highly expressive classifiers that can learn nonlinear decision boundaries such as decision trees.
To compare the effect of irrelevant attributes on the performance of nonlinear SVMs and decision trees, consider the two-dimensional data set shown in Figure 4.41(a) containing 500+ and 500o instances, where the two classes can be easily separated using a nonlinear decision boundary. We incrementally add irrelevant attributes to this data set and compare the performance of two classifiers: decision tree and nonlinear SVM (using radial basis function kernel), using 70% of the data for training and the rest for testing. Figure 4.41(b) shows the test error rates of the two classifiers as we increase the number of irrelevant attributes. We can see that the test error rate of decision trees swiftly reaches 0.5 (same as random guessing) in the presence of even a small number of irrelevant attributes. This can be attributed to the problem of multiple comparisons while choosing splitting attributes at internal nodes as
discussed in Example 3.7 of the previous chapter. On the other hand, nonlinear SVM shows a more robust and steady performance even after adding a moderately large number of irrelevant attributes. Its test error rate gradually declines and eventually reaches close to 0.5 after adding 125 irrelevant attributes, at which point it becomes difficult to discern the discriminative information in the original two attributes from the noise in the remaining attributes for learning nonlinear decision boundaries.
Figure 4.41. Comparing the effect of adding irrelevant attributes on the performance of nonlinear SVMs and decision trees.
Figure 4.41. Full Alternative Text
4. SVM can be applied to categorical data by introducing dummy variables for each categorical attribute value present in the data. For example, if Marital Status has three values {Single,Married,Divorced}, we can
introduce a binary variable for each of the attribute values.
5. The SVM formulation presented in this chapter is for binary class problems. However, multiclass extensions of SVM have also been proposed.
6. Although the training time of an SVM model can be large, the learned parameters can be succinctly represented with the help of a small number of support vectors, making the classification of test instances quite fast.
4.10 Ensemble Methods This section presents techniques for improving classification accuracy by aggregating the predictions of multiple classifiers. These techniques are known as ensemble or classifier combination methods. An ensemble method constructs a set of base classifiers from training data and performs classification by taking a vote on the predictions made by each base classifier. This section explains why ensemble methods tend to perform better than any single classifier and presents techniques for constructing the classifier ensemble.
4.10.1 Rationale for Ensemble Method The following example illustrates how an ensemble method can improve a classifier's performance.
Example 4.8. Consider an ensemble of 25 binary classifiers, each of which has an error rate of ∈=0.35. The ensemble classifier predicts the class label of a test example by taking a majority vote on the predictions made by the base classifiers. If the base classifiers are identical, then all the base classifiers will commit the same mistakes. Thus, the error rate of the ensemble remains 0.35. On the other hand, if the base classifiers are independent—i.e., their errors are uncorrelated—then the ensemble makes a wrong prediction only if more than half of the base classifiers predict incorrectly. In this case, the error rate of the ensemble classifier is
eensemble=∑i=1325(25i)∈i(1−∈)25−i=0.06, (4.101)
which is considerably lower than the error rate of the base classifiers.
Figure 4.42 shows the error rate of an ensemble of 25 binary classifiers (eensemble) for different base classifier error rates (∈). The diagonal line represents the case in which the base classifiers are identical, while the solid line represents the case in which the base classifiers are independent. Observe that the ensemble classifier performs worse than the base classifiers when ∈ is larger than 0.5.
The preceding example illustrates two necessary conditions for an ensemble classifier to perform better than a single classifier: (1) the base classifiers should be independent of each other, and (2) the base classifiers should do better than a classifier that performs random guessing. In practice, it is difficult to ensure total independence among the base classifiers. Nevertheless, improvements in classification accuracies have been observed in ensemble methods in which the base classifiers are somewhat correlated.
4.10.2 Methods for Constructing an Ensemble Classifier A logical view of the ensemble method is presented in Figure 4.43. The basic idea is to construct multiple classifiers from the original data and then aggregate their predictions when classifying unknown examples. The ensemble of classifiers can be constructed in many ways:
1. By manipulating the training set. In this approach, multiple training sets are created by resampling the original data according to some sampling distribution and constructing a classifier from each training set. The sampling distribution determines how likely it is that an example will be selected for training, and it may vary from one trial to another. Bagging and boosting are two examples of ensemble methods that manipulate their training sets. These methods are described in further detail in Sections 4.10.4 and 4.10.5.
Figure 4.42. Comparison between errors of base classifiers and errors of the ensemble classifier.
Figure 4.43. A logical view of the ensemble learning method.
Figure 4.43. Full Alternative Text
2. By manipulating the input features. In this approach, a subset of input features is chosen to form each training set. The subset can be either chosen randomly or based on the recommendation of domain experts. Some studies have shown that this approach works very well with data sets that contain highly redundant features. Random forest, which is described in Section 4.10.6, is an ensemble method that manipulates its input features and uses decision trees as its base classifiers.
3. By manipulating the class labels. This method can be used when the number of classes is sufficiently large. The training data is transformed into a binary class problem by randomly partitioning the class labels into
two disjoint subsets, A0 and A1. Training examples whose class label belongs to the subset A0 are assigned to class 0, while those that belong to the subset A1 are assigned to class 1. The relabeled examples are then used to train a base classifier. By repeating this process multiple times, an ensemble of base classifiers is obtained. When a test example is presented, each base classifier Ci is used to predict its class label. If the test example is predicted as class 0, then all the classes that belong to A0 will receive a vote. Conversely, if it is predicted to be class 1, then all the classes that belong to A1 will receive a vote. The votes are tallied and the class that receives the highest vote is assigned to the test example. An example of this approach is the error-correcting output coding method described on page 331.
4. By manipulating the learning algorithm. Many learning algorithms can be manipulated in such a way that applying the algorithm several times on the same training data will result in the construction of different classifiers. For example, an artificial neural network can change its network topology or the initial weights of the links between neurons. Similarly, an ensemble of decision trees can be constructed by injecting randomness into the tree-growing procedure. For example, instead of choosing the best splitting attribute at each node, we can randomly choose one of the top k attributes for splitting.
The first three approaches are generic methods that are applicable to any classifier, whereas the fourth approach depends on the type of classifier used. The base classifiers for most of these approaches can be generated sequentially (one after another) or in parallel (all at once). Once an ensemble of classifiers has been learned, a test example x is classified by combining the predictions made by the base classifiers Ci(x):
C*(x)=f(C1(x), C2(x), …, Ck(x)).
where f is the function that combines the ensemble responses. One simple approach for obtaining C*(x) is to take a majority vote of the individual predictions. An alternate approach is to take a weighted majority vote, where the weight of a base classifier denotes its accuracy or relevance.
Ensemble methods show the most improvement when used with unstable
classifiers, i.e., base classifiers that are sensitive to minor perturbations in the training set, because of high model complexity. Although unstable classifiers may have a low bias in finding the optimal decision boundary, their predictions have a high variance for minor changes in the training set or model selection. This trade-off between bias and variance is discussed in detail in the next section. By aggregating the responses of multiple unstable classifiers, ensemble learning attempts to minimize their variance without worsening their bias.
4.10.3 Bias-Variance Decomposition Bias-variance decomposition is a formal method for analyzing the generalization error of a predictive model. Although the analysis is slightly different for classification than regression, we first discuss the basic intuition of this decomposition by using an analogue of a regression problem.
Consider the illustrative task of reaching a target y by firing projectiles from a starting position x, as shown in Figure 4.44. The target corresponds to the desired output at a test instance, while the starting position corresponds to its observed attributes. In this analogy, the projectile represents the model used for predicting the target using the observed attributes. Let y^ denote the point where the projectile hits the ground, which is analogous of the prediction of the model.
Figure 4.44. Bias-variance decomposition.
Figure 4.44. Full Alternative Text
Ideally, we would like our predictions to be as close to the true target as possible. However, note that different trajectories of projectiles are possible based on differences in the training data or in the approach used for model selection. Hence, we can observe a variance in the predictions y^ over different runs of projectile. Further, the target in our example is not fixed but has some freedom to move around, resulting in a noise component in the true target. This can be understood as the non-deterministic nature of the output variable, where the same set of attributes can have different output values. Let y^avg represent the average prediction of the projectile over multiple runs, and yavg denote the average target value. The difference between y^avg and yavg is known as the bias of the model.
In the context of classification, it can be shown that the generalization error of a classification model m can be decomposed into terms involving the bias, variance, and noise components of the model in the following way:
gen.error(m)=c1×noise+bias(m)+c2×variance(m)
where c1 and c2 are constants that depend on the characteristics of training and test sets. Note that while the noise term is intrinsic to the target class, the bias and variance terms depend on the choice of the classification model. The bias of a model represents how close the average prediction of the model is to the average target. Models that are able to learn complex decision boundaries, e.g., models produced by k-nearest neighbor and multi-layer ANN, generally show low bias. The variance of a model captures the stability of its predictions in response to minor perturbations in the training set or the model selection approach.
We can say that a model shows better generalization performance if it has a lower bias and lower variance. However, if the complexity of a model is high but the training size is small, we generally expect to see a lower bias but
higher variance, resulting in the phenomena of overfitting. This phenomena is pictorially represented in Figure 4.45(a). On the other hand, an overly simplistic model that suffers from underfitting may show a lower variance but would suffer from a high bias, as shown in Figure 4.45(b). Hence, the trade-off between bias and variance provides a useful way for interpreting the effects of underfitting and overfitting on the generalization performance of a model.
Figure 4.45. Plots showing the behavior of two-dimensional solutions with constant L2 and L1 norms.
Figure 4.45. Full Alternative Text
The bias-variance trade-off can be used to explain why ensemble learning improves the generalization performance of unstable classifiers. If a base classifier show low bias but high variance, it can become susceptible to overfitting, as even a small change in the training set will result in different predictions. However, by combining the responses of multiple base classifiers, we can expect to reduce the overall variance. Hence, ensemble learning methods show better performance primarily by lowering the variance in the predictions, although they can even help in reducing the bias. One of the simplest approaches for combining predictions and reducing their variance is to compute their average. This forms the basis of the bagging method, described in the following subsection.
4.10.4 Bagging Bagging, which is also known as bootstrap aggregating, is a technique that repeatedly samples (with replacement) from a data set according to a uniform probability distribution. Each bootstrap sample has the same size as the original data. Because the sampling is done with replacement, some instances may appear several times in the same training set, while others may be omitted from the training set. On average, a bootstrap sample Di contains approximately 63% of the original training data because each sample has a probability 1−(1−1/N)N of being selected in each Di. If N is sufficiently large, this probability converges to 1−1/e≃0.632. The basic procedure for bagging is summarized in Algorithm 4.5. After training the k classifiers, a test instance is assigned to the class that receives the highest number of votes.
To illustrate how bagging works, consider the data set shown in Table 4.4. Let x denote a one-dimensional attribute and y denote the class label. Suppose we use only one-level binary decision trees, with a test condition x≤k, where k is a split point chosen to minimize the entropy of the leaf nodes. Such a tree is also known as a decision stump.
Table 4.4. Example of data set
used to construct an ensemble of bagging classifiers. x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y 1 1 1 −1 −1 −1 −1 1 1 1
Without bagging, the best decision stump we can produce splits the instances at either x≤0.35 or x≤0.75. Either way, the accuracy of the tree is at most 70%. Suppose we apply the bagging procedure on the data set using 10 bootstrap samples. The examples chosen for training in each bagging round are shown in Figure 4.46. On the right-hand side of each table, we also describe the decision stump being used in each round.
We classify the entire data set given in Table 4.4 by taking a majority vote among the predictions made by each base classifier. The results of the predictions are shown in Figure 4.47. Since the class labels are either −1 or +1, taking the majority vote is equivalent to summing up the predicted values of y and examining the sign of the resulting sum (refer to the second to last row in Figure 4.47). Notice that the ensemble classifier perfectly classifies all 10 examples in the original data.
Algorithm 4.5 Bagging algorithm. 1: Let k be the number of bootstrap samples. 2: for i = 1 to k do 3: Create a bootstrap sample of size N, Di. 4: Train a base classifier Ci on the bootstrap sample Di. 5: end for 6: C*(x)=argmaxy∑iδ(Ci(x)=y). {δ(⋅)=1 if its argument is true and 0 otherwise.}
Figure 4.46. Example of bagging.
Figure 4.46. Full Alternative Text
The preceding example illustrates another advantage of using ensemble methods in terms of enhancing the representation of the target function. Even though each base classifier is a decision stump, combining the classifiers can lead to a decision boundary that mimics a decision tree of depth 2.
Bagging improves generalization error by reducing the variance of the base classifiers. The performance of bagging depends on the stability of the base classifier. If a base classifier is unstable, bagging helps to reduce the errors associated with random fluctuations in the training data. If a base classifier is stable, i.e., robust to minor perturbations in the training set, then the error of the ensemble is primarily caused by bias in the base classifier. In this situation, bagging may not be able to improve the performance of the base classifiers significantly. It may even degrade the classifier's performance because the effective size of each training set is about 37% smaller than the original data.
Figure 4.47. Example of combining classifiers constructed using the bagging approach.
4.10.5 Boosting Boosting is an iterative procedure used to adaptively change the distribution of training examples for learning base classifiers so that they increasingly focus on examples that are hard to classify. Unlike bagging, boosting assigns a weight to each training example and may adaptively change the weight at the end of each boosting round. The weights assigned to the training examples can be used in the following ways:
1. They can be used to inform the sampling distribution used to draw a set of bootstrap samples from the original data.
2. They can be used to learn a model that is biased toward examples with higher weight.
This section describes an algorithm that uses weights of examples to determine the sampling distribution of its training set. Initially, the examples are assigned equal weights, 1/N, so that they are equally likely to be chosen for training. A sample is drawn according to the sampling distribution of the training examples to obtain a new training set. Next, a classifier is built from the training set and used to classify all the examples in the original data. The weights of the training examples are updated at the end of each boosting round. Examples that are classified incorrectly will have their weights increased, while those that are classified correctly will have their weights decreased. This forces the classifier to focus on examples that are difficult to classify in subsequent iterations.
The following table shows the examples chosen during each boosting round, when applied to the data shown in Table 4.4.
Boosting (Round 1): 7 3 2 8 7 9 4 10 6 3 Boosting (Round 2): 5 4 9 4 2 5 1 7 4 2 Boosting (Round 3): 4 4 8 10 4 5 4 6 3 4
Initially, all the examples are assigned the same weights. However, some examples may be chosen more than once, e.g., examples 3 and 7, because the sampling is done with replacement. A classifier built from the data is then used to classify all the examples. Suppose example 4 is difficult to classify. The weight for this example will be increased in future iterations as it gets misclassified repeatedly. Meanwhile, examples that were not chosen in the previous round, e.g., examples 1 and 5, also have a better chance of being selected in the next round since their predictions in the previous round were likely to be wrong. As the boosting rounds proceed, examples that are the hardest to classify tend to become even more prevalent. The final ensemble is obtained by aggregating the base classifiers obtained from each boosting round.
Over the years, several implementations of the boosting algorithm have been developed. These algorithms differ in terms of (1) how the weights of the training examples are updated at the end of each boosting round, and (2) how the predictions made by each classifier are combined. An implementation called AdaBoost is explored in the next section.
AdaBoost Let {(xj, yj)|j=1, 2, …, N} denote a set of N training examples. In the AdaBoost algorithm, the importance of a base classifier Ci depends on its
Figure 4.48. Plot of α as a function of training error ∈.
error rate, which is defined as
∈i=1N[∑j=1Nwj I(Ci(xj)≠yj) ], (4.102)
where I(p)=1 if the predicate p is true, and 0 otherwise. The importance of a classifier Ci is given by the following parameter,
αi=12ln (1−∈i∈i).
Note that αi has a large positive value if the error rate is close to 0 and a large negative value if the error rate is close to 1, as shown in Figure 4.48.
The αi parameter is also used to update the weight of the training examples. To illustrate, let wi(j) denote the weight assigned to example (xi, yi) during the jth boosting round. The weight update mechanism for AdaBoost is given
by the equation:
wi(j+1)=wi(j)Zj×{e−αjif Cj(xi)=yi,eαjif Cj(xi)≠yi (4.103)
where Zj is the normalization factor used to ensure that ∑iwi(j+1)=1. The weight update formula given in Equation 4.103 increases the weights of incorrectly classified examples and decreases the weights of those classified correctly.
Instead of using a majority voting scheme, the prediction made by each classifier Cj is weighted according to αj. This approach allows AdaBoost to penalize models that have poor accuracy, e.g., those generated at the earlier boosting rounds. In addition, if any intermediate rounds produce an error rate higher than 50%, the weights are reverted back to their original uniform values, wi=1/N, and the resampling procedure is repeated. The AdaBoost algorithm is summarized in Algorithm 4.6.
Algorithm 4.6 AdaBoost algorithm. 1: w = {wj = 1/N | j = 1, 2,…,N}. {Initialize the weights for all 2: Let k be the number of boosting rounds. 3: for i = 1 to k do 4: Create training set Di by sampling (with replacement) from 5: Train a base classifier Ci on Di. 6: Apply Ci to all examples in the original training set, D. 7: ∈i=1N[ ∑jwj δ(Ci(xj)≠yj) ] {Calculate the weighted error.} 8: if ∈i > 0.5 then 9: w = {wj = 1/N | j = 1, 2,…,N}. {Reset the weights for all 10: Go back to Step 4. 11: end if 12: αi=12ln1−∈i∈i. 13: Update the weight of each example according to Equation 4.103 14: end for 15: C*(x)=argmaxy∑j=1Tαjδ(Cj(x)=y))..
Let us examine how the boosting approach works on the data set shown in Table 4.4. Initially, all the examples have identical weights. After three boosting rounds, the examples chosen for training are shown in Figure 4.49(a). The weights for each example are updated at the end of each
boosting round using Equation 4.103, as shown in Figure 4.50(b).
Without boosting, the accuracy of the decision stump is, at best, 70%. With AdaBoost, the results of the predictions are given in Figure 4.50(b). The final prediction of the ensemble classifier is obtained by taking a weighted average of the predictions made by each base classifier, which is shown in the last row of Figure 4.50(b). Notice that AdaBoost perfectly classifies all the examples in the training data.
Figure 4.49. Example of boosting.
Figure 4.49. Full Alternative Text
An important analytical result of boosting shows that the training error of the
ensemble is bounded by the following expression:
eensemble≤∏i[∈i(1−∈i) ], (4.104)
where ∈i is the error rate of each base classifier i. If the error rate of the base classifier is less than 50%, we can write ∈i=0.5 −γi, where γi measures how much better the classifier is than random guessing. The bound on the training error of the ensemble becomes
eensemble≤∏i1−4γi2≤exp(−2∑iγi2). (4.105)
Hence, the training error of the ensemble decreases exponentially, which leads to the fast convergence of the algorithm. By focusing on examples that are difficult to classify by base classifiers, it is able to reduce the bias of the final predictions along with the variance. AdaBoost has been shown to provide significant improvements in performance over base classifiers on a range of data sets. Nevertheless, because of its tendency to focus on training examples that are wrongly classified, the boosting technique can be susceptible to overfitting, resulting in poor generalization performance in some scenarios.
Figure 4.50. Example of combining classifiers constructed using the AdaBoost approach.
4.10.6 Random Forests Random forests attempt to improve the generalization performance by constructing an ensemble of decorrelated decision trees. Random forests build on the idea of bagging to use a different bootstrap sample of the training data for learning decision trees. However, a key distinguishing feature of random forests from bagging is that at every internal node of a tree, the best splitting criterion is chosen among a small set of randomly selected attributes. In this way, random forests construct ensembles of decision trees by not only manipulating training instances (by using bootstrap samples
similar to bagging), but also the input attributes (by using different subsets of attributes at every internal node).
Given a training set D consisting of n instances and d attributes, the basic procedure of training a random forest classifier can be summarized using the following steps:
1. Construct a bootstrap sample Di of the training set by randomly sampling n instances (with replacement) from D.
2. Use Di to learn a decision tree Ti as follows. At every internal node of Ti, randomly sample a set of p attributes and choose an attribute from this subset that shows the maximum reduction in an impurity measure for splitting. Repeat this procedure till every leaf is pure, i.e., containing instances from the same class.
Once an ensemble of decision trees have been constructed, their average prediction (majority vote) on a test instance is used as the final prediction of the random forest. Note that the decision trees involved in a random forest are unpruned trees, as they are allowed to grow to their largest possible size till every leaf is pure. Hence, the base classifiers of random forest represent unstable classifiers that have low bias but high variance, because of their large size.
Another property of the base classifiers learned in random forests is the lack of correlation among their model parameters and test predictions. This can be attributed to the use of an independently sampled data set Di for learning every decision tree Ti, similar to the bagging approach. However, random forests have the additional advantage of choosing a splitting criterion at every internal node using a different (and randomly selected) subset of attributes. This property significantly helps in breaking the correlation structure, if any, among the decision trees Ti.
To realize this, consider a training set involving a large number of attributes, where only a small subset of attributes are strong predictors of the target class, whereas other attributes are weak indicators. Given such a training set, even if we consider different bootstrap samples Di for learning Ti, we would mostly be choosing the same attributes for splitting at internal nodes, because
the weak attributes would be largely overlooked when compared with the strong predictors. This can result in a considerable correlation among the trees. However, if we restrict the choice of attributes at every internal node to a random subset of attributes, we can ensure the selection of both strong and weak predictors, thus promoting diversity among the trees. This principle is utilized by random forests for creating decorrelated decision trees.
By aggregating the predictions of an ensemble of strong and decorrelated decision trees, random forests are able to reduce the variance of the trees without negatively impacting their low bias. This makes random forests quite robust to overfitting. Additionally, because of their ability to consider only a small subset of attributes at every internal node, random forests are computationally fast and robust even in high-dimensional settings.
The number of attributes to be selected at every node, p, is a hyper-parameter of the random forest classifier. A small value of p can reduce the correlation among the classifiers but may also reduce their strength. A large value can improve their strength but may result in correlated trees similar to bagging. Although common suggestions for p in the literature include d and log2d+1, a suitable value of p for a given training set can always be selected by tuning it over a validation set, as described in the previous chapter. However, there is an alternative way for selecting hyper-parameters in random forests, which does not require using a separate validation set. It involves computing a reliable estimate of the generalization error rate directly during training, known as the out-of-bag (oob) error estimate. The oob estimate can be computed for any generic ensemble learning method that builds independent base classifiers using bootstrap samples of the training set, e.g., bagging and random forests. The approach for computing oob estimate can be described as follows.
Consider an ensemble learning method that uses an independent base classifier Ti built on a bootstrap sample of the training set Di. Since every training instance x will be used for training approximately 63% of base classifiers, we can call x as an out-of-bag sample for the remaining 27% of base classifiers that did not use it for training. If we use these remaining 27% classifiers to make predictions on x, we can obtain the oob error on x by taking their majority vote and comparing it with its class label. Note that the
oob error estimates the error of 27% classifiers on an instance that was not used for training those classifiers. Hence, the oob error can be considered as a reliable estimate of generalization error. By taking the average of oob errors of all training instances, we can compute the overall oob error estimate. This can be used as an alternative to the validation error rate for selecting hyper- parameters. Hence, random forests do not need to use a separate partition of the training set for validation, as it can simultaneously train the base classifiers and compute generalization error estimates on the same data set.
Random forests have been empirically found to provide significant improvements in generalization performance that are often comparable, if not superior, to the improvements provided by the AdaBoost algorithm. Random forests are also more robust to overfitting and run much faster than the AdaBoost algorithm.
4.10.7 Empirical Comparison among Ensemble Methods Table 4.5 shows the empirical results obtained when comparing the performance of a decision tree classifier against bagging, boosting, and random forest. The base classifiers used in each ensemble method consist of 50 decision trees. The classification accuracies reported in this table are obtained from tenfold cross-validation. Notice that the ensemble classifiers generally outperform a single decision tree classifier on many of the data sets.
Table 4.5. Comparing the accuracy of a decision tree classifier against three ensemble methods.
Data Set Number of (Attributes,
Classes, Instances)
Decision Tree (%) Bagging(%) Boosting(%) RF(%)
Anneal (39, 6,898) 92.09 94.43 95.43 95.43
Australia (15, 2,690) 85.51 87.10 85.22 85.80
Auto (26, 7,205) 81.95 85.37 85.37 84.39
Breast (11, 2,699) 95.14 96.42 97.28 96.14
Cleve (14, 2,303) 76.24 81.52 82.18 82.18
Credit (16, 2,690) 85.8 86.23 86.09 85.8
Diabetes (9, 2, 768) 72.40 76.30 73.18 75.13
German (21, 2,1000) 70.90 73.40 73.00 74.5
Glass (10, 7,214) 67.29 76.17 77.57 78.04
Heart (14, 2,270) 80.00 81.48 80.74 83.33
Hepatitis (20, 2,155) 81.94 81.29 83.87 83.23
Horse (23, 2,368) 85.33 85.87 81.25 85.33
Ionosphere (35, 2,351) 89.17 92.02 93.73 93.45
Iris (5, 3, 150) 94.67 94.67 94.00 93.33 Labor (17, 2, 57) 78.95 84.21 89.47 84.21
Led7 (8, 10,3200) 73.34 73.66 73.34 73.06
Lymphography (19, 4, 77.03 79.05 85.14 82.43
148) Pima (9, 2, 768) 74.35 76.69 73.44 77.60
Sonar (61, 2,208) 78.85 78.85 84.62 85.58
Tic-tac-toe (10, 2,958) 83.72 93.84 98.54 95.82
Vehicle (19, 4,846) 71.04 74.11 78.25 74.94
Waveform (22, 3,5000) 76.44 83.30 83.90 84.04
Wine (14, 3,178) 94.38 96.07 97.75 97.75
Zoo (17, 7,101) 93.07 93.07 95.05 97.03
4.11 Class Imbalance Problem In many data sets there are a disproportionate number of instances that belong to different classes, a property known as skew or class imbalance.For example, consider a health-care application where diagnostic reports are used to decide whether a person has a rare disease. Because of the infrequent nature of the disease, we can expect to observe a smaller number of subjects who are positively diagnosed. Similarly, in credit card fraud detection, fraudulent transactions are greatly outnumbered by legitimate transactions.
The degree of imbalance between the classes varies across different applications and even across different data sets from the same application. For example, the risk for a rare disease may vary across different populations of subjects depending on their dietary and lifestyle choices. However, despite their infrequent occurrences, a correct classification of the rare class often has greater value than a correct classification of the majority class. For example, it may be more dangerous to ignore a patient suffering from a disease than to misdiagnose a healthy person.
More generally, class imbalance poses two challenges for classification. First, it can be difficult to find sufficiently many labeled samples of a rare class. Note that many of the classification methods discussed so far work well only when the training set has a balanced representation of both classes. Although some classifiers are more effective at handling imbalance in the training data than others, e.g., rule-based classifiers and k-NN, they are all impacted if the minority class is not well-represented in the training set. In general, a classifier trained over an imbalanced data set shows a bias toward improving its performance over the majority class, which is often not the desired behavior. As a result, many existing classification models, when trained on an imbalanced data set, may not effectively detect instances of the rare class.
Second, accuracy, which is the traditional measure for evaluating classification performance, is not well-suited for evaluating models in the presence of class imbalance in the test data. For example, if 1% of the credit card transactions are fraudulent, then a trivial model that predicts every
transaction as legitimate will have an accuracy of 99% even though it fails to detect any of the fraudulent activities. Thus, there is a need to use alternative evaluation metrics that are sensitive to the skew and can capture different criteria of performance than accuracy.
In this section, we first present some of the generic methods for building classifiers when there is class imbalance in the training set. We then discuss methods for evaluating classification performance and adapting classification decisions in the presence of a skewed test set. In the remainder of this section, we will consider binary classification problems for simplicity, where the minority class is referred as the positive (+) class while the majority class is referred as the negative (−) class.
4.11.1 Building Classifiers with Class Imbalance There are two primary considerations for building classifiers in the presence of class imbalance in the training set. First, we need to ensure that the learning algorithm is trained over a data set that has adequate representation of both the majority as well as the minority classes. Some common approaches for ensuring this includes the methodologies of oversampling and undersampling the training set. Second, having learned a classification model, we need a way to adapt its classification decisions (and thus create an appropriately tuned classifier) to best match the requirements of the imbalanced test set. This is typically done by converting the outputs of the classification model to real-valued scores, and then selecting a suitable threshold on the classification score to match the needs of a test set. Both these considerations are discussed in detail in the following.
Oversampling and Undersampling The first step in learning with imbalanced data is to transform the training set to a balanced training set, where both classes have nearly equal
representation. The balanced training set can then be used with any of the existing classification techniques (without making any modifications in the learning algorithm) to learn a model that gives equal emphasis to both classes. In the following, we present some of the common techniques for transforming an imbalanced training set to a balanced one.
A basic approach for creating balanced training sets is to generate a sample of training instances where the rare class has adequate representation. There are two types of sampling methods that can be used to enhance the representation of the minority class: (a) undersampling, where the frequency of the majority class is reduced to match the frequency of the minority class, and (b) oversampling, where artificial examples of the minority class are created to make them equal in proportion to the number of negative instances.
To illustrate undersampling, consider a training set that contains 100 positive examples and 1000 negative examples. To overcome the skew among the classes, we can select a random sample of 100 examples from the negative class and use them with the 100 positive examples to create a balanced training set. A classifier built over the resultant balanced set will then be unbiased toward both classes. However, one limitation of undersampling is that some of the useful negative examples (e.g., those closer to the actual decision boundary) may not be chosen for training, therefore, resulting in an inferior classification model. Another limitation is that the smaller sample of 100 negative instances may have a higher variance than the larger set of 1000.
Oversampling attempts to create a balanced training set by artificially generating new positive examples. A simple approach for oversampling is to duplicate every positive instance n−/n+ times, where n+ and n− are the numbers of positive and negative training instances, respectively. Figure 4.51 illustrates the effect of oversampling on the learning of a decision boundary using a classifier such as a decision tree. Without oversampling, only the positive examples at the bottom right-hand side of Figure 4.51(a) are classified correctly. The positive example in the middle of the diagram is misclassified because there are not enough examples to justify the creation of a new decision boundary to separate the positive and negative instances. Oversampling provides the additional examples needed to ensure that the
decision boundary surrounding the positive example is not pruned, as illustrated in Figure 4.51(b). Note that duplicating a positive instance is analogous to doubling its weight during the training stage. Hence, the effect of oversampling can be alternatively achieved by assigning higher weights to positive instances than negative instances. This method of weighting instances can be used with a number of classifiers such as logistic regression, ANN, and SVM.
Figure 4.51. Illustrating the effect of oversampling of the rare class.
One limitation of the duplication method for oversampling is that the replicated positive examples have an artificially lower variance when compared with their true distribution in the overall data. This can bias the classifier to the specific distribution of training instances, which may not be representative of the overall distribution of test instances, leading to poor generalizability. To overcome this limitation, an alternative approach for oversampling is to generate synthetic positive instances in the neighborhood of existing positive instances. In this approach, called the Synthetic Minority Oversampling Technique (SMOTE), we first determine the k-nearest positive
neighbors of every positive instance x, and then generate a synthetic positive instance at some intermediate point along the line segment joining x to one of its randomly chosen k-nearest neighbor, xk. This process is repeated until the desired number of positive instances is reached. However, one limitation of this approach is that it can only generate new positive instances in the convex hull of the existing positive class. Hence, it does not help improve the representation of the positive class outside the boundary of existing positive instances. Despite their complementary strengths and weaknesses, undersampling and oversampling provide useful directions for generating balanced training sets in the presence of class imbalance.
Assigning Scores to Test Instances If a classifier returns an ordinal score s(x)for every test instance x such that a higher score denotes a greater likelihood of x belonging to the positive class, then for every possible value of score threshold, sT, we can create a new binary classifier where a test instance x is classified positive only if s(x)>sT. Thus, every choice of sT can potentially lead to a different classifier, and we are interested in finding the classifier that is best suited for our needs.
Ideally, we would like the classification score to vary monotonically with the actual posterior probability of the positive class, i.e., if s(x1) and s(x2) are the scores of any two instances, x1 and x2, then s(x1)≥s(x2)⇒P(y=1|x1)≥P(y=1|x2). However, this is difficult to guarantee in practice as the properties of the classification score depends on several factors such as the complexity of the classification algorithm and the representative power of the training set. In general, we can only expect the classification score of a reasonable algorithm to be weakly related to the actual posterior probability of the positive class, even though the relationship may not be strictly monotonic. Most classifiers can be easily modified to produce such a real valued score. For example, the signed distance of an instance from the positive margin hyperplane of SVM can be used as a classification score. As another example, test instances belonging to a leaf in a decision tree can be assigned a score based on the fraction of training instances labeled as positive in the leaf. Also, probabilistic classifiers such as naïve Bayes, Bayesian networks, and logistic regression naturally output
estimates of posterior probabilities, P(y=1|x). Next, we discuss some evaluation measures for assessing the goodness of a classifier in the presence of class imbalance.
Table 4.6. A confusion matrix for a binary classification problem in which the classes are not equally important.
Predicted Class + −
Actual class
+ f++ (TP) f+− (FN) − f−+ (FP) f−− (TN)
4.11.2 Evaluating Performance with Class Imbalance The most basic approach for representing a classifier's performance on a test set is to use a confusion matrix, as shown in Table 4.6. This table is essentially the same as Table 3.4, which was introduced in the context of evaluating classification performance in Section 3.2. A confusion matrix summarizes the number of instances predicted correctly or incorrectly by a classifier using the following four counts:
True positive (TP) or f++, which corresponds to the number of positive examples correctly predicted by the classifier.
False positive (FP) or f−+ (also known as Type I error), which corresponds to the number of negative examples wrongly predicted as
positive by the classifier.
False negative (FN) or f+− (also known as Type II error), which corresponds to the number of positive examples wrongly predicted as negative by the classifier.
True negative (TN) or f−−, which corresponds to the number of negative examples correctly predicted by the classifier.
The confusion matrix provides a concise representation of classification performance on a given test data set. However, it is often difficult to interpret and compare the performance of classifiers using the four-dimensional representations (corresponding to the four counts) provided by their confusion matrices. Hence, the counts in the confusion matrix are often summarized using a number of evaluation measures. Accuracy is an example of one such measure that combines these four counts into a single value, which is used extensively when classes are balanced. However, the accuracy measure is not suitable for handling data sets with imbalanced class distributions as it tends to favor classifiers that correctly classify the majority class. In the following, we describe other possible measures that capture different criteria of performance when working with imbalanced classes.
A basic evaluation measure is the true positive rate (TPR), which is defined as the fraction of positive test instances correctly predicted by the classifier:
TPR=TPTP+FN.
In the medical community, TPR is also known as sensitivity, while in the information retrieval literature, it is also called recall (r). A classifier with a high TPR has a high chance of correctly identifying the positive instances of the data.
Analogously to TPR, the true negative rate (TNR) (also known as specificity) is defined as the fraction of negative test instances correctly predicted by the classifier, i.e.,
TNR=TNFP+TN.
A high TNR value signifies that the classifier correctly classifies any randomly chosen negative instance in the test set. A commonly used evaluation measure that is closely related to TNR is the false positive rate (FPR), which is defined as 1−TNR.
FPR=FPFP+TN.
Similarly, we can define false negative rate (FNR) as 1−TPR.
FNR=FNFN+TP.
Note that the evaluation measures defined above do not take into account the skew among the classes, which can be formally defined as α=P/(P+N), where P and N denote the number of actual positives and actual negatives, respectively. As a result, changing the relative numbers of P and N will have no effect on TPR, TNR, FPR, or FNR, since they depend only on the fraction of correct classifications for every class, independently of the other class. Furthermore, knowing the values of TPR and TNR (and consequently FNR and FPR) does not by itself help us uniquely determine all four entries of the confusion matrix. However, together with information about the skew factor, α, and the total number of instances, N, we can compute the entire confusion matrix using TPR and TNR, as shown in Table 4.7.
Table 4.7. Entries of the confusion matrix in terms of the TPR, TNR, skew, α, and total number of instances, N. Predicted + Predicted − Actual + TPR×α×N (1−TPR)×α×N α×N Actual − (1−TNR)×(1−α)×N TNR×(1−α)×N (1−α)×N N
An evaluation measure that is sensitive to the skew is precision, which can be defined as the fraction of correct predictions of the positive class over the total number of positive predictions, i.e.,
Precision, p=TPTP+FP.
Precision is also referred as the positive predicted value (PPV). A classifier that has a high precision is likely to have most of its positive predictions correct. Precision is a useful measure for highly skewed test sets where the positive predictions, even though small in numbers, are required to be mostly correct. A measure that is closely related to precision is the false discovery rate (FDR), which can be defined as 1−p.
FDR=FPTP+FP.
Although both FDR and FPR focus on FP, they are designed to capture different evaluation objectives and thus can take quite contrasting values, especially in the presence of class imbalance. To illustrate this, consider a classifier with the following confusion matrix.
Predicted Class + −
Actual Class
+ 100 0 − 100 900
Since half of the positive predictions made by the classifier are incorrect, it has a FDR value of 100/(100+100)=0.5. However, its FPR is equal to 100/(100+900)=0.1, which is quite low. This example shows that in the presence of high skew (i.e., very small value of α), even a small FPR can result in high FDR. See Section 10.6 for further discussion of this issue.
Note that the evaluation measures defined above provide an incomplete representation of performance, because they either only capture the effect of false positives (e.g., FPR and precision) or the effect of false negatives (e.g., TPR or recall), but not both. Hence, if we optimize only one of these evaluation measures, we may end up with a classifier that shows low FN but high FP, or vice-versa. For example, a classifier that declares every instance
to be positive will have a perfect recall, but high FPR and very poor precision. On the other hand, a classifier that is very conservative in classifying an instance as positive (to reduce FP) may end up having high precision but very poor recall. We thus need evaluation measures that account for both types of misclassifications, FP and FN. Some examples of such evaluation measures are summarized by the following definitions.
Positive Likelihood Ratio=TPRFPR.F1 measure=2rpr+p=2×TP2×TP+FP+FN. (TP+FN).
While some of these evaluation measures are invariant to the skew (e.g., the positive likelihood ratio), others (e.g., precision and the F1 measure) are sensitive to skew. Further, different evaluation measures capture the effects of different types of misclassification errors in various ways. For example, the F1 measure represents a harmonic mean between recall and precision, i.e.,
F1=21r+1p.
Because the harmonic mean of two numbers tends to be closer to the smaller of the two numbers, a high value of F1-measure ensures that both precision and recall are reasonably high. Similarly, the G measure represents the geometric mean between recall and precision. A comparison among harmonic, geometric, and arithmetic means is given in the next example.
Example 4.9. Consider two positive numbers a=1 and b=5. Their arithmetic mean is μa= (a+b)/2=3 and their geometric mean is μg=ab=2.236. Their harmonic mean is μh=(2×1×5)/6=1.667, which is closer to the smaller value between a and b than the arithmetic and geometric means.
A generic extension of the F1 measure is the Fβ measure, which can be defined as follows.
Fβ=(β2+1)rpr+β2p=(β2+1)×TP(β2+1)TP+β2FP+FN. (4.106)
Both precision and recall can be viewed as special cases of Fβ by setting β=0
and β=∞, respectively. Low values of β make Fβ closer to precision, and high values make it closer to recall.
A more general measure that captures Fβ as well as accuracy is the weighted accuracy measure, which is defined by the following equation:
Weighted accuracy=w1TP+w4TNw1TP+w2FP+w3FN+w4TN. (4.107)
The relationship between weighted accuracy and other performance measures is summarized in the following table:
Measure w1 w2 w3 w4 Recall 1 1 0 0 Precision 1 0 1 0 Fβ β2+1 β2 1 0 Accuracy 1 1 1 1
4.11.3 Finding an Optimal Score Threshold Given a suitably chosen evaluation measure E and a distribution of classification scores, s(x), on a validation set, we can obtain the optimal score threshold s* on the validation set using the following approach:
1. Sort the scores in increasing order of their values.
2. For every unique value of score, s, consider the classification model that assigns an instance x as positive only if s(x)>s. Let E(s) denote the performance of this model on the validation set.
3. Find s* that maximizes the evaluation measure, E(s).
s*=argmaxs E(s).
Note that s* can be treated as a hyper-parameter of the classification
algorithm that is learned during model selection. Using s*, we can assign a positive label to a future test instance x only if s(x)>s*. If the evaluation measure E is skew invariant (e.g., Positive Likelihood Ratio), then we can select s* without knowing the skew of the test set, and the resultant classifier formed using s* can be expected to show optimal performance on the test set (with respect to the evaluation measure E). On the other hand, if E is sensitive to the skew (e.g., precision or F1-measure), then we need to ensure that the skew of the validation set used for selecting s* is similar to that of the test set, so that the classifier formed using s* shows optimal test performance with respect to E. Alternatively, given an estimate of the skew of the test data, α, we can use it along with the TPR and TNR on the validation set to estimate all entries of the confusion matrix (see Table 4.7), and thus the estimate of any evaluation measure E on the test set. The score threshold s* selected using this estimate of E can then be expected to produce optimal test performance with respect to E. Furthermore, the methodology of selecting s* on the validation set can help in comparing the test performance of different classification algorithms, by using the optimal values of s* for each algorithm.
4.11.4 Aggregate Evaluation of Performance Although the above approach helps in finding a score threshold s* that provides optimal performance with respect to a desired evaluation measure and a certain amount of skew, α, sometimes we are interested in evaluating the performance of a classifier on a number of possible score thresholds, each corresponding to a different choice of evaluation measure and skew value. Assessing the performance of a classifier over a range of score thresholds is called aggregate evaluation of performance. In this style of analysis, the emphasis is not on evaluating the performance of a single classifier corresponding to the optimal score threshold, but to assess the general quality of ranking produced by the classification scores on the test set. In general, this helps in obtaining robust estimates of classification performance that are not sensitive to specific choices of score thresholds.
ROC Curve One of the widely-used tools for aggregate evaluation is the receiver operating characteristic (ROC) curve. An ROC curve is a graphical approach for displaying the trade-off between TPR and FPR of a classifier, over varying score thresholds. In an ROC curve, the TPR is plotted along the y-axis and the FPR is shown on the x-axis. Each point along the curve corresponds to a classification model generated by placing a threshold on the test scores produced by the classifier. The following procedure describes the generic approach for computing an ROC curve:
1. Sort the test instances in increasing order of their scores.
2. Select the lowest ranked test instance (i.e., the instance with lowest score). Assign the selected instance and those ranked above it to the positive class. This approach is equivalent to classifying all the test instances as positive class. Because all the positive examples are classified correctly and the negative examples are misclassified, TPR=FPR=1.
3. Select the next test instance from the sorted list. Classify the selected instance and those ranked above it as positive, while those ranked below it as negative. Update the counts of TP and FP by examining the actual class label of the selected instance. If this instance belongs to the positive class, the TP count is decremented and the FP count remains the same as before. If the instance belongs to the negative class, the FP count is decremented and TP count remains the same as before.
4. Repeat Step 3 and update the TP and FP counts accordingly until the highest ranked test instance is selected. At this final threshold, TPR=FPR=0, as all instances are labeled as negative.
5. Plot the TPR against FPR of the classifier.
Example 4.10. [Generating ROC
Curve] Figure 4.52 shows an example of how to compute the TPR and FPR values for every choice of score threshold. There are five positive examples and five negative examples in the test set. The class labels of the test instances are shown in the first row of the table, while the second row corresponds to the sorted score values for each instance. The next six rows contain the counts of TP , FP , TN, and FN, along with their corresponding TPR and FPR. The table is then filled from left to right. Initially, all the instances are predicted to be positive. Thus, TP=FP=5 and TPR=FPR=1. Next, we assign the test instance with the lowest score as the negative class. Because the selected instance is actually a positive example, the TP count decreases from 5 to 4 and the FP count is the same as before. The FPR and TPR are updated accordingly. This process is repeated until we reach the end of the list, where TPR=0 and FPR=0. The ROC curve for this example is shown in Figure 4.53.
Figure 4.52. Computing the TPR and FPR at every score threshold.
Figure 4.53. ROC curve for the data shown in Figure 4.52.
Note that in an ROC curve, the TPR monotonically increases with FPR, because the inclusion of a test instance in the set of predicted positives can either increase the TPR or the FPR. The ROC curve thus has a staircase pattern. Furthermore, there are several critical points along an ROC curve that have well-known interpretations:
(TPR=0, FPR=0): Model predicts every instance to be a negative class.
(TPR=1, FPR=1): Model predicts every instance to be a positive class.
(TPR=1, FPR=0): The perfect model with zero misclassifications.
A good classification model should be located as close as possible to the upper left corner of the diagram, while a model that makes random guesses should reside along the main diagonal, connecting the points (TPR=0, FPR=0) and (TPR=1, FPR=1). Random guessing means that an instance is classified as a positive class with a fixed probability p, irrespective
of its attribute set. For example, consider a data set that contains n+ positive instances and n− negative instances. The random classifier is expected to correctly classify pn+ of the positive instances and to misclassify pn− of the negative instances. Therefore, the TPR of the classifier is (pn+)/n+=p, while its FPR is (pn−)/p=p. Hence, this random classifier will reside at the point (p, p) in the ROC curve along the main diagonal.
Figure 4.54. ROC curves for two different classifiers.
Figure 4.54. Full Alternative Text
Since every point on the ROC curve represents the performance of a classifier generated using a particular score threshold, they can be viewed as different operating points of the classifier. One may choose one of these operating
points depending on the requirements of the application. Hence, an ROC curve facilitates the comparison of classifiers over a range of operating points. For example, Figure 4.54 compares the ROC curves of two classifiers, M1 and M2, generated by varying the score thresholds. We can see that M1 is better than M2 when FPR is less than 0.36, as M1 shows better TPR than M2 for this range of operating points. On the other hand, M2 is superior when FPR is greater than 0.36, since the TPR of M2 is higher than that of M1 for this range. Clearly, neither of the two classifiers dominates (is strictly better than) the other, i.e., shows higher values of TPR and lower values of FPR over all operating points.
To summarize the aggregate behavior across all operating points, one of the commonly used measures is the area under the ROC curve (AUC). If the classifier is perfect, then its area under the ROC curve will be equal 1. If the algorithm simply performs random guessing, then its area under the ROC curve will be equal to 0.5.
Although the AUC provides a useful summary of aggregate performance, there are certain caveats in using the AUC for comparing classifiers. First, even if the AUC of algorithm A is higher than the AUC of another algorithm B, this does not mean that algorithm A is always better than B, i.e., the ROC curve of A dominates that of B across all operating points. For example, even though M1 shows a slightly lower AUC than M2 in Figure 4.54, we can see that both M1 and M2 are useful over different ranges of operating points and none of them are strictly better than the other across all possible operating points. Hence, we cannot use the AUC to determine which algorithm is better, unless we know that the ROC curve of one of the algorithms dominates the other.
Second, although the AUC summarizes the aggregate performance over all operating points, we are often interested in only a small range of operating points in most applications. For example, even though M1 shows slightly lower AUC than M2, it shows higher TPR values than M2 for small FPR values (smaller than 0.36). In the presence of class imbalance, the behavior of an algorithm over small FPR values (also termed as early retrieval) is often more meaningful for comparison than the performance over all FPR values. This is because, in many applications, it is important to assess the TPR
achieved by a classifier in the first few instances with highest scores, without incurring a large FPR. Hence, in Figure 4.54, due to the high TPR values of M1 during early retrieval (FPR<0.36), we may prefer M1 over M2 for imbalanced test sets, despite the lower AUC of M1. Hence, care must be taken while comparing the AUC values of different classifiers, usually by visualizing their ROC curves rather than just reporting their AUC.
A key characteristic of ROC curves is that they are agnostic to the skew in the test set, because both the evaluation measures used in constructing ROC curves (TPR and FPR) are invariant to class imbalance. Hence, ROC curves are not suitable for measuring the impact of skew on classification performance. In particular, we will obtain the same ROC curve for two test data sets that have very different skew.
Figure 4.55.
PR curves for two different classifiers.
Figure 4.55. Full Alternative Text
Precision-Recall Curve An alternate tool for aggregate evaluation is the precision recall curve (PR curve). The PR curve plots the precision and recall values of a classifier on the y and x axes respectively, by varying the threshold on the test scores. Figure 4.55 shows an example of PR curves for two hypothetical classifiers, M1 and M2. The approach for generating a PR curve is similar to the approach described above for generating an ROC curve. However, there are some key distinguishing features in the PR curve:
1. PR curves are sensitive to the skew factor α=P/(P+N), and different PR curves are generated for different values of α.
2. When the score threshold is lowest (every instance is labeled as positive), the precision is equal to α while recall is 1. As we increase the score threshold, the number of predicted positives can stay the same or decrease. Hence, the recall monotonically declines as the score threshold increases. In general, the precision may increase or decrease for the same value of recall, upon addition of an instance into the set of predicted positives. For example, if the kth ranked instance belongs to the negative class, then including it will result in a drop in the precision without affecting the recall. The precision may improve at the next step, which adds the (k+1)th ranked instance, if this instance belongs to the positive class. Hence, the PR curve is not a smooth, monotonically increasing curve like the ROC curve, and generally has a zigzag pattern. This pattern is more prominent in the left part of the curve, where even a small change in the number of false positives can cause a large change in precision.
3. As, as we increase the imbalance among the classes (reduce the value of α), the rightmost points of all PR curves will move downwards. At and near the leftmost point on the PR curve (corresponding to larger values
of score threshold), the recall is close to zero, while the precision is equal to the fraction of positives in the top ranked instances of the algorithm. Hence, different classifiers can have different values of precision at the leftmost points of the PR curve. Also, if the classification score of an algorithm monotonically varies with the posterior probability of the positive class, we can expect the PR curve to gradually decrease from a high value of precision on the leftmost point to a constant value of α at the rightmost point, albeit with some ups and downs. This can be observed in the PR curve of algorithm M1 in Figure 4.55, which starts from a higher value of precision on the left that gradually decreases as we move towards the right. On the other hand, the PR curve of algorithm M2 starts from a lower value of precision on the left and shows more drastic ups and downs as we move right, suggesting that the classification score of M2 shows a weaker monotonic relationship with the posterior probability of the positive class.
4. A random classifier that assigns an instance to be positive with a fixed probability p has a precision of α and a recall of p. Hence, a classifier that performs random guessing has a horizontal PR curve with y=α, as shown using a dashed line in Figure 4.55. Note that the random baseline in PR curves depends on the skew in the test set, in contrast to the fixed main diagonal of ROC curves that represents random classifiers.
5. Note that the precision of an algorithm is impacted more strongly by false positives in the top ranked test instances than the FPR of the algorithm. For this reason, the PR curve generally helps to magnify the differences between classifiers in the left portion of the PR curve. Hence, in the presence of class imbalance in the test data, analyzing the PR curves generally provides a deeper insight into the performance of classifiers than the ROC curves, especially in the early retrieval range of operating points.
6. The classifier corresponding to (precision=1, recall=1) represents the perfect classifier. Similar to AUC, we can also compute the area under the PR curve of an algorithm, known as AUC-PR. The AUC-PR of a random classifier is equal to α, while that of a perfect algorithm is equal
to 1. Note that AUC-PR varies with changing skew in the test set, in contrast to the area under the ROC curve, which is insensitive to the skew. The AUC-PR helps in accentuating the differences between classification algorithms in the early retrieval range of operating points. Hence, it is more suited for evaluating classification performance in the presence of class imbalance than the area under the ROC curve. However, similar to ROC curves, a higher value of AUC-PR does not guarantee the superiority of a classification algorithm over another. This is because the PR curves of two algorithms can easily cross each other, such that they both show better performances in different ranges of operating points. Hence, it is important to visualize the PR curves before comparing their AUC-PR values, in order to ensure a meaningful evaluation.
4.12 Multiclass Problem Some of the classification techniques described in this chapter are originally designed for binary classification problems. Yet there are many real-world problems, such as character recognition, face identification, and text classification, where the input data is divided into more than two categories. This section presents several approaches for extending the binary classifiers to handle multiclass problems. To illustrate these approaches, let Y= {y1, y2, … ,yK} be the set of classes of the input data.
The first approach decomposes the multiclass problem into K binary problems. For each class yi∈Y, a binary problem is created where all instances that belong to yi are considered positive examples, while the remaining instances are considered negative examples. A binary classifier is then constructed to separate instances of class yi from the rest of the classes. This is known as the one-against-rest (1-r) approach.
The second approach, which is known as the one-against-one (1-1) approach, constructs K(K −1)/2 binary classifiers, where each classifier is used to distinguish between a pair of classes, (yi, yj). Instances that do not belong to either yi or yj are ignored when constructing the binary classifier for (yi, yj). In both 1-r and 1-1 approaches, a test instance is classified by combining the predictions made by the binary classifiers. A voting scheme is typically employed to combine the predictions, where the class that receives the highest number of votes is assigned to the test instance. In the 1-r approach, if an instance is classified as negative, then all classes except for the positive class receive a vote. This approach, however, may lead to ties among the different classes. Another possibility is to transform the outputs of the binary classifiers into probability estimates and then assign the test instance to the class that has the highest probability.
Example 4.11.
Consider a multiclass problem where Y={y1, y2, y3, y4}. Suppose a test instance is classified as (+, −, −, −) according to the 1-r approach. In other words, it is classified as positive when y1 is used as the positive class and negative when y2, y3, and y4 are used as the positive class. Using a simple majority vote, notice that y1 receives the highest number of votes, which is four, while the remaining classes receive only three votes. The test instance is therefore classified as y1.
Example 4.12. Suppose the test instance is classified using the 1-1 approach as follows:
Binary pair of classes +:y1−:y2 +:y1−:y3 +:y1−:y4 +:y2−:y3 +:y2−:y4 +:y3−:y4
Classification + + − + − +
The first two rows in this table correspond to the pair of classes (yi, yj) chosen to build the classifier and the last row represents the predicted class for the test instance. After combining the predictions, y1 and y4 each receive two votes, while y2 and y3 each receives only one vote. The test instance is therefore classified as either y1 or y4, depending on the tie-breaking procedure.
Error-Correcting Output Coding A potential problem with the previous two approaches is that they may be sensitive to binary classification errors. For the 1-r approach given in Example
4.12, if at least of one of the binary classifiers makes a mistake in its prediction, then the classifier may end up declaring a tie between classes or making a wrong prediction. For example, suppose the test instance is classified as (+, −, +, −) due to misclassification by the third classifier. In this case, it will be difficult to tell whether the instance should be classified as y1
or y3, unless the probability associated with each class prediction is taken into account.
The error-correcting output coding (ECOC) method provides a more robust way for handling multiclass problems. The method is inspired by an information-theoretic approach for sending messages across noisy channels.
The idea behind this approach is to add redundancy into the transmitted message by means of a codeword, so that the receiver may detect errors in the received message and perhaps recover the original message if the number of errors is small.
For multiclass learning, each class yi is represented by a unique bit string of length n known as its codeword. We then train n binary classifiers to predict each bit of the codeword string. The predicted class of a test instance is given by the codeword whose Hamming distance is closest to the codeword produced by the binary classifiers. Recall that the Hamming distance between a pair of bit strings is given by the number of bits that differ.
Example 4.13. Consider a multiclass problem where Y={y1, y2, y3, y4}. Suppose we encode the classes using the following seven bit codewords:
Class Codeword y1 1 1 1 1 1 1 1 y2 0 0 0 0 1 1 1 y3 0 0 1 1 0 0 1 y4 0 1 0 1 0 1 0
Each bit of the codeword is used to train a binary classifier. If a test instance is classified as (0,1,1,1,1,1,1) by the binary classifiers, then the Hamming distance between the codeword and y1 is 1, while the Hamming distance to the remaining classes is 3. The test instance is therefore classified as y1.
An interesting property of an error-correcting code is that if the minimum Hamming distance between any pair of codewords is d, then any ⌊ (d−1)/2) ⌋ errors in the output code can be corrected using its nearest codeword. In Example 4.13, because the minimum Hamming distance between any pair of codewords is 4, the classifier may tolerate errors made by one of the seven binary classifiers. If there is more than one classifier that makes a mistake, then the classifier may not be able to compensate for the error.
An important issue is how to design the appropriate set of codewords for different classes. From coding theory, a vast number of algorithms have been developed for generating n-bit codewords with bounded Hamming distance. However, the discussion of these algorithms is beyond the scope of this book. It is worthwhile mentioning that there is a significant difference between the design of error-correcting codes for communication tasks compared to those used for multiclass learning. For communication, the codewords should maximize the Hamming distance between the rows so that error correction can be performed. Multiclass learning, however, requires that both the row- wise and column-wise distances of the codewords must be well separated. A larger column-wise distance ensures that the binary classifiers are mutually independent, which is an important requirement for ensemble learning methods.
4.13 Bibliographic Notes Mitchell [278] provides excellent coverage on many classification techniques from a machine learning perspective. Extensive coverage on classification can also be found in Aggarwal [195], Duda et al. [229], Webb [307], Fukunaga [237], Bishop [204], Hastie et al. [249], Cherkassky and Mulier [215], Witten and Frank [310], Hand et al. [247], Han and Kamber [244], and Dunham [230].
Direct methods for rule-based classifiers typically employ the sequential covering scheme for inducing classification rules. Holte's 1R [255] is the simplest form of a rule-based classifier because its rule set contains only a single rule. Despite its simplicity, Holte found that for some data sets that exhibit a strong one-to-one relationship between the attributes and the class label, 1R performs just as well as other classifiers. Other examples of rule- based classifiers include IREP [234], RIPPER [218], CN2 [216, 217], AQ [276], RISE [224], and ITRULE [296]. Table 4.8 shows a comparison of the characteristics of four of these classifiers.
Table 4.8. Comparison of various rule-based classifiers.
RIPPER CN2(unordered) CN2
(ordered) AQR
Rule- growing strategy
General-to- specific
General-to- specific
General-to- specific
General-to- specific
(seeded by a positive example)
Evaluation metric
FOIL's Info gain Laplace
Entropy and likelihood
ratio
Number of true
positives
Stopping condition forrule- growing
All examples belong to the same class
No performance
gain
No performance
gain
Rules cover only
positive class
Rule pruning Reduced errorpruning None None None
Instance elimination
Positive and negative
Positive only
Positive only
Positive and
negative Stopping condition for adding rules
Error>50% orbased on
MDL
No performance
gain
No performance
gain
All positive examples
are covered Rule setp runing
Replace or modify rules
Statistical tests None None
Search strategy Greedy Beam search
Beam search
Beam search
For rule-based classifiers, the rule antecedent can be generalized to include any propositional or first-order logical expression (e.g., Horn clauses). Readers who are interested in first-order logic rule-based classifiers may refer to references such as [278] or the vast literature on inductive logic programming [279]. Quinlan [287] proposed the C4.5rules algorithm for extracting classification rules from decision trees. An indirect method for extracting rules from artificial neural networks was given by Andrews et al. in [198].
Cover and Hart [220] presented an overview of the nearest neighbor classification method from a Bayesian perspective. Aha provided both theoretical and empirical evaluations for instance-based methods in [196]. PEBLS, which was developed by Cost and Salzberg [219], is a nearest neighbor classifier that can handle data sets containing nominal attributes. Each training example in PEBLS is also assigned a weight factor that depends on the number of times the example helps make a correct prediction. Han et al. [243] developed a weight-adjusted nearest neighbor algorithm, in which the feature weights are learned using a greedy, hill-climbing optimization algorithm. A more recent survey of k-nearest neighbor
classification is given by Steinbach and Tan [298].
Naïve Bayes classifiers have been investigated by many authors, including Langley et al. [267], Ramoni and Sebastiani [288], Lewis [270], and Domingos and Pazzani [227]. Although the independence assumption used in naïve Bayes classifiers may seem rather unrealistic, the method has worked surprisingly well for applications such as text classification. Bayesian networks provide a more flexible approach by allowing some of the attributes to be interdependent. An excellent tutorial on Bayesian networks is given by Heckerman in [252] and Jensen in [258]. Bayesian networks belong to a broader class of models known as probabilistic graphical models. A formal introduction to the relationships between graphs and probabilities was presented in Pearl [283]. Other great resources on probabilistic graphical models include books by Bishop [205], and Jordan [259]. Detailed discussions of concepts such as d-separation and Markov blankets are provided in Geiger et al. [238] and Russell and Norvig [291].
Generalized linear models (GLM) are a rich class of regression models that have been extensively studied in the statistical literature. They were formulated by Nelder and Wedderburn in 1972 [280] to unify a number of regression models such as linear regression, logistic regression, and Poisson regression, which share some similarities in their formulations. An extensive discussion of GLMs is provided in the book by McCullagh and Nelder [274].
Artificial neural networks (ANN) have witnessed a long and winding history of developments, involving multiple phases of stagnation and resurgence. The idea of a mathematical model of a neural network was first introduced in 1943 by McCulloch and Pitts [275]. This led to a series of computational machines to simulate a neural network based on the theory of neural plasticity [289]. The perceptron, which is the simplest prototype of modern ANNs, was developed by Rosenblatt in 1958 [290]. The perceptron uses a single layer of processing units that can perform basic mathematical operations such as addition and multiplication. However, the perceptron can only learn linear decision boundaries and is guaranteed to converge only when the classes are linearly separable. Despite the interest in learning multi- layer networks to overcome the limitations of perceptron, progress in this area remain halted until the invention of the backpropagation algorithm by
Werbos in 1974 [309], which allowed for the quick training of multi-layer ANNs using the gradient descent method. This led to an upsurge of interest in the artificial intelligence (AI) community to develop multi-layer ANN models, a trend that continued for more than a decade. Historically, ANNs mark a paradigm shift in AI from approaches based on expert systems (where knowledge is encoded using if-then rules) to machine learning approaches (where the knowledge is encoded in the parameters of a computational model). However, there were still a number of algorithmic and computational challenges in learning large ANN models, which remained unresolved for a long time. This hindered the development of ANN models to the scale necessary for solving real-world problems. Slowly, ANNs started getting outpaced by other classification models such as support vector machines, which provided better performance as well as theoretical guarantees of convergence and optimality. It is only recently that the challenges in learning deep neural networks have been circumvented, owing to better computational resources and a number of algorithmic improvements in ANNs since 2006. This re-emergence of ANN has been dubbed as “deep learning,” which has often outperformed existing classification models and gained wide-spread popularity.
Deep learning is a rapidly evolving area of research with a number of potentially impactful contributions being made every year. Some of the landmark advancements in deep learning include the use of large-scale restricted Boltzmann machines for learning generative models of data [201, 253], the use of autoencoders and its variants (denoising autoencoders) for learning robust feature representations [199, 305, 306], and sophistical architectures to promote sharing of parameters across nodes such as convolutional neural networks for images [265, 268] and recurrent neural networks for sequences [241, 242, 277]. Other major improvements include the approach of unsupervised pretraining for initializing ANN models [232], the dropout technique for regularization [254, 297], batch normalization for fast learning of ANN parameters [256], and maxout networks for effective usage of the dropout technique [240]. Even though the discussions in this chapter on learning ANN models were centered around the gradient descent method, most of the modern ANN models involving a large number of hidden layers are trained using the stochastic gradient descent method since it is highly scalable [207]. An extensive survey of deep learning approaches has
been presented in review articles by Bengio [200], LeCun et al. [269], and Schmidhuber [293]. An excellent summary of deep learning approaches can also be obtained from recent books by Goodfellow et al. [239] and Nielsen [281].
Vapnik [303, 304] has written two authoritative books on Support Vector Machines (SVM). Other useful resources on SVM and kernel methods include the books by Cristianini and Shawe-Taylor [221] and Schölkopf and Smola [294]. There are several survey articles on SVM, including those written by Burges [212], Bennet et al. [202], Hearst [251], and Mangasarian [272]. SVM can also be viewed as an L2 norm regularizer of the hinge loss function, as described in detail by Hastie et al. [249]. The L1 norm regularizer of the square loss function can be obtained using the least absolute shrinkage and selection operator (Lasso), which was introduced by Tibshirani in 1996 [301]. The Lasso has several interesting properties such as the ability to simultaneously perform feature selection as well as regularization, so that only a subset of features are selected in the final model. An excellent review of Lasso can be obtained from a book by Hastie et al. [250].
A survey of ensemble methods in machine learning was given by Diet-terich [222]. The bagging method was proposed by Breiman [209]. Freund and Schapire [236] developed the AdaBoost algorithm. Arcing, which stands for adaptive resampling and combining, is a variant of the boosting algorithm proposed by Breiman [210]. It uses the non-uniform weights assigned to training examples to resample the data for building an ensemble of training sets. Unlike AdaBoost, the votes of the base classifiers are not weighted when determining the class label of test examples. The random forest method was introduced by Breiman in [211]. The concept of bias-variance decomposition is explained in detail by Hastie et al. [249]. While the bias- variance decomposition was initially proposed for regression problems with squared loss function, a unified framework for classification problems involving 0–1 losses was introduced by Domingos [226].
Related work on mining rare and imbalanced data sets can be found in the survey papers written by Chawla et al. [214] and Weiss [308]. Sampling- based methods for mining imbalanced data sets have been investigated by
many authors, such as Kubat and Matwin [266], Japkowitz [257], and Drummond and Holte [228]. Joshi et al. [261] discussed the limitations of boosting algorithms for rare class modeling. Other algorithms developed for mining rare classes include SMOTE [213], PNrule [260], and CREDOS [262].
Various alternative metrics that are well-suited for class imbalanced problems are available. The precision, recall, and F1-measure are widely-used metrics in information retrieval [302]. ROC analysis was originally used in signal detection theory for performing aggregate evaluation over a range of score thresholds. A method for comparing classifier performance using the convex hull of ROC curves was suggested by Provost and Fawcett in [286]. Bradley [208] investigated the use of area under the ROC curve (AUC) as a performance metric for machine learning algorithms. Despite the vast body of literature on optimizing the AUC measure in machine learning models, it is well-known that AUC suffers from certain limitations. For example, the AUC can be used to compare the quality of two classifiers only if the ROC curve of one classifier strictly dominates the other. However, if the ROC curves of two classifiers intersect at any point, then it is difficult to assess the relative quality of classifiers using the AUC measure. An in-depth discussion of the pitfalls in using AUC as a performance measure can be obtained in works by Hand [245, 246], and Powers [284]. The AUC has also been considered to be an incoherent measure of performance, i.e., it uses different scales while comparing the performance of different classifiers, although a coherent interpretation of AUC has been provided by Ferri et al. [235]. Berrar and Flach [203] describe some of the common caveats in using the ROC curve for clinical microarray research. An alternate approach for measuring the aggregate performance of a classifier is the precision-recall (PR) curve, which is especially useful in the presence of class imbalance [292].
An excellent tutorial on cost-sensitive learning can be found in a review article by Ling and Sheng [271]. The properties of a cost matrix had been studied by Elkan in [231]. Margineantu and Dietterich [273] examined various methods for incorporating cost information into the C4.5 learning algorithm, including wrapper methods, class distribution-based methods, and loss-based methods. Other cost-sensitive learning methods that are algorithm- independent include AdaCost [233], MetaCost [225], and costing [312].
Extensive literature is also available on the subject of multiclass learning. This includes the works of Hastie and Tibshirani [248], Allwein et al. [197], Kong and Dietterich [264], and Tax and Duin [300]. The error-correcting output coding (ECOC) method was proposed by Dietterich and Bakiri [223]. They had also investigated techniques for designing codes that are suitable for solving multiclass problems.
Apart from exploring algorithms for traditional classification settings where every instance has a single set of features with a unique categorical label, there has been a lot of recent interest in non-traditional classification paradigms, involving complex forms of inputs and outputs. For example, the paradigm of multi-label learning allows for an instance to be assigned multiple class labels rather than just one. This is useful in applications such as object recognition in images, where a photo image may include more than one classification object, such as, grass, sky, trees, and mountains. A survey on multi-label learning can be found in [313]. As another example, the paradigm of multi-instance learning considers the problem where the instances are available in the form of groups called bags, and training labels are available at the level of bags rather than individual instances. Multi- instance learning is useful in applications where an object can exist as multiple instances in different states (e.g., the different isomers of a chemical compound), and even if a single instance shows a specific characteristic, the entire bag of instances associated with the object needs to be assigned the relevant class. A survey on multi-instance learning is provided in [314].
In a number of real-world applications, it is often the case that the training labels are scarce in quantity, because of the high costs associated with obtaining gold-standard supervision. However, we almost always have abundant access to unlabeled test instances, which do not have supervised labels but contain valuable information about the structure or distribution of instances. Traditional learning algorithms, which only make use of the labeled instances in the training set for learning the decision boundary, are unable to exploit the information contained in unlabeled instances. In contrast, learning algorithms that make use of the structure in the unlabeled data for learning the classification model are known as semi-supervised learning algorithms [315, 316]. The use of unlabeled data is also explored in the paradigm of multi-view learning [299, 311], where every object is
observed in multiple views of the data, involving diverse sets of features. A common strategy used by multi-view learning algorithms is co-training [206], where a different model is learned for every view of the data, but the model predictions from every view are constrained to be identical to each other on the unlabeled test instances.
Another learning paradigm that is commonly explored in the paucity of training data is the framework of active learning, which attempts to seek the smallest set of label annotations to learn a reasonable classification model. Active learning expects the annotator to be involved in the process of model learning, so that the labels are requested incrementally over the most relevant set of instances, given a limited budget of label annotations. For example, it may be useful to obtain labels over instances closer to the decision boundary that can play a bigger role in fine-tuning the boundary. A review on active learning approaches can be found in [285, 295].
In some applications, it is important to simultaneously solve multiple learning tasks together, where some of the tasks may be similar to one another. For example, if we are interested in translating a passage written in English into different languages, the tasks involving lexically similar languages (such as Spanish and Portuguese) would require similar learning of models. The paradigm of multi-task learning helps in simultaneously learning across all tasks while sharing the learning among related tasks. This is especially useful when some of the tasks do not contain sufficiently many training samples, in which case borrowing the learning from other related tasks helps in the learning of robust models. A special case of multi-task learning is transfer learning, where the learning from a source task (with sufficient number of training samples) has to be transferred to a destination task (with paucity of training data). An extensive survey of transfer learning approaches is provided by Pan et al. [282].
Most classifiers assume every data instance must belong to a class, which is not always true for some applications. For example, in malware detection, due to the ease in which new malwares are created, a classifier trained on existing classes may fail to detect new ones even if the features for the new malwares are considerably different than those for existing malwares. Another example is in critical applications such as medical diagnosis, where
prediction errors are costly and can have severe consequences. In this situation, it would be better for the classifier to refrain from making any prediction on a data instance if it is unsure of its class. This approach, known as classifier with reject option, does not need to classify every data instance unless it determines the prediction is reliable (e.g., if the class probability exceeds a user-specified threshold). Instances that are unclassified can be presented to domain experts for further determination of their true class labels.
Classifiers can also be distinguished in terms of how the classification model is trained. A batch classifier assumes all the labeled instances are available during training. This strategy is applicable when the training set size is not too large and for stationary data, where the relationship between the attributes and classes does not vary over time. An online classifier, on the other hand, trains an initial model using a subset of the labeled data [263]. The model is then updated incrementally as more labeled instances become available. This strategy is effective when the training set is too large or when there is concept drift due to changes in the distribution of the data over time.
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4.14 Exercises 1. 1. Consider a binary classification problem with the following set of
attributes and attribute values:
Air Conditioner={Working, Broken}
Engine={Good, Bad}
Mileage={High, Medium, Low}
Rust={Yes, No}
Suppose a rule-based classifier produces the following rule set:
Mileage=High→Mileage=HighMileage=Low→Value=HighAir Conditioner
1. Are the rules mutually exclusive?
2. Is the rule set exhaustive?
3. Is ordering needed for this set of rules?
4. Do you need a default class for the rule set?
2. 2. The RIPPER algorithm (by Cohen [218]) is an extension of an earlier algorithm called IREP (by Fürnkranz and Widmer [234]). Both algorithms apply the reduced-error pruning method to determine whether a rule needs to be pruned. The reduced error pruning method uses a validation set to estimate the generalization error of a classifier. Consider the following pair of rules:
R1:A→CR2:A∧B→C
R2 is obtained by adding a new conjunct, B, to the left-hand side of R1. For this question, you will be asked to determine whether R2 is
preferred over R1 from the perspectives of rule-growing and rule- pruning. To determine whether a rule should be pruned, IREP computes the following measure:
vIREP=p+(N−n)P+N,
where P is the total number of positive examples in the validation set, N is the total number of negative examples in the validation set, p is the number of positive examples in the validation set covered by the rule, and n is the number of negative examples in the validation set covered by the rule. vIREP is actually similar to classification accuracy for the validation set. IREP favors rules that have higher values of vIREP. On the other hand, RIPPER applies the following measure to determine whether a rule should be pruned:
vRIPPER=p−nP+n.
1. Suppose R1 is covered by 350 positive examples and 150 negative examples, while R2 is covered by 300 positive examples and 50 negative examples. Compute the FOIL's information gain for the rule R2 with respect to R1.
2. Consider a validation set that contains 500 positive examples and 500 negative examples. For R1, suppose the number of positive examples covered by the rule is 200, and the number of negative examples covered bytheruleis50. For R2, suppose the number of positive examples covered by the rule is 100 and the number of negative examples is 5. Compute vIREP for both rules. Which rule does IREP prefer?
3. Compute vRIPPER for the previous problem. Which rule does RIPPER prefer?
3. 3. C4.5rules is an implementation of an indirect method for generating rules from a decision tree. RIPPER is an implementation of a direct method for generating rules directly from data.
1. Discuss the strengths and weaknesses of both methods.
2. Consider a data set that has a large difference in the class size (i.e., some classes are much bigger than others). Which method (between C4.5rules and RIPPER) is better in terms of finding high accuracy rules for the small classes?
4. 4. Consider a training set that contains 100 positive examples and 400 negative examples. For each of the following candidate rules,
R1:A→+(covers 4 positive and 1 negative examples),R2:B→+ (covers 30 positive and 10 negative examples),R3:C→+ (covers 100 positive and 90 negative examples),
determine which is the best and worst candidate rule according to:
1. Rule accuracy.
2. FOIL's information gain.
3. The likelihood ratio statistic.
4. The Laplace measure.
5. The m-estimate measure (with k=2 and p+=0.2).
5. 5. Figure 4.3 illustrates the coverage of the classification rules R1, R2, and R3. Determine which is the best and worst rule according to:
1. The likelihood ratio statistic.
2. The Laplace measure.
3. The m-estimate measure (with k=2 and p+=0.58).
4. The rule accuracy after R1 has been discovered, where none of the examples covered by R1 are discarded.
5. The rule accuracy after R1 has been discovered, where only the positive examples covered by R1 are discarded.
6. The rule accuracy after R1 has been discovered, where both positive and negative examples covered by R1 are discarded.
6. 6.
1. Suppose the fraction of undergraduate students who smoke is 15% and the fraction of graduate students who smoke is 23%. If one- fifth of the college students are graduate students and the rest are undergraduates, what is the probability that a student who smokes is a graduate student?
2. Given the information in part (a), is a randomly chosen college student more likely to be a graduate or undergraduate student?
3. Repeat part (b) assuming that the student is a smoker.
4. Suppose 30% of the graduate students live in a dorm but only 10% of the undergraduate students live in a dorm. If a student smokes and lives in the dorm, is he or she more likely to be a graduate or undergraduate student? You can assume independence between students who live in a dorm and those who smoke.
7. 7. Consider the data set shown in Table 4.9
Table 4.9. Data set for Exercise 7. Instance A B C Class
1 0 0 0 + 2 0 0 1 − 3 0 1 1 − 4 0 1 1 − 5 0 0 1 + 6 1 0 1 +
7 1 0 1 − 8 1 0 1 − 9 1 1 1 + 10 1 0 1 +
1. Estimate the conditional probabilities for P(A|+), P(B|+), P(C|+), P(A|−), P(B|−), and P(C|−).
2. Use the estimate of conditional probabilities given in the previous question to predict the class label for a test sample (A=0, B=1, C=0) using the naïve Bayes approach.
3. Estimate the conditional probabilities using the m-estimate approach, with p=1/2 and m=4.
4. Repeat part (b) using the conditional probabilities given in part (c).
5. Compare the two methods for estimating probabilities. Which method is better and why?
8. 8. Consider the data set shown in Table 4.10.
Table 4.10. Data set for Exercise 8. Instance A B C Class
1 0 0 1 − 2 1 0 1 + 3 0 1 0 − 4 1 0 0 − 5 1 0 1 + 6 0 0 1 + 7 1 1 0 −
8 0 0 0 − 9 0 1 0 + 10 1 1 1 +
1. Estimate the conditional probabilities for P(A=1|+), P(B=1|+), P(C=1|+), P(A=1|−), P(B=1|−), and P(C=1|−) using the same approach as in the previous problem.
2. Use the conditional probabilities in part (a) to predict the class label for a test sample (A=1, B=1, C=1) using the naïve Bayes approach.
3. Compare P(A=1), P(B=1), and P(A=1, B=1). State the relationships between A and B.
4. Repeat the analysis in part (c) using P(A=1), P(B=0), and P(A=1, B=0).
5. Compare P(A=1, B=1|Class=+) against P(A=1|Class=+) and P(B=1|Class=+). Are the variables conditionally independent given the class?
9. 9.
1. Explain how naïve Bayes performs on the data set shown in Figure 4.56.
2. If each class is further divided such that there are four classes (A1, A2, B1, and B2), will naïve Bayes perform better?
3. How will a decision tree perform on this data set (for the two-class problem)? What if there are four classes?
10. 10. Figure 4.57 illustrates the Bayesian network for the data set shown in Table 4.11. (Assume that all the attributes are binary).
1. Draw the probability table for each node in the network.
2. Use the Bayesian network to compute
P(Engine=Bad, Air Conditioner=Broken).
11. 11. Given the Bayesian network shown in Figure 4.58, compute the following probabilities:
Figure 4.56. Data set for Exercise 9.
Figure 4.56. Full Alternative Text
Figure 4.57. Bayesian network.
1. P(B=good,F=empty, G=empty, S=yes).
2. P(B=bad,F=empty, G=not empty, S=no).
3. Given that the battery is bad, compute the probability that the car will start.
12. 12. Consider the one-dimensional data set shown in Table 4.12.
1. Classify the data point x=5.0 according to its 1-, 3-, 5-, and 9- nearest neighbors (using majority vote).
2. Repeat the previous analysis using the distance-weighted voting approach described in Section 4.3.1.
Table 4.11. Data set for Exercise 10.
Mileage Engine AirConditioner
Number of Instances
with Car Value=Hi
Number of Instances with Car Value=Lo
Hi Good Working 3 4 Hi Good Broken 1 2 Hi Bad Working 1 5 Hi Bad Broken 0 4 Lo Good Working 9 0 Lo Good Broken 5 1 Lo Bad Working 1 2 Lo Bad Broken 0 2
Figure 4.58. Bayesian network for Exercise 11.
Figure 4.58. Full Alternative Text
13. 13. The nearest neighbor algorithm described in Section 4.3 can be extended to handle nominal attributes. A variant of the algorithm called PEBLS (Parallel Exemplar-Based Learning System) by Cost and Salzberg [219] measures the distance between two values of a nominal attribute using the modified value difference metric (MVDM). Given a pair of nominal attribute values, V1 and V2, the distance between them is defined as follows:
d(V1, V2)=∑i=1k| ni1n1−ni2n2, | (4.108)
where nij is the number of examples from class i with attribute value Vj and nj is the number of examples with attribute value Vj.
Table 4.12. Data set for Exercise 12. x 0.5 3.0 4.5 4.6 4.9 5.2 5.3 5.5 7.0 9.5 y − − + + + − − + − −
Consider the training set for the loan classification problem shown in Figure 4.8. Use the MVDM measure to compute the distance between every pair of attribute values for the Home Owner and Marital Status attributes.
14. 14. For each of the Boolean functions given below, state whether the problem is linearly separable.
1. A AND B AND C
2. NOT A AND B
3. (A OR B) AND (A OR C)
4. (A XOR B) AND (A OR B)
15. 15.
1. Demonstrate how the perceptron model can be used to represent the AND and OR functions between a pair of Boolean variables.
2. Comment on the disadvantage of using linear functions as activation functions for multi-layer neural networks.
16. 16. You are asked to evaluate the performance of two classification models, M1 and M2. The test set you have chosen contains 26 binary attributes, labeled as A through Z. Table 4.13 shows the posterior probabilities obtained by applying the models to the test set. (Only the posterior probabilities for the positive class are shown). As this is a two- class problem, P(−)=1−P(+) and P(−|A, …, Z)=1−P(+|A, …, Z). Assume that we are mostly interested in detecting instances from the positive class.
1. Plot the ROC curve for both M1 and M2. (You should plot them on the same graph.) Which model do you think is better? Explain your reasons.
2. For model M1, suppose you choose the cutoff threshold to be t=0.5. In other words, any test instances whose posterior probability is greater than t will be classified as a positive example. Compute the precision, recall, and F-measure for the model at this threshold value.
3. Repeat the analysis for part (b) using the same cutoff threshold on model M2. Compare the F-measure results for both models. Which model is better? Are the results consistent with what you expect from the ROC curve?
4. Repeat part (b) for model M1 using the threshold t=0.1. Which threshold do you prefer, t=0.5 or t=0.1? Are the results consistent with what you expect from the ROC curve?
Table 4.13. Posterior probabilities for Exercise 16.
Instance TrueClass P(+|A, …, Z, M1) P(+|A, …, Z, M2)
1 + 0.73 0.61 2 + 0.69 0.03 3 − 0.44 0.68 4 − 0.55 0.31 5 + 0.67 0.45 6 + 0.47 0.09 7 − 0.08 0.38 8 − 0.15 0.05 9 + 0.45 0.01 10 − 0.35 0.04
17. 17. Following is a data set that contains two attributes, X and Y , and two class labels, “+” and “−”. Each attribute can take three different values: 0, 1, or 2.
X Y Number of Instances
+ − 0 0 0 100 1 0 0 0 2 0 0 100 0 1 10 100 1 1 10 0 2 1 10 100 0 2 0 100
1 2 0 0 2 2 0 100
The concept for the “+” class is Y=1 and the concept for the “−” class is X=0∨X=2.
1. Build a decision tree on the data set. Does the tree capture the “+” and “−” concepts?
2. What are the accuracy, precision, recall, and F1-measure of the decision tree? (Note that precision, recall, and F1-measure are defined with respect to the “+” class.)
3. Build a new decision tree with the following cost function:
C(i, j)={ 0,if i=j;1,if i=+, j=−;Number of − instancesNumber of+ instancesif i=−, j=+;
(Hint: only the leaves of the old decision tree need to be changed.) Does the decision tree capture the “+” concept?
4. What are the accuracy, precision, recall, and F1-measure of the new decision tree?
18. 18. Consider the task of building a classifier from random data, where the attribute values are generated randomly irrespective of the class labels. Assume the data set contains instances from two classes, “+” and “−.” Half of the data set is used for training while the remaining half is used for testing.
1. Suppose there are an equal number of positive and negative instances in the data and the decision tree classifier predicts every test instance to be positive. What is the expected error rate of the classifier on the test data?
2. Repeat the previous analysis assuming that the classifier predicts each test instance to be positive class with probability 0.8 and negative class with probability 0.2.
3. Suppose two-thirds of the data belong to the positive class and the remaining one-third belong to the negative class. What is the expected error of a classifier that predicts every test instance to be positive?
4. Repeat the previous analysis assuming that the classifier predicts each test instance to be positive class with probability 2/3 and negative class with probability 1/3.
19. 19. Derive the dual Lagrangian for the linear SVM with non-separable data where the objective function is
f(w)=ǁ w ǁ22+C(∑i=1Nξi)2.
20. 20. Consider the XOR problem where there are four training points:
(1, 1, −), (1, 0, +), (0, 1, +), (0, 0, −).
Transform the data into the following feature space:
φ=(1, 2x1, 2x2, 2x1x2, x12, x22).
Find the maximum margin linear decision boundary in the transformed space.
21. 21. Given the data sets shown in Figures 4.59, explain how the decision tree, naïve Bayes, and k-nearest neighbor classifiers would perform on these data sets.
Figure 4.59. Data set for Exercise 21.
Figure 4.59. Full Alternative Text
5 Association Analysis: Basic Concepts and Algorithms Many business enterprises accumulate large quantities of data from their day- to-day operations. For example, huge amounts of customer purchase data are collected daily at the checkout counters of grocery stores. Table 5.1 gives an example of such data, commonly known as market basket transactions. Each row in this table corresponds to a transaction, which contains a unique identifier labeled TID and a set of items bought by a given customer. Retailers are interested in analyzing the data to learn about the purchasing behavior of their customers. Such valuable information can be used to support a variety of business-related applications such as marketing promotions, inventory management, and customer relationship management.
Table 5.1. An example of market basket transactions. TID Items
1 {Bread, Milk} 2 {Bread, Diapers, Beer, Eggs} 3 {Milk, Diapers, Beer, Cola} 4 {Bread, Milk, Diapers, Beer} 5 {Bread, Milk, Diapers, Cola}
This chapter presents a methodology known as association analysis, which is useful for discovering interesting relationships hidden in large data sets. The uncovered relationships can be represented in the form of sets of items present in many transactions, which are known as frequent itemsets, or association rules, that represent relationships between two itemsets. For example, the following rule can be extracted from the data set shown in Table
5.1:
{Diapers}→{Beer}.
The rule suggests a relationship between the sale of diapers and beer because many customers who buy diapers also buy beer. Retailers can use these types of rules to help them identify new opportunities for cross-selling their products to the customers.
Besides market basket data, association analysis is also applicable to data from other application domains such as bioinformatics, medical diagnosis, web mining, and scientific data analysis. In the analysis of Earth science data, for example, association patterns may reveal interesting connections among the ocean, land, and atmospheric processes. Such information may help Earth scientists develop a better understanding of how the different elements of the Earth system interact with each other. Even though the techniques presented here are generally applicable to a wider variety of data sets, for illustrative purposes, our discussion will focus mainly on market basket data.
There are two key issues that need to be addressed when applying association analysis to market basket data. First, discovering patterns from a large transaction data set can be computationally expensive. Second, some of the discovered patterns may be spurious (happen simply by chance) and even for non-spurious patterns, some are more interesting than others. The remainder of this chapter is organized around these two issues. The first part of the chapter is devoted to explaining the basic concepts of association analysis and the algorithms used to efficiently mine such patterns. The second part of the chapter deals with the issue of evaluating the discovered patterns in order to help prevent the generation of spurious results and to rank the patterns in terms of some interestingness measure.
5.1 Preliminaries This section reviews the basic terminology used in association analysis and presents a formal description of the task.
Binary Representation Market basket data can be represented in a binary format as shown in Table 5.2, where each row corresponds to a transaction and each column corresponds to an item. An item can be treated as a binary variable whose value is one if the item is present in a transaction and zero otherwise. Because the presence of an item in a transaction is often considered more important than its absence, an item is an asymmetric binary variable. This representation is a simplistic view of real market basket data because it ignores important aspects of the data such as the quantity of items sold or the price paid to purchase them. Methods for handling such non-binary data will be explained in Chapter 6.
Table 5.2. A binary 0/1 representation of market basket data. TID Bread Milk Diapers Beer Eggs Cola
1 1 1 0 0 0 0 2 1 0 1 1 1 0 3 0 1 1 1 0 1 4 1 1 1 1 0 0 5 1 1 1 0 0 1
Itemset and Support Count Let I={i1, i2, … , id} be the set of all items in a market basket data and T= {t1, t2, …, tN} be the set of all transactions. Each transaction, ti contains a subset of items chosen from I. In association analysis, a collection of zero or more items is termed an itemset. If an itemset contains k items, it is called a k-itemset. For instance, {Beer, Diapers, Milk} is an example of a 3-itemset. The null (or empty) set is an itemset that does not contain any items.
A transaction tj is said to contain an itemset X if X is a subset of tj. For example, the second transaction shown in Table 5.2 contains the itemset {Bread, Diapers} but not {Bread, Milk}. An important property of an itemset is its support count, which refers to the number of transactions that contain a particular itemset. Mathematically, the support count, σ(X), for an itemset X can be stated as follows:
σ(X)=|{ti|X⊆ti, ti∈T}|,
where the symbol |⋅| denotes the number of elements in a set. In the data set shown in Table 5.2, the support count for {Beer, Diapers, Milk} is equal to two because there are only two transactions that contain all three items.
Often, the property of interest is the support, which is fraction of transactions in which an itemset occurs:
s(X)=σ(X)/N.
An itemset X is called frequent if s(X) is greater than some user-defined threshold, minsup.
Association Rule An association rule is an implication expression of the form X→Y, where X and Y are disjoint itemsets, i.e., X∩Y=∅. The strength of an association rule can be measured in terms of its support and confidence. Support determines
how often a rule is applicable to a given data set, while confidence determines how frequently items in Y appear in transactions that contain X. The formal definitions of these metrics are
Support, s(X→Y)=σ(X∪Y)N; (5.1) Confidence, c(X→Y)=σ(X∪Y)σ(X). (5.2)
Example 5.1. Consider the rule {Milk, Diapers}→{Beer}. Because the support count for {Milk, Diapers, Beer} is 2 and the total number of transactions is 5, the rule's support is 2/5=0.4. The rule's confidence is obtained by dividing the support count for {Milk, Diapers, Beer} by the support count for {Milk, Diapers}. Since there are 3 transactions that contain milk and diapers, the confidence for this rule is 2/3=0.67.
Why Use Support and Confidence? Support is an important measure because a rule that has very low support might occur simply by chance. Also, from a business perspective a low support rule is unlikely to be interesting because it might not be profitable to promote items that customers seldom buy together (with the exception of the situation described in Section 5.8). For these reasons, we are interested in finding rules whose support is greater than some user-defined threshold. As will be shown in Section 5.2.1, support also has a desirable property that can be exploited for the efficient discovery of association rules.
Confidence, on the other hand, measures the reliability of the inference made by a rule. For a given rule X→Y, the higher the confidence, the more likely it is for Y to be present in transactions that contain X. Confidence also provides an estimate of the conditional probability of Y given X.
Association analysis results should be interpreted with caution. The inference made by an association rule does not necessarily imply causality. Instead, it can sometimes suggest a strong co-occurrence relationship between items in
the antecedent and consequent of the rule. Causality, on the other hand, requires knowledge about which attributes in the data capture cause and effect, and typically involves relationships occurring over time (e.g., greenhouse gas emissions lead to global warming). See Section 5.7.1 for additional discussion.
Formulation of the Association Rule Mining Problem The association rule mining problem can be formally stated as follows:
Definition 5.1. (Association Rule Discovery.) Given a set of transactions T , find all the rules having support ≥ minsup and confidence ≥ minconf, where minsup and minconf are the corresponding support and confidence thresholds.
A brute-force approach for mining association rules is to compute the support and confidence for every possible rule. This approach is prohibitively expensive because there are exponentially many rules that can be extracted from a data set. More specifically, assuming that neither the left nor the right- hand side of the rule is an empty set, the total number of possible rules, R, extracted from a data set that contains d items is
R=3d−2d+1+1. (5.3)
The proof for this equation is left as an exercise to the readers (see Exercise 5 on page 440). Even for the small data set shown in Table 5.1, this approach requires us to compute the support and confidence for 36−27+1=602 rules. More than 80% of the rules are discarded after applying minsup=20% and mincof=50%, thus wasting most of the computations. To avoid performing
needless computations, it would be useful to prune the rules early without having to compute their support and confidence values.
An initial step toward improving the performance of association rule mining algorithms is to decouple the support and confidence requirements. From Equation 5.1, notice that the support of a rule X→Y is the same as the support of its corresponding itemset, X∪Y. For example, the following rules have identical support because they involve items from the same itemset,
{Beer, Diapers, Milk}:
{Beer, Diapers}→{Milk},{Beer, Milk}→{Diapers},{Diapers, Milk} →{Beer},{Beer}→{Diapers, Milk},{Milk}→{Beer, Diapers},{Diapers} →{Beer, Milk}.
If the itemset is infrequent, then all six candidate rules can be pruned immediately without our having to compute their confidence values.
Therefore, a common strategy adopted by many association rule mining algorithms is to decompose the problem into two major subtasks:
1. Frequent Itemset Generation, whose objective is to find all the itemsets that satisfy the minsup threshold.
2. Rule Generation, whose objective is to extract all the high confidence rules from the frequent itemsets found in the previous step. These rules are called strong rules.
The computational requirements for frequent itemset generation are generally more expensive than those of rule generation. Efficient techniques for generating frequent itemsets and association rules are discussed in Sections 5.2 and 5.3, respectively.
5.2 Frequent Itemset Generation A lattice structure can be used to enumerate the list of all possible itemsets. Figure 5.1 shows an itemset lattice for I={a, b, c, d, e}. In general, a data set that contains k items can potentially generate up to 2k−1 frequent itemsets, excluding the null set. Because k can be very large in many practical applications, the search space of itemsets that need to be explored is exponentially large.
Figure 5.1. An itemset lattice.
Figure 5.1. Full Alternative Text
A brute-force approach for finding frequent itemsets is to determine the support count for every candidate itemset in the lattice structure. To do this, we need to compare each candidate against every transaction, an operation that is shown in Figure 5.2. If the candidate is contained in a transaction, its support count will be incremented. For example, the support for {Bread, Milk} is incremented three times because the itemset is contained in transactions 1, 4, and 5. Such an approach can be very expensive because it requires O(NMw) comparisons, where N is the number of transactions, M=2k −1 is the number of candidate itemsets, and w is the maximum transaction width. Transaction width is the number of items present in a transaction.
Figure 5.2. Counting the support of candidate itemsets.
There are three main approaches for reducing the computational complexity
of frequent itemset generation.
1. Reduce the number of candidate itemsets (M). The Apriori principle, described in the next Section, is an effective way to eliminate some of the candidate itemsets without counting their support values.
2. Reduce the number of comparisons. Instead of matching each candidate itemset against every transaction, we can reduce the number of comparisons by using more advanced data structures, either to store the candidate itemsets or to compress the data set. We will discuss these strategies in Sections 5.2.4 and 5.6, respectively.
3. Reduce the number of transactions (N). As the size of candidate itemsets increases, fewer transactions will be supported by the itemsets. For instance, since the width of the first transaction in Table 5.1 is 2, it would be advantageous to remove this transaction before searching for frequent itemsets of size 3 and larger. Algorithms that employ such a strategy are discussed in the Bibliographic Notes.
5.2.1 The Apriori Principle This Section describes how the support measure can be used to reduce the number of candidate itemsets explored during frequent itemset generation. The use of support for pruning candidate itemsets is guided by the following principle.
Theorem 5.1 (Apriori Principle). If an itemset is frequent, then all of its subsets must also be frequent.
To illustrate the idea behind the Apriori principle, consider the itemset lattice shown in Figure 5.3. Suppose {c, d, e} is a frequent itemset. Clearly, any transaction that contains {c, d, e} must also contain its subsets, {c, d}, {c, e}, {d, e}, {c}, {d}, and {e}. As a result, if {c, d, e} is frequent, then all subsets of {c, d, e} (i.e., the shaded itemsets in this figure) must also be frequent.
Figure 5.3. An illustration of the Apriori principle. If {c, d, e} is frequent, then all subsets of this itemset are frequent.
Conversely, if an itemset such as {a, b} is infrequent, then all of its supersets must be infrequent too. As illustrated in Figure 5.4, the entire subgraph containing the supersets of {a, b} can be pruned immediately once {a, b} is found to be infrequent. This strategy of trimming the exponential search space based on the support measure is known as support-based pruning. Such a pruning strategy is made possible by a key property of the support
measure, namely, that the support for an itemset never exceeds the support for its subsets. This property is also known as the anti-monotone property of the support measure.
Figure 5.4. An illustration of support-based pruning. If {a, b} is infrequent, then all supersets of {a, b} are infrequent.
Definition 5.2. (Anti-monotone Property.) A measure f possesses the anti-monotone property if for every itemset X that is a proper subset of itemset Y, i.e. X⊂Y, we have f(Y)≤f(X).
More generally, a large number of measures—see Section 5.7.1—can be applied to itemsets to evaluate various properties of itemsets. As will be shown in the next Section, any measure that has the anti-monotone property can be incorporated directly into an itemset mining algorithm to effectively prune the exponential search space of candidate itemsets.
5.2.2 Frequent Itemset Generation in the Apriori Algorithm Apriori is the first association rule mining algorithm that pioneered the use of support-based pruning to systematically control the exponential growth of candidate itemsets. Figure 5.5 provides a high-level illustration of the frequent itemset generation part of the Apriori algorithm for the transactions shown in Table 5.1. We assume that the support threshold is 60%, which is equivalent to a minimum support count equal to 3.
Figure 5.5. Illustration of frequent itemset generation using the Apriori algorithm.
Figure 5.5. Full Alternative Text
Initially, every item is considered as a candidate 1-itemset. After counting their supports, the candidate itemsets {Cola} and {Eggs} are discarded because they appear in fewer than three transactions. In the next iteration, candidate 2-itemsets are generated using only the frequent 1-itemsets because the Apriori principle ensures that all supersets of the infrequent 1-itemsets must be infrequent. Because there are only four frequent 1-itemsets, the number of candidate 2-itemsets generated by the algorithm is (42)=6. Two of these six candidates, {Beer, Bread} and {Beer, Milk}, are subsequently
found to be infrequent after computing their support values. The remaining four candidates are frequent, and thus will be used to generate candidate 3- itemsets. Without support-based pruning, there are(63)=20 candidate 3- itemsets that can be formed using the six items given in this example. With the Apriori principle, we only need to keep candidate 3-itemsets whose subsets are frequent. The only candidate that has this property is {Bread, Diapers, Milk}. However, even though the subsets of {Bread, Diapers, Milk} are frequent, the itemset itself is not.
The effectiveness of the Apriori pruning strategy can be shown by counting the number of candidate itemsets generated. A brute-force strategy of enumerating all itemsets (up to size 3) as candidates will produce
(61)+(62)+(63)=6+15+20=41
candidates. With the Apriori principle, this number decreases to
(61)+(42)+1=6+6+1=13
candidates, which represents a 68% reduction in the number of candidate itemsets even in this simple example.
The pseudocode for the frequent itemset generation part of the Apriori algorithm is shown in Algorithm 5.1. Let Ck denote the set of candidate k- itemsets and Fk denote the set of frequent k-itemsets:
The algorithm initially makes a single pass over the data set to determine the support of each item. Upon completion of this step, the set of all frequent 1-itemsets, F1, will be known (steps 1 and 2).
Next, the algorithm will iteratively generate new candidate k-itemsets and prune unnecessary candidates that are guaranteed to be infrequent given the frequent (k−1)-itemsets found in the previous iteration (steps 5 and 6). Candidate generation and pruning is implemented using the functions candidate-gen and candidate-prune, which are described in Section 5.2.3.
To count the support of the candidates, the algorithm needs to make an
additional pass over the data set (steps 7–12). The subset function is used to determine all the candidate itemsets in Ck that are contained in each transaction t. The implementation of this function is described in Section 5.2.4.
After counting their supports, the algorithm eliminates all candidate itemsets whose support counts are less than N×minsup (step 13).
The algorithm terminates when there are no new frequent itemsets generated, i.e., Fk=∅ (step 14).
The frequent itemset generation part of the Apriori algorithm has two important characteristics. First, it is a level-wise algorithm; i.e., it traverses the itemset lattice one level at a time, from frequent 1-itemsets to the maximum size of frequent itemsets. Second, it employs a generate-and-test strategy for finding frequent itemsets. At each iteration (level), new candidate itemsets are generated from the frequent itemsets found in the previous iteration. The support for each candidate is then counted and tested against the minsup threshold. The total number of iterations needed by the algorithm is kmax+1, where kmax is the maximum size of the frequent itemsets.
5.2.3 Candidate Generation and Pruning The candidate-gen and candidate-prune functions shown in Steps 5 and 6 of Algorithm 5.1 generate candidate itemsets and prunes unnecessary ones by performing the following two operations, respectively:
1. Candidate Generation. This operation generates new candidate k- itemsets based on the frequent (k−1)-itemsets found in the previous iteration.
Algorithm 5.1 Frequent itemset
generation of the Apriori algorithm. 1: k = 1. 2: Fk = {i | i ∈ I ∧ σ({i}) ≥ N × minsup}. {Find all frequent 1- 3: repeat 4: k = k + 1. 5: Ck = candidate-gen(Fk − 1). {Generate candidate itemsets.} 6: Ck = candidate-prune(Ck, Fk − 1). {Prune candidate itemsets.} 7: for each transaction t ∈ T do 8: Ct = subset(Ck, t). {Identify all candidates that belong to 9: for each candidate itemset c ∈ Ct do 10: σ(c) = σ(c) + 1. {Increment support count.} 11: end for 12: end for 13: Fk = {c | c ∈ Ck ∧ σ(c) ≥ N × minsup}. {Extract the frequent 14: until Fk = ∅ 15: Result = ∪Fk.
2. Candidate Pruning. This operation eliminates some of the candidate k- itemsets using support-based pruning, i.e. by removing k-itemsets whose subsets are known to be infrequent in previous iterations. Note that this pruning is done without computing the actual support of these k-itemsets (which could have required comparing them against each transaction).
Candidate Generation In principle, there are many ways to generate candidate itemsets. An effective candidate generation procedure must be complete and non-redundant. A candidate generation procedure is said to be complete if it does not omit any frequent itemsets. To ensure completeness, the set of candidate itemsets must subsume the set of all frequent itemsets, i.e., ∀k:Fk⊆Ck. A candidate generation procedure is non-redundant if it does not generate the same candidate itemset more than once. For example, the candidate itemset {a, b, c, d} can be generated in many ways—by merging {a, b, c} with {d}, {b, d} with {a, c}, {c} with {a, b, d}, etc. Generation of duplicate candidates leads
to wasted computations and thus should be avoided for efficiency reasons. Also, an effective candidate generation procedure should avoid generating too many unnecessary candidates. A candidate itemset is unnecessary if at least one of its subsets is infrequent, and thus, eliminated in the candidate pruning step.
Next, we will briefly describe several candidate generation procedures, including the one used by the candidate-gen function.
Brute-Force Method The brute-force method considers every k-itemset as a potential candidate and then applies the candidate pruning step to remove any unnecessary candidates whose subsets are infrequent (see Figure 5.6). The number of candidate itemsets generated at level k is equal to (dk), where d is the total number of items. Although candidate generation is rather trivial, candidate pruning becomes extremely expensive because a large number of itemsets must be examined.
Figure 5.6. A brute-force method for generating candidate 3-itemsets.
Fk−1×F1 Method An alternative method for candidate generation is to extend each frequent (k −1)-itemset with frequent items that are not part of the (k−1)-itemset. Figure 5.7 illustrates how a frequent 2-itemset such as {Beer, Diapers} can be augmented with a frequent item such as Bread to produce a candidate 3- itemset {Beer, Diapers, Bread}.
Figure 5.7. Generating and pruning candidate k-itemsets by merging a frequent (k−1)-itemset with a frequent item. Note that some of the candidates are unnecessary because their subsets are infrequent.
The procedure is complete because every frequent k-itemset is composed of a frequent (k−1)-itemset and a frequent 1-itemset. Therefore, all frequent k- itemsets are part of the candidate k-itemsets generated by this procedure. Figure 5.7 shows that the Fk−1×F1 candidate generation method only produces four candidate 3-itemsets, instead of the
(63)=20 itemsets produced by the brute-force method. The Fk−1×F1 method generates lower number of candidates because every candidate is guaranteed to contain at least one frequent (k−1)-itemset. While this procedure is a substantial improvement over the brute-force method, it can still produce a large number of unnecessary candidates, as the remaining subsets of a candidate itemset can still be infrequent.
Note that the approach discussed above does not prevent the same candidate
itemset from being generated more than once. For instance, {Bread, Diapers, Milk} can be generated by merging {Bread, Diapers} with {Milk}, {Bread, Milk} with {Diapers}, or {Diapers, Milk} with {Bread}. One way to avoid generating duplicate candidates is by ensuring that the items in each frequent itemset are kept sorted in their lexicographic order. For example, itemsets such as {Bread, Diapers}, {Bread, Diapers, Milk}, and {Diapers, Milk} follow lexicographic order as the items within every itemset are arranged alphabetically. Each frequent (k−1)-itemset X is then extended with frequent items that are lexicographically larger than the items in X. For example, the itemset {Bread, Diapers} can be augmented with {Milk} because Milk is lexicographically larger than Bread and Diapers. However, we should not augment {Diapers, Milk} with {Bread} nor {Bread, Milk} with {Diapers} because they violate the lexicographic ordering condition. Every candidate k- itemset is thus generated exactly once, by merging the lexicographically largest item with the remaining k−1 items in the itemset. If the Fk−1×F1 method is used in conjunction with lexicographic ordering, then only two candidate 3-itemsets will be produced in the example illustrated in Figure 5.7. {Beer, Bread, Diapers} and {Beer, Bread, Milk} will not be generated because {Beer, Bread} is not a frequent 2-itemset.
Fk−1×Fk−1 Method This candidate generation procedure, which is used in the candidate-gen function of the Apriori algorithm, merges a pair of frequent (k−1)-itemsets only if their first k−2 items, arranged in lexicographic order, are identical. Let A={a1, a2, …, ak−1} and B={b1, b2, …, bk−1} be a pair of frequent (k−1)- itemsets, arranged lexicographically. A and B are merged if they satisfy the following conditions:
ai=bi (for i=1, 2, …, k−2).
Note that in this case, ak−1≠bk−1 because A and B are two distinct itemsets. The candidate k-itemset generated by merging A and B consists of the first k −2 common items followed by ak−1 and bk−1 in lexicographic order. This candidate generation procedure is complete, because for every lexicographically ordered frequent k-itemset, there exists two
lexicographically ordered frequent (k−1)-itemsets that have identical items in the first k−2 positions.
In Figure 5.8, the frequent itemsets {Bread, Diapers} and {Bread, Milk} are merged to form a candidate 3-itemset {Bread, Diapers, Milk}. The algorithm does not have to merge {Beer, Diapers} with {Diapers, Milk} because the first item in both itemsets is different. Indeed, if {Beer, Diapers, Milk} is a viable candidate, it would have been obtained by merging {Beer, Diapers} with {Beer, Milk} instead. This example illustrates both the completeness of the candidate generation procedure and the advantages of using lexicographic ordering to prevent duplicate candidates. Also, if we order the frequent (k−1)- itemsets according to their lexicographic rank, itemsets with identical first k −2 items would take consecutive ranks. As a result, the Fk−1×Fk−1 candidate generation method would consider merging a frequent itemset only with ones that occupy the next few ranks in the sorted list, thus saving some computations.
Figure 5.8.
Generating and pruning candidate k-itemsets by merging pairs of frequent (k−1)-itemsets.
Figure 5.8 shows that the Fk−1×Fk−1 candidate generation procedure results in only one candidate 3-itemset. This is a considerable reduction from the four candidate 3-itemsets generated by the Fk−1×F1 method. This is because the Fk−1×Fk−1 method ensures that every candidate k-itemset contains at least two frequent (k−1)-itemsets, thus greatly reducing the number of candidates that are generated in this step.
Note that there can be multiple ways of merging two frequent (k−1)-itemsets in the Fk−1×Fk−1 procedure, one of which is merging if their first k−2 items are identical. An alternate approach could be to merge two frequent (k−1)- itemsets A and B if the last k−2 items of A are identical to the first k−2 itemsets of B. For example, {Bread, Diapers} and {Diapers, Milk} could be merged using this approach to generate the candidate 3-itemset {Bread, Diapers, Milk}. As we will see later, this alternate Fk−1×Fk−1 procedure is useful in generating sequential patterns, which will be discussed in Chapter 6.
Candidate Pruning To illustrate the candidate pruning operation for a candidate k-itemset, X= {i1, i2, …, ik}, consider its k proper subsets, X−{ij}(∀j=1, 2, …, k). If any of them are infrequent, then X is immediately pruned by using the Apriori principle. Note that we don't need to explicitly ensure that all subsets of X of size less than k−1 are frequent (see Exercise 7). This approach greatly reduces the number of candidate itemsets considered during support counting. For the brute-force candidate generation method, candidate pruning requires checking only k subsets of size k−1 for each candidate k-itemset. However, since the Fk−1×F1 candidate generation strategy ensures that at least one of the (k−1)-size subsets of every candidate k-itemset is frequent, we only need to check for the remaining k−1 subsets. Likewise, the Fk−1×Fk −1 strategy requires examining only k−2 subsets of every candidate k-itemset, since two of its (k−1)-size subsets are already known to be frequent in the candidate generation step.
5.2.4 Support Counting Support counting is the process of determining the frequency of occurrence for every candidate itemset that survives the candidate pruning step. Support counting is implemented in steps 6 through 11 of Algorithm 5.1. A brute- force approach for doing this is to compare each transaction against every candidate itemset (see Figure 5.2) and to update the support counts of candidates contained in a transaction. This approach is computationally expensive, especially when the numbers of transactions and candidate itemsets are large.
An alternative approach is to enumerate the itemsets contained in each transaction and use them to update the support counts of their respective candidate itemsets. To illustrate, consider a transaction t that contains five items, {1, 2, 3, 5, 6}. There are (53)=10 itemsets of size 3 contained in this transaction. Some of the itemsets may correspond to the candidate 3-itemsets under investigation, in which case, their support counts are incremented. Other subsets of t that do not correspond to any candidates can be ignored.
Figure 5.9 shows a systematic way for enumerating the 3-itemsets contained in t. Assuming that each itemset keeps its items in increasing lexicographic order, an itemset can be enumerated by specifying the smallest item first, followed by the larger items. For instance, givent={1, 2, 3, 5, 6}, all the 3- itemsets contained in t must begin with item 1, 2, or 3. It is not possible to construct a 3-itemset that begins with items 5 or 6 because there are only two items in t whose labels are greater than or equal to 5. The number of ways to specify the first item of a 3-itemset contained in t is illustrated by the Level 1 prefix tree structure depicted in Figure 5.9. For instance, 1 2 3 5 6 represents a 3-itemset that begins with item 1, followed by two more items chosen from the set {2, 3, 5, 6}.
Figure 5.9. Enumerating subsets of three items from a transaction t.
Figure 5.9. Full Alternative Text
After fixing the first item, the prefix tree structure at Level 2 represents the number of ways to select the second item. For example, 1 2 3 5 6 corresponds to itemsets that begin with the prefix (1 2) and are followed by the items 3, 5, or 6. Finally, the prefix tree structure at Level 3 represents the complete set of 3-itemsets contained in t. For example, the 3-itemsets that begin with prefix {1 2} are {1, 2, 3}, {1, 2, 5}, and {1, 2, 6}, while those that begin with prefix {2 3} are {2, 3, 5} and {2, 3, 6}.
The prefix tree structure shown in Figure 5.9 demonstrates how itemsets contained in a transaction can be systematically enumerated, i.e., by
specifying their items one by one, from the leftmost item to the rightmost item. We still have to determine whether each enumerated 3-itemset corresponds to an existing candidate itemset. If it matches one of the candidates, then the support count of the corresponding candidate is incremented. In the next Section, we illustrate how this matching operation can be performed efficiently using a hash tree structure.
Support Counting Using a Hash Tree* In the Apriori algorithm, candidate itemsets are partitioned into different buckets and stored in a hash tree. During support counting, itemsets contained in each transaction are also hashed into their appropriate buckets. That way, instead of comparing each itemset in the transaction with every candidate itemset, it is matched only against candidate itemsets that belong to the same bucket, as shown in Figure 5.10.
Figure 5.10.
Counting the support of itemsets using hash structure.
Figure 5.11 shows an example of a hash tree structure. Each internal node of the tree uses the following hash function, h(p)=(p−1) mod 3,, where mode refers to the modulo (remainder) operator, to determine which branch of the current node should be followed next. For example, items 1, 4, and 7 are hashed to the same branch (i.e., the leftmost branch) because they have the same remainder after dividing the number by 3. All candidate itemsets are stored at the leaf nodes of the hash tree. The hash tree shown in Figure 5.11 contains 15 candidate 3-itemsets, distributed across 9 leaf nodes.
Figure 5.11. Hashing a transaction at the root node of a hash tree.
Figure 5.11. Full Alternative Text
Consider the transaction, t={1, 2, 3, 4, 5, 6}. To update the support counts of the candidate itemsets, the hash tree must be traversed in such a way that all the leaf nodes containing candidate 3-itemsets belonging to t must be visited at least once. Recall that the 3-itemsets contained in t must begin with items 1, 2, or 3, as indicated by the Level 1 prefix tree structure shown in Figure 5.9. Therefore, at the root node of the hash tree, the items 1, 2, and 3 of the transaction are hashed separately. Item 1 is hashed to the left child of the root node, item 2 is hashed to the middle child, and item 3 is hashed to the right child. At the next level of the tree, the transaction is hashed on the second item listed in the Level 2 tree structure shown in Figure 5.9. For example, after hashing on item 1 at the root node, items 2, 3, and 5 of the transaction are hashed. Based on the hash function, items 2 and 5 are hashed to the middle child, while item 3 is hashed to the right child, as shown in Figure 5.12. This process continues until the leaf nodes of the hash tree are reached. The candidate itemsets stored at the visited leaf nodes are compared against the transaction. If a candidate is a subset of the transaction, its support count is incremented. Note that not all the leaf nodes are visited while traversing the hash tree, which helps in reducing the computational cost. In this example, 5 out of the 9 leaf nodes are visited and 9 out of the 15 itemsets are compared against the transaction.
Figure 5.12. Subset operation on the leftmost subtree of the root of a candidate hash tree.
5.2.5 Computational Complexity The computational complexity of the Apriori algorithm, which includes both its runtime and storage, can be affected by the following factors.
Support Threshold
Lowering the support threshold often results in more itemsets being declared as frequent. This has an adverse effect on the computational complexity of the algorithm because more candidate itemsets must be generated and counted at every level, as shown in Figure 5.13. The maximum size of frequent itemsets also tends to increase with lower support thresholds. This increases the total number of iterations to be performed by the Apriori algorithm, further increasing the computational cost.
Figure 5.13. Effect of support threshold on the number of candidate and frequent itemsets obtained from a benchmark data set.
Figure 5.13. Full Alternative Text
Number of Items (Dimensionality) As the number of items increases, more space will be needed to store the support counts of items. If the number of frequent items also grows with the
dimensionality of the data, the runtime and storage requirements will increase because of the larger number of candidate itemsets generated by the algorithm.
Number of Transactions Because the Apriori algorithm makes repeated passes over the transaction data set, its run time increases with a larger number of transactions.
Average Transaction Width For dense data sets, the average transaction width can be very large. This affects the complexity of the Apriori algorithm in two ways. First, the maximum size of frequent itemsets tends to increase as the average transaction width increases. As a result, more candidate itemsets must be examined during candidate generation and support counting, as illustrated in Figure 5.14. Second, as the transaction width increases, more itemsets are contained in the transaction. This will increase the number of hash tree traversals performed during support counting.
A detailed analysis of the time complexity for the Apriori algorithm is presented next.
Figure 5.14. Effect of average transaction width on the number of candidate and frequent itemsets obtained from a synthetic data set.
Figure 5.14. Full Alternative Text
Generation of frequent 1-itemsets For each transaction, we need to update the support count for every item present in the transaction. Assuming that w is the average transaction width, this operation requires O(Nw) time, where N is the total number of transactions.
Candidate generation To generate candidate k-itemsets, pairs of frequent (k−1)-itemsets are merged
to determine whether they have at least k−2 items in common. Each merging operation requires at most k−2 equality comparisons. Every merging step can produce at most one viable candidate k-itemset, while in the worst-case, the algorithm must try to merge every pair of frequent (k−1)-itemsets found in the previous iteration. Therefore, the overall cost of merging frequent itemsets is
∑k=2w(k−2)|Ck|<Cost of merging<∑k=2w(k−2)|Fk−1|2,
where w is the maximum transaction width. A hash tree is also constructed during candidate generation to store the candidate itemsets. Because the maximum depth of the tree is k, the cost for populating the hash tree with candidate itemsets is O(∑k=2wk|Ck|). During candidate pruning, we need to verify that the k−2 subsets of every candidate k-itemset are frequent. Since the cost for looking up a candidate in a hash tree is O(k), the candidate pruning step requires O(∑k=2wk(k−2)|Ck|) time.
Support counting Each transaction of width |t| produces (|t|k) itemsets of size k. This is also the effective number of hash tree traversals performed for each transaction. The cost for support counting is O(N∑k(wk)αk), where w is the maximum transaction width and αk is the cost for updating the support count of a candidate k-itemset in the hash tree.
5.3 Rule Generation This Section describes how to extract association rules efficiently from a given frequent itemset. Each frequent k-itemset, Y, can produce up to 2k−2 association rules, ignoring rules that have empty antecedents or consequents ∅→Y or Y→∅). An association rule can be extracted by partitioning the itemset Y into two non-empty subsets, X and Y−X, such that X→Y−X satisfies the confidence threshold. Note that all such rules must have already met the support threshold because they are generated from a frequent itemset.
Example 5.2. Let X={a, b, c} be a frequent itemset. There are six candidate association rules that can be generated from X:{a, b}→{c}, {a, c}→{b}, {b, c} →{a}, {a}→{b, c}, {b}→{a, c}, and {c}→{a, b}. As each of their support is identical to the support for X, all the rules satisfy the support threshold.
Computing the confidence of an association rule does not require additional scans of the transaction data set. Consider the rule {1, 2}→{3}, which is generated from the frequent itemset X={1, 2, 3}. The confidence for this rule is σ{(1, 2, 3})/σ({1, 2}). Because {1, 2, 3} is frequent, the anti-monotone property of support ensures that {1, 2} must be frequent, too. Since the support counts for both itemsets were already found during frequent itemset generation, there is no need to read the entire data set again.
5.3.1 Confidence-Based Pruning Confidence does not show the anti-monotone property in the same way as the support measure. For example, the confidence for X→Y can be larger, smaller, or equal to the confidence for another rule X˜→Y˜, where X˜⊆X and Y˜⊆Y (see Exercise 3 on page 439). Nevertheless, if we compare rules generated from the same frequent itemset Y, the following theorem holds for
the confidence measure.
Theorem 5.2. Let Y be an itemset and X is a subset of Y. If a rule X→Y−X does not satisfy the confidence threshold, then any rule X˜→Y−X˜, where X˜ is a subset of X, must not satisfy the confidence threshold as well.
To prove this theorem, consider the following two rules: X˜→Y−X˜ and X→Y−X, where X˜⊂X. The confidence of the rules are σ(Y)/σ(X˜) and σ(Y)/σ(X), respectively. Since X˜ is a subset of X, σ(X˜)/σ(X). Therefore, the former rule cannot have a higher confidence than the latter rule.
5.3.2 Rule Generation in Apriori Algorithm The Apriori algorithm uses a level-wise approach for generating association rules, where each level corresponds to the number of items that belong to the rule consequent. Initially, all the high confidence rules that have only one item in the rule consequent are extracted. These rules are then used to generate new candidate rules. For example, if {acd}→{b} and {abd}→{c} are high confidence rules, then the candidate rule {ad}→{bc} is generated by merging the consequents of both rules. Figure 5.15 shows a lattice structure for the association rules generated from the frequent itemset {a, b, c, d}. If any node in the lattice has low confidence, then according to Theorem 5.2, the entire subgraph spanned by the node can be pruned immediately. Suppose the confidence for {bcd}→{a} is low. All the rules containing item a in its consequent, including {cd}→{ab}, {bd}→{ac}, {bc}→{ad}, and {d} →{abc} can be discarded.
Figure 5.15. Pruning of association rules using the confidence measure.
Figure 5.15. Full Alternative Text
A pseudocode for the rule generation step is shown in Algorithms 5.2 and 5.3. Note the similarity between the ap-genrules procedure given in Algorithm 5.3 and the frequent itemset generation procedure given in Algorithm 5.1. The only difference is that, in rule generation, we do not have to make additional passes over the data set to compute the confidence of the candidate rules. Instead, we determine the confidence of each rule by using the support counts computed during frequent itemset generation.
Algorithm 5.2 Rule generation of
the Apriori algorithm. 1: for each frequent k-itemset fk, k ≥ 2 do 2: H1 = {i | i ∈ fk} {1-item consequents of the rule.} 3: call ap-genrules(fk, H1.) 4: end for
Algorithm 5.3 Procedure ap- genrules (fk, Hm). 1: k = |fk| {size of frequent itemset.} 2: m = |Hm| {size of rule consequent.} 3: if k > m + 1 then 4: Hm + 1 = candidate-gen(Hm). 5: Hm + 1 = candidate-prune(Hm + 1,Hm). 6: for each hm + 1 ∈ Hm + 1 do 7: conf = σ(fk)/σ(fk − hm + 1). 8: if conf ≥ minconf then 9: output the rule (fk − hm + 1) → hm + 1. 10: else 11: delete hm + 1 from Hm + 1. 12: end if 13: end for 14: call ap-genrules(fk, Hm + 1.) 15: end if
5.3.3 An Example: Congressional Voting Records This Section demonstrates the results of applying association analysis to the voting records of members of the United States House of Representatives. The data is obtained from the 1984 Congressional Voting Records Database, which is available at the UCI machine learning data repository. Each
transaction contains information about the party affiliation for a representative along with his or her voting record on 16 key issues. There are 435 transactions and 34 items in the data set. The set of items are listed in Table 5.3.
Table 5.3. List of binary attributes from the 1984 United States Congressional Voting Records. Source: The UCI machine learning repository.
1. Republican
2. Democrat
3. handicapped-infants=yes
4. handicapped-infants=no
5. water project cost sharing=yes
6. water project cost sharing=no
7. budget-resolution=yes
8. budget-resolution=no
9. physician fee freeze=yes
10. physician fee freeze=no
11. aid to El Salvador=yes
12. aid to El Salvador=no
13. religious groups in schools=yes
14. religious groups in schools=no
15. anti-satellite test ban=yes
16. anti-satellite test ban=no
17. aid to Nicaragua=yes
18. aid to Nicaragua=no
19. MX-missile=yes
20. MX-missile=no
21. immigration=yes
22. immigration=no
23. synfuel corporation cutback=yes
24. synfuel corporation cutback=no
25. education spending=yes
26. education spending=no
27. right-to-sue=yes
28. right-to-sue=no
29. crime=yes
30. crime=no
31. duty-free-exports=yes
32. duty-free-exports=no
33. export administration act=yes
34. export administration act=no
The Apriori algorithm is then applied to the data set with minsup=30% and minconf=90%. Some of the high confidence rules extracted by the algorithm are shown in Table 5.4. The first two rules suggest that most of the members who voted yes for aid to El Salvador and no for budget resolution and MX missile are Republicans; while those who voted no for aid to El Salvador and yes for budget resolution and MX missile are Democrats. These high confidence rules show the key issues that divide members from both political parties.
Table 5.4. Association rules extracted from the 1984 United States Congressional Voting Records. Association Rule Confidence {budget resolution=no, MX- missile=no, aid to El Salvador=yes }→{Republican} 91.0%
{budget resolution=yes, MX- missile=yes, aid to El Salvador=no }→{Democrat} 97.5%
{crime=yes, right-to- sue=yes, physician fee freeze=yes }→{Republican} 93.5%
{crime=no, right-to-sue=no, physician fee freeze=no } →{Democrat} 100%
5.4 Compact Representation of Frequent Itemsets In practice, the number of frequent itemsets produced from a transaction data set can be very large. It is useful to identify a small representative set of frequent itemsets from which all other frequent itemsets can be derived. Two such representations are presented in this Section in the form of maximal and closed frequent itemsets.
5.4.1 Maximal Frequent Itemsets
Definition 5.3. (Maximal Frequent Itemset.) A frequent itemset is maximal if none of its immediate supersets are frequent.
To illustrate this concept, consider the itemset lattice shown in Figure 5.16. The itemsets in the lattice are divided into two groups: those that are frequent and those that are infrequent. A frequent itemset border, which is represented by a dashed line, is also illustrated in the diagram. Every itemset located above the border is frequent, while those located below the border (the shaded nodes) are infrequent. Among the itemsets residing near the border, {a, d}, {a, c, e}, and {b, c, d, e} are maximal frequent itemsets because all of their immediate supersets are infrequent. For example, the itemset {a, d} is maximal frequent because all of its immediate supersets, {a, b, d}, {a, c, d}, and {a, d, e}, are infrequent. In contrast, {a, c} is non-maximal because one of its immediate supersets, {a, c, e}, is frequent.
Figure 5.16. Maximal frequent itemset.
Figure 5.16. Full Alternative Text
Maximal frequent itemsets effectively provide a compact representation of frequent itemsets. In other words, they form the smallest set of itemsets from which all frequent itemsets can be derived. For example, every frequent itemset in Figure 5.16 is a subset of one of the three maximal frequent itemsets, {a, d}, {a, c, e}, and {b, c, d, e}. If an itemset is not a proper subset of any of the maximal frequent itemsets, then it is either infrequent (e.g., {a, d, e}) or maximal frequent itself (e.g., {b, c, d, e}). Hence, the maximal
frequent itemsets {a, c, e}, {a, d}, and {b, c, d, e} provide a compact representation of the frequent itemsets shown in Figure 5.16. Enumerating all the subsets of maximal frequent itemsets generates the complete list of all frequent itemsets.
Maximal frequent itemsets provide a valuable representation for data sets that can produce very long, frequent itemsets, as there are exponentially many frequent itemsets in such data. Nevertheless, this approach is practical only if an efficient algorithm exists to explicitly find the maximal frequent itemsets. We briefly describe one such approach in Section 5.5.
Despite providing a compact representation, maximal frequent itemsets do not contain the support information of their subsets. For example, the support of the maximal frequent itemsets {a, c, e}, {a, d}, and {b, c, d, e} do not provide any information about the support of their subsets except that it meets the support threshold. An additional pass over the data set is therefore needed to determine the support counts of the non-maximal frequent itemsets. In some cases, it is desirable to have a minimal representation of itemsets that preserves the support information. We describe such a representation in the next Section.
5.4.2 Closed Itemsets Closed itemsets provide a minimal representation of all itemsets without losing their support information. A formal definition of a closed itemset is presented below.
Definition 5.4. (Closed Itemset.) An itemset X is closed if none of its immediate supersets has exactly the same support count as X.
Put another way, X is not closed if at least one of its immediate supersets has the same support count as X. Examples of closed itemsets are shown in Figure 5.17. To better illustrate the support count of each itemset, we have
associated each node (itemset) in the lattice with a list of its corresponding transaction IDs. For example, since the node {b, c} is associated with transaction IDs 1, 2, and 3, its support count is equal to three. From the transactions given in this diagram, notice that the support for {b} is identical to {b, c}.This is because every transaction that contains b also contains c. Hence, {b} is not a closed itemset. Similarly, since c occurs in every transaction that contains both a and d, the itemset {a, d} is not closed as it has the same support as its superset {a, c, d}. On the other hand, {b, c} is a closed itemset because it does not have the same support count as any of its supersets.
Figure 5.17.
An example of the closed frequent itemsets (with minimum support equal to 40%).
Figure 5.17. Full Alternative Text
An interesting property of closed itemsets is that if we know their support counts, we can derive the support count of every other itemset in the itemset lattice without making additional passes over the data set. For example, consider the 2-itemset {b, e} in Figure 5.17. Since {b, e} is not closed, its support must be equal to the support of one of its immediate supersets, {a, b, e}, {b, c, e}, and {b, d, e}. Further, none of the supersets of {b, e} can have a support greater than the support of {b, e}, due to the anti-monotone nature of the support measure. Hence, the support of {b, e} can be computed by examining the support counts of all of its immediate supersets of size three and taking their maximum value. If an immediate superset is closed (e.g., {b, c, e}), we would know its support count. Otherwise, we can recursively compute its support by examining the supports of its immediate supersets of size four. In general, the support count of any non-closed (k−1)-itemset can be determined as long as we know the support counts of all k-itemsets. Hence, one can devise an iterative algorithm that computes the support counts of itemsets at level k−1 using the support counts of itemsets at level k, starting from the level kmax, where kmax is the size of the largest closed itemset.
Even though closed itemsets provide a compact representation of the support counts of all itemsets, they can still be exponentially large in number. Moreover, for most practical applications, we only need to determine the support count of all frequent itemsets. In this regard, closed frequent item- sets provide a compact representation of the support counts of all frequent itemsets, which can be defined as follows.
Definition 5.5. (Closed Frequent Itemset.) An itemset is a closed frequent itemset if it is closed and its support is greater
than or equal to minsup.
In the previous example, assuming that the support threshold is 40%, {b, c} is a closed frequent itemset because its support is 60%. In Figure 5.17, the closed frequent itemsets are indicated by the shaded nodes.
Algorithms are available to explicitly extract closed frequent itemsets from a given data set. Interested readers may refer to the Bibliographic Notes at the end of this chapter for further discussions of these algorithms. We can use closed frequent itemsets to determine the support counts for all non-closed frequent itemsets. For example, consider the frequent itemset {a, d} shown in Figure 5.17. Because this itemset is not closed, its support count must be equal to the maximum support count of its immediate supersets, {a, b, d}, {a, c, d}, and {a, d, e}. Also, since {a, d} is frequent, we only need to consider the support of its frequent supersets. In general, the support count of every non-closed frequent k-itemset can be obtained by considering the support of all its frequent supersets of size k+1. For example, since the only frequent superset of {a, d} is {a, c, d}, its support is equal to the support of {a, c, d}, which is 2. Using this methodology, an algorithm can be developed to compute the support for every frequent itemset. The pseudocode for this algorithm is shown in Algorithm 5.4. The algorithm proceeds in a specific-to- general fashion, i.e., from the largest to the smallest frequent itemsets. This is because, in order to find the support for a non-closed frequent itemset, the support for all of its supersets must be known. Note that the set of all frequent itemsets can be easily computed by taking the union of all subsets of frequent closed itemsets.
Algorithm 5.4 Support counting using closed frequent itemsets. 1: Let C denote the set of closed frequent itemsets and F denote the set of all frequent itemsets. 2: Let kmax denote the maximum size of closed frequent itemsets 3: Fkmax = {f|f ∈ C, |f| = kmax} {Find all frequent itemsets of size 4: for k = kmax − 1 down to 1 do 5: Fk = {f|f ∈ F, |f| = k} {Find all frequent itemsets of size
6: for each f ∈ Fk do 7: if f ∉ C then 8: f⋅support=max{f′⋅support|f′∈Fk+1, f⊂f′} 9: end if 10: end for 11: end for
To illustrate the advantage of using closed frequent itemsets, consider the data set shown in Table 5.5, which contains ten transactions and fifteen items. The items can be divided into three groups: (1) Group A, which contains items a1 through a5; (2) Group B, which contains items b1 through b5; and (3) Group C, which contains items c1 through c5. Assuming that the support threshold is 20%, itemsets involving items from the same group are frequent, but itemsets involving items from different groups are infrequent. The total number of frequent itemsets is thus 3×(25−1)=93. However, there are only four closed frequent itemsets in the data: ({a3, a4}, {a1, a2, a3, a4, a5}, {b1,b2,b3,b4,b5}, and {c1, c2, c3, c4, c5}). It is often sufficient to present only the closed frequent itemsets to the analysts instead of the entire set of frequent itemsets.
Table 5.5. A transaction data set for mining closed itemsets. TID a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 c1 c2 c3 c4 c5
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 3 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 5 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 6 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 8 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 9 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
Finally, note that all maximal frequent itemsets are closed because none of the maximal frequent itemsets can have the same support count as their immediate supersets. The relationships among frequent, closed, closed frequent, and maximal frequent itemsets are shown in Figure 5.18.
Figure 5.18. Relationships among frequent, closed, closed frequent, and maximal frequent itemsets.
5.5 Alternative Methods for Generating Frequent Itemsets* Apriori is one of the earliest algorithms to have successfully addressed the combinatorial explosion of frequent itemset generation. It achieves this by applying the Apriori principle to prune the exponential search space. Despite its significant performance improvement, the algorithm still incurs considerable I/O overhead since it requires making several passes over the transaction data set. In addition, as noted in Section 5.2.5, the performance of the Apriori algorithm may degrade significantly for dense data sets because of the increasing width of transactions. Several alternative methods have been developed to overcome these limitations and improve upon the efficiency of the Apriori algorithm. The following is a high-level description of these methods.
Traversal of Itemset Lattice A search for frequent itemsets can be conceptually viewed as a traversal on the itemset lattice shown in Figure 5.1. The search strategy employed by an algorithm dictates how the lattice structure is traversed during the frequent itemset generation process. Some search strategies are better than others, depending on the configuration of frequent itemsets in the lattice. An overview of these strategies is presented next.
General-to-Specific versus Specific-to-General: The Apriori algorithm uses a general-to-specific search strategy, where pairs of frequent (k−1)- itemsets are merged to obtain candidate k-itemsets. This general-to- specific search strategy is effective, provided the maximum length of a frequent itemset is not too long. The configuration of frequent itemsets that works best with this strategy is shown in Figure 5.19(a), where the darker nodes represent infrequent itemsets. Alternatively, a specificto- general search strategy looks for more specific frequent itemsets first,
before finding the more general frequent itemsets. This strategy is useful to discover maximal frequent itemsets in dense transactions, where the frequent itemset border is located near the bottom of the lattice, as shown in Figure 5.19(b). The Apriori principle can be applied to prune all subsets of maximal frequent itemsets. Specifically, if a candidate k- itemset is maximal frequent, we do not have to examine any of its subsets of size k−1. However, if the candidate k-itemset is infrequent, we need to check all of its k−1 subsets in the next iteration. Another approach is to combine both general-to-specific and specific-to-general search strategies. This bidirectional approach requires more space to store the candidate itemsets, but it can help to rapidly identify the frequent itemset border, given the configuration shown in Figure 5.19(c).
Figure 5.19. General-to-specific, specific-to-general, and bidirectional search.
Figure 5.19. Full Alternative Text
Equivalence Classes: Another way to envision the traversal is to first partition the lattice into disjoint groups of nodes (or equivalence classes). A frequent itemset generation algorithm searches for frequent itemsets within a particular equivalence class first before moving to another equivalence class. As an example, the level-wise strategy used in the Apriori algorithm can be considered to be partitioning the lattice on the basis of itemset sizes; i.e., the algorithm discovers all frequent 1- itemsets first before proceeding to larger-sized itemsets. Equivalence classes can also be defined according to the prefix or suffix labels of an itemset. In this case, two itemsets belong to the same equivalence class if they share a common prefix or suffix of length k. In the prefix-based approach, the algorithm can search for frequent itemsets starting with the prefix a before looking for those starting with prefixes b, c, and so on. Both prefix-based and suffix-based equivalence classes can be demonstrated using the tree-like structure shown in Figure 5.20.
Figure 5.20.
Equivalence classes based on the prefix and suffix labels of itemsets.
Figure 5.20. Full Alternative Text
Breadth-First versus Depth-First: The Apriori algorithm traverses the lattice in a breadth-first manner, as shown in Figure 5.21(a). It first discovers all the frequent 1-itemsets, followed by the frequent 2- itemsets, and so on, until no new frequent itemsets are generated. The itemset lattice can also be traversed in a depth-first manner, as shown in Figures 5.21(b) and 5.22. The algorithm can start from, say, node a in Figure 5.22, and count its support to determine whether it is frequent. If so, the algorithm progressively expands the next level of nodes, i.e., ab, abc, and so on, until an infrequent node is reached, say, abcd. It then backtracks to another branch, say, abce, and continues the search from there.
Figure 5.21. Breadth-first and depth-first traversals.
Figure 5.21. Full Alternative Text
Figure 5.22. Generating candidate itemsets using the depth-first approach.
Figure 5.22. Full Alternative Text
The depth-first approach is often used by algorithms designed to find maximal frequent itemsets. This approach allows the frequent itemset border to be detected more quickly than using a breadth-first approach.
Once a maximal frequent itemset is found, substantial pruning can be performed on its subsets. For example, if the node bcde shown in Figure 5.22 is maximal frequent, then the algorithm does not have to visit the subtrees rooted at bd, be, c, d, and e because they will not contain any maximal frequent itemsets. However, if abc is maximal frequent, only the nodes such as ac and bc are not maximal frequent (but the subtrees of ac and bc may still contain maximal frequent itemsets). The depth- first approach also allows a different kind of pruning based on the
support of itemsets. For example, suppose the support for {a, b, c} is identical to the support for {a, b}. The subtrees rooted at abd and abe can be skipped because they are guaranteed not to have any maximal frequent itemsets. The proof of this is left as an exercise to the readers.
Representation of Transaction Data Set There are many ways to represent a transaction data set. The choice of representation can affect the I/O costs incurred when computing the support of candidate itemsets. Figure 5.23 shows two different ways of representing market basket transactions. The representation on the left is called a horizontal data layout, which is adopted by many association rule mining algorithms, including Apriori. Another possibility is to store the list of transaction identifiers (TID-list) associated with each item. Such a representation is known as the vertical data layout. The support for each candidate itemset is obtained by intersecting the TID-lists of its subset items. The length of the TID-lists shrinks as we progress to larger sized itemsets. However, one problem with this approach is that the initial set of TID-lists might be too large to fit into main memory, thus requiring more sophisticated techniques to compress the TID-lists. We describe another effective approach to represent the data in the next Section.
Figure 5.23. Horizontal and vertical data format.
Horizontal Data Layout
Figure 5.23. Full Alternative Text
5.6 FP-Growth Algorithm* This Section presents an alternative algorithm called FP-growth that takes a radically different approach to discovering frequent itemsets. The algorithm does not subscribe to the generate-and-test paradigm of Apriori. Instead, it encodes the data set using a compact data structure called an FP-tree and extracts frequent itemsets directly from this structure. The details of this approach are presented next.
5.6.1 FP-Tree Representation An FP-tree is a compressed representation of the input data. It is constructed by reading the data set one transaction at a time and mapping each transaction onto a path in the FP-tree. As different transactions can have several items in common, their paths might overlap. The more the paths overlap with one another, the more compression we can achieve using the FP-tree structure. If the size of the FP-tree is small enough to fit into main memory, this will allow us to extract frequent itemsets directly from the structure in memory instead of making repeated passes over the data stored on disk.
Figure 5.24 shows a data set that contains ten transactions and five items. The structures of the FP-tree after reading the first three transactions are also depicted in the diagram. Each node in the tree contains the label of an item along with a counter that shows the number of transactions mapped onto the given path. Initially, the FP-tree contains only the root node represented by the null symbol. The FP-tree is subsequently extended in the following way:
Figure 5.24. Construction of an FP-tree.
Figure 5.24. Full Alternative Text
1. The data set is scanned once to determine the support count of each item. Infrequent items are discarded, while the frequent items are sorted in decreasing support counts inside every transaction of the data set. For the data set shown in Figure 5.24, a is the most frequent item, followed by b, c, d, and e.
2. The algorithm makes a second pass over the data to construct the FP- tree. After reading the first transaction, {a, b}, the nodes labeled as a and b are created. A path is then formed from null →a→b to encode the transaction. Every node along the path has a frequency count of 1.
3. After reading the second transaction, {b, c, d}, a new set of nodes is created for items b, c, and d. A path is then formed to represent the transaction by connecting the nodes null →b→c→d. Every node along this path also has a frequency count equal to one. Although the first two transactions have an item in common, which is b, their paths are disjoint because the transactions do not share a common prefix.
4. The third transaction, {a, c, d, e}, shares a common prefix item (which is a) with the first transaction. As a result, the path for the third transaction, null →a→c→d→e, overlaps with the path for the first transaction, null →a→b. Because of their overlapping path, the frequency count for node a is incremented to two, while the frequency counts for the newly created nodes, c, d, and e, are equal to one.
5. This process continues until every transaction has been mapped onto one of the paths given in the FP-tree. The resulting FP-tree after reading all the transactions is shown at the bottom of Figure 5.24.
The size of an FP-tree is typically smaller than the size of the uncompressed data because many transactions in market basket data often share a few items in common. In the best-case scenario, where all the transactions have the same set of items, the FP-tree contains only a single branch of nodes. The worst-case scenario happens when every transaction has a unique set of items. As none of the transactions have any items in common, the size of the FP-tree is effectively the same as the size of the original data. However, the
physical storage requirement for the FP-tree is higher because it requires additional space to store pointers between nodes and counters for each item.
The size of an FP-tree also depends on how the items are ordered. The notion of ordering items in decreasing order of support counts relies on the possibility that the high support items occur more frequently across all paths and hence must be used as most commonly occurring prefixes. For example, if the ordering scheme in the preceding example is reversed, i.e., from lowest to highest support item, the resulting FP-tree is shown in Figure 5.25. The tree appears to be denser because the branching factor at the root node has increased from 2 to 5 and the number of nodes containing the high support items such as a and b has increased from 3 to 12. Nevertheless, ordering by decreasing support counts does not always lead to the smallest tree, especially when the high support items do not occur frequently together with the other items. For example, suppose we augment the data set given in Figure 5.24 with 100 transactions that contain {e}, 80 transactions that contain {d}, 60 transactions that contain {c}, and 40 transactions that contain {b}. Item e is now most frequent, followed by d, c, b, and a. With the augmented transactions, ordering by decreasing support counts will result in an FP-tree similar to Figure 5.25, while a scheme based on increasing support counts produces a smaller FP-tree similar to Figure 5.24(iv).
Figure 5.25.
An FP-tree representation for the data set shown in Figure 5.24 with a different item ordering scheme.
Figure 5.25. Full Alternative Text
An FP-tree also contains a list of pointers connecting nodes that have the same items. These pointers, represented as dashed lines in Figures 5.24 and 5.25, help to facilitate the rapid access of individual items in the tree. We explain how to use the FP-tree and its corresponding pointers for frequent itemset generation in the next Section.
5.6.2 Frequent Itemset Generation in FP-Growth Algorithm FP-growth is an algorithm that generates frequent itemsets from an FP-tree by exploring the tree in a bottom-up fashion. Given the example tree shown in Figure 5.24, the algorithm looks for frequent itemsets ending in e first, followed by d, c, b, and finally, a. This bottom-up strategy for finding frequent itemsets ending with a particular item is equivalent to the suffix- based approach described in Section 5.5. Since every transaction is mapped onto a path in the FP-tree, we can derive the frequent itemsets ending with a particular item, say, e, by examining only the paths containing node e. These paths can be accessed rapidly using the pointers associated with node e. The extracted paths are shown in Figure 5.26 (a). Similar paths for itemsets ending in d, c, b, and a are shown in Figures 5.26 (b), (c), (d), and (e), respectively.
Figure 5.26. Decomposing the frequent itemset generation problem into multiple subproblems, where each subproblem involves finding frequent itemsets ending in e, d, c, b, and a.
Figure 5.26. Full Alternative Text
FP-growth finds all the frequent itemsets ending with a particular suffix by employing a divide-and-conquer strategy to split the problem into smaller subproblems. For example, suppose we are interested in finding all frequent itemsets ending in e. To do this, we must first check whether the itemset {e} itself is frequent. If it is frequent, we consider the subproblem of finding frequent itemsets ending in de,followedby ce, be,and ae. In turn, each of these subproblems are further decomposed into smaller subproblems. By
merging the solutions obtained from the subproblems, all the frequent itemsets ending in e can be found. Finally, the set of all frequent itemsets can be generated by merging the solutions to the subproblems of finding frequent itemsets ending in e, d, c, b, and a. This divide-and-conquer approach is the key strategy employed by the FP-growth algorithm.
For a more concrete example on how to solve the subproblems, consider the task of finding frequent itemsets ending with e.
1. The first step is to gather all the paths containing node e. These initial paths are called prefix paths and are shown in Figure 5.27(a).
Figure 5.27. Example of applying the FP-growth algorithm to find frequent itemsets ending in e.
Figure 5.27. Full Alternative Text
2. From the prefix paths shown in Figure 5.27(a), the support count for e is obtained by adding the support counts associated with node e. Assuming that the minimum support count is 2, {e} is declared a frequent itemset because its support count is 3.
3. Because {e} is frequent, the algorithm has to solve the subproblems of finding frequent itemsets ending in de, ce, be,and ae. Before solving these subproblems, it must first convert the prefix paths into a conditional FP-tree, which is structurally similar to an FP-tree, except it is used to find frequent itemsets ending with a particular suffix. A conditional FP-tree is obtained in the following way:
1. First, the support counts along the prefix paths must be updated because some of the counts include transactions that do not contain item e. For example, the rightmost path shown in Figure 5.27(a), null → b:2 → c:2 → e:1, includes a transaction {b, c} that does not contain item e. The counts along the prefix path must therefore be adjusted to 1 to reflect the actual number of transactions containing {b, c, e}.
2. The prefix paths are truncated by removing the nodes for e. These nodes can be removed because the support counts along the prefix paths have been updated to reflect only transactions that contain e and the subproblems of finding frequent itemsets ending in de, ce, be, and ae no longer need information about node e.
3. After updating the support counts along the prefix paths, some of the items may no longer be frequent. For example, the node b appears only once and has a support count equal to 1, which means that there is only one transaction that contains both b and e.Item b
can be safely ignored from subsequent analysis because all itemsets ending in be must be infrequent.
The conditional FP-tree for e is shown in Figure 5.27(b). The tree looks different than the original prefix paths because the frequency counts have been updated and the nodes b and e have been eliminated.
4. FP-growth uses the conditional FP-tree for e to solve the subproblems of finding frequent itemsets ending in de, ce,and ae. To find the frequent itemsets ending in de, the prefix paths for d are gathered from the conditional FP-tree for e (Figure 5.27(c)). By adding the frequency counts associated with node d, we obtain the support count for {d, e}. Since the support count is equal to 2, {d, e} is declared a frequent itemset. Next, the algorithm constructs the conditional FP-tree for de using the approach described in step 3. After updating the support counts and removing the infrequent item c, the conditional FP-tree for de is shown in Figure 5.27(d). Since the conditional FP-tree contains only one item, a, whose support is equal to minsup, the algorithm extracts the frequent itemset {a, d, e} and moves on to the next subproblem, which is to generate frequent itemsets ending in ce. After processing the prefix paths for c, {c, e} is found to be frequent. However, the conditional FP-tree for ce will have no frequent items and thus will be eliminated. The algorithm proceeds to solve the next subproblem and finds {a, e} to be the only frequent itemset remaining.
This example illustrates the divide-and-conquer approach used in the FP- growth algorithm. At each recursive step, a conditional FP-tree is constructed by updating the frequency counts along the prefix paths and removing all infrequent items. Because the subproblems are disjoint, FP-growth will not generate any duplicate itemsets. In addition, the counts associated with the nodes allow the algorithm to perform support counting while generating the common suffix itemsets.
FP-growth is an interesting algorithm because it illustrates how a compact representation of the transaction data set helps to efficiently generate frequent itemsets. In addition, for certain transaction data sets, FP-growth outperforms the standard Apriori algorithm by several orders of magnitude. The run-time performance of FP-growth depends on the compaction factor of the data set.
If the resulting conditional FP-trees are very bushy (in the worst case, a full prefix tree), then the performance of the algorithm degrades significantly because it has to generate a large number of subproblems and merge the results returned by each subproblem.
5.7 Evaluation of Association Patterns Although the Apriori principle significantly reduces the exponential search space of candidate itemsets, association analysis algorithms still have the potential to generate a large number of patterns. For example, although the data set shown in Table 5.1 contains only six items, it can produce hundreds of association rules at particular support and confidence thresholds. As the size and dimensionality of real commercial databases can be very large, we can easily end up with thousands or even millions of patterns, many of which might not be interesting. Identifying the most interesting patterns from the multitude of all possible ones is not a trivial task because “one person's trash might be another person's treasure.” It is therefore important to establish a set of well-accepted criteria for evaluating the quality of association patterns.
The first set of criteria can be established through a data-driven approach to define objective interestingness measures. These measures can be used to rank patterns—itemsets or rules—and thus provide a straightforward way of dealing with the enormous number of patterns that are found in a data set. Some of the measures can also provide statistical information, e.g., itemsets that involve a set of unrelated items or cover very few transactions are considered uninteresting because they may capture spurious relationships in the data and should be eliminated. Examples of objective interestingness measures include support, confidence, and correlation.
The second set of criteria can be established through subjective arguments. A pattern is considered subjectively uninteresting unless it reveals unexpected information about the data or provides useful knowledge that can lead to profitable actions. For example, the rule {Butter}→{Bread} may not be interesting, despite having high support and confidence values, because the relationship represented by the rule might seem rather obvious. On the other hand, the rule {Diapers}→{Beer} is interesting because the relationship is quite unexpected and may suggest a new cross-selling opportunity for retailers. Incorporating subjective knowledge into pattern evaluation is a
difficult task because it requires a considerable amount of prior information from domain experts. Readers interested in subjective interestingness measures may refer to resources listed in the bibliography at the end of this chapter.
5.7.1 Objective Measures of Interestingness An objective measure is a data-driven approach for evaluating the quality of association patterns. It is domain-independent and requires only that the user specifies a threshold for filtering low-quality patterns. An objective measure is usually computed based on the frequency counts tabulated in a contingency table. Table 5.6 shows an example of a contingency table for a pair of binary variables, A and B.We use the notation A¯(B¯) to indicate that A (B)isabsent from a transaction. Each entry fij in this 2×2 table denotes a frequency count. For example, f11 is the number of times A and B appear together in the same transaction, while f01 is the number of transactions that contain B but not A. The row sum f1+ represents the support count for A, while the column sum f+1 represents the support count for B. Finally, even though our discussion focuses mainly on asymmetric binary variables, note that contingency tables are also applicable to other attribute types such as symmetric binary, nominal, and ordinal variables.
Table 5.6. A 2-way contingency table for variables A and B.
B B¯ A f11 f10 f1+
A¯ f01 f00 f0+ f+1 f+0 N
Limitations of the Support- Confidence Framework The classical association rule mining formulation relies on the support and confidence measures to eliminate uninteresting patterns. The drawback of support, which is described more fully in Section 5.8, is that many potentially interesting patterns involving low support items might be eliminated by the support threshold. The drawback of confidence is more subtle and is best demonstrated with the following example.
Example 5.3. Suppose we are interested in analyzing the relationship between people who drink tea and coffee. We may gather information about the beverage preferences among a group of people and summarize their responses into a contingency table such as the one shown in Table 5.7.
Table 5.7. Beverage preferences among a group of 1000 people.
Coffee Coffee¯ Tea 150 50 200 Tea¯ 650 150 800
800 200 1000
The information given in this table can be used to evaluate the association rule {Tea}→{Coffee}. At first glance, it may appear that people who drink tea also tend to drink coffee because the rule's support (15%) and confidence (75%) values are reasonably high. This argument would have been acceptable except that the fraction of people who drink coffee, regardless of whether
they drink tea, is 80%, while the fraction of tea drinkers who drink coffee is only 75%. Thus knowing that a person is a tea drinker actually decreases her probability of being a coffee drinker from 80% to 75%! The rule {Tea} →{Coffee} is therefore misleading despite its high confidence value.
Now consider a similar problem where we are interested in analyzing the relationship between people who drink tea and people who use honey in their beverage. Table 5.8 summarizes the information gathered over the same group of people about their preferences for drinking tea and using honey. If we evaluate the association rule {Tea}→{Honey} using this information, we will find that the confidence value of this rule is merely 50%, which might be easily rejected using a reasonable threshold on the confidence value, say 70%. One thus might consider that the preference of a person for drinking tea has no influence on her preference for using honey. However, the fraction of people who use honey, regardless of whether they drink tea, is only 12%. Hence, knowing that a person drinks tea significantly increases her probability of using honey from 12% to 50%. Further, the fraction of people who do not drink tea but use honey is only 2.5%! This suggests that there is definitely some information in the preference of a person of using honey given that she drinks tea. The rule {Tea}→{Honey} may therefore be falsely rejected if confidence is used as the evaluation measure.
Table 5.8. Information about people who drink tea and people who use honey in their beverage.
Honey Honey¯ Tea 100 100 200 Tea¯ 20 780 800
120 880 1000
Note that if we take the support of coffee drinkers into account, we would not be surprised to find that many of the people who drink tea also drink coffee, since the overall number of coffee drinkers is quite large by itself. What is more surprising is that the fraction of tea drinkers who drink coffee is actually less than the overall fraction of people who drink coffee, which points to an inverse relationship between tea drinkers and coffee drinkers. Similarly, if we account for the fact that the support of using honey is inherently small, it is easy to understand that the fraction of tea drinkers who use honey will naturally be small. Instead, what is important to measure is the change in the fraction of honey users, given the information that they drink tea.
The limitations of the confidence measure are well-known and can be understood from a statistical perspective as follows. The support of a variable measures the probability of its occurrence, while the support s(A, B) of a pair of a variables A and B measures the probability of the two variables occurring together. Hence, the joint probability P (A, B) can be written as
P(A, B)=s(A, B)=f11N.
If we assume A and B are statistically independent, i.e. there is no relationship between the occurrences of A and B, then P(A, B)=P(A)×P(B). Hence, under the assumption of statistical independence between A and B, the support sindep(A, B) of A and B can be written as
sindep(A, B)=s(A)×s(B)or equivalently,sindep(A, B)=f1+N×f+1N. (5.4)
If the support between two variables, s(A, B) is equal to sindep(A, B), then A and B can be considered to be unrelated to each other. However, if s(A, B) is widely different from sindep(A, B), then A and B are most likely dependent. Hence, any deviation of s(A, B) from s(A)×s(B) can be seen as an indication of a statistical relationship between A and B. Since the confidence measure only considers the deviance of s(A, B) from s(A) and not from s(A)×s(B), it fails to account for the support of the consequent, namely s(B). This results in the detection of spurious patterns (e.g., {Tea}→{Coffee}) and the rejection of truly interesting patterns (e.g., {Tea}→{Honey}), as illustrated in the previous example.
Various objective measures have been used to capture the deviance of s(A, B) from sindep(A, B), that are not susceptible to the limitations of the confidence measure. Below, we provide a brief description of some of these measures and discuss some of their properties.
Interest Factor The interest factor, which is also called as the “lift,” can be defined as follows:
I(A, B)=s(A, B)s(A)×s(B)=Nf11f1+f+1. (5.5)
Notice that s(A)×s(B)=sindep(A, B). Hence, the interest factor measures the ratio of the support of a pattern s(A, B) against its baseline support sindep(A, B) computed under the statistical independence assumption. Using Equations 5.5 and 5.4, we can interpret the measure as follows:
I(A, B)={=1,if A and B are independent;>1,if A and B are positively related; <1,if A and B are negatively related. (5.6)
For the tea-coffee example shown in Table 5.7,I=0.150.2×0.8=0.9375, thus suggesting a slight negative relationship between tea drinkers and coffee drinkers. Also, for the tea-honey example shown in Table 5.8, I=0.10.12×0.2=4.1667, suggesting a strong positive relationship between people who drink tea and people who use honey in their beverage. We can thus see that the interest factor is able to detect meaningful patterns in the tea- coffee and tea-honey examples. Indeed, the interest factor has a number of statistical advantages over the confidence measure that make it a suitable measure for analyzing statistical independence between variables.
Piatesky-Shapiro (PS) Measure Instead of computing the ratio between s(A, B) and sindep(A, B)=s(A)×s(B), the PS measure considers the difference between s(A, B) and s(A)×s(B) as follows.
PS=s(A, B)−s(A)×s(B)=f11N−f1+f+1N2 (5.7)
The PS value is 0 when A and B are mutually independent of each other. Otherwise, PS>0 when there is a positive relationship between the two variables, and PS<0 when there is a negative relationship.
Correlation Analysis Correlation analysis is one of the most popular techniques for analyzing relationships between a pair of variables. For continuous variables, correlation is defined using Pearson's correlation coefficient (see Equation 2.10 on page 83). For binary variables, correlation can be measured using the ϕ-co
