Datamining assignment

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Data Mining: Data

Lecture Notes for Chapter 2

Introduction to Data Mining

Tan, Steinbach, Kumar

What is Data?

Collection of data objects and their attributes

An attribute is a property or characteristic of an object
Examples: eye color of a person, temperature, etc.
Attribute is also known as variable, field, characteristic, or feature
A collection of attributes describe an object
Object is also known as record, point, case, sample, entity, or instance

Attributes

Objects

Attribute Values

Attribute values are numbers or symbols assigned to an attribute

Distinction between attributes and attribute values
Same attribute can be mapped to different attribute values
Example: height can be measured in feet or meters

Different attributes can be mapped to the same set of values
Example: Attribute values for ID and age are integers
But properties of attribute values can be different

ID has no limit but age has a maximum and minimum value

Measurement of Length

The way you measure an attribute is somewhat may not match the attributes properties.

Types of Attributes

There are different types of attributes
Nominal
Examples: ID numbers, eye color, zip codes
Ordinal
Examples: rankings (e.g., taste of potato chips on a scale from 1-10), grades, height in {tall, medium, short}
Interval
Examples: calendar dates, temperatures in Celsius or Fahrenheit.
Ratio
Examples: temperature in Kelvin, length, time, counts

Properties of Attribute Values

The type of an attribute depends on which of the following properties it possesses:
Distinctness: = 
Order: < >
Addition: + -
Multiplication: * /

Nominal attribute: distinctness
Ordinal attribute: distinctness & order
Interval attribute: distinctness, order & addition
Ratio attribute: all 4 properties

Attribute Type

Description

Examples

Operations

Nominal

The values of a nominal attribute are just different names, i.e., nominal attributes provide only enough information to distinguish one object from another. (=, )

zip codes, employee ID numbers, eye color, sex: {male, female}

mode, entropy, contingency correlation, 2 test

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

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

Attribute Level

Transformation

Comments

Nominal

Any permutation of values

If all employee ID numbers were reassigned, would it make any difference?

Ordinal

An order preserving change of values, i.e.,
new_value = f(old_value)
where f is a monotonic function.

An attribute encompassing the notion of good, better best can be represented equally well by the values {1, 2, 3} or by { 0.5, 1, 10}.

Interval

new_value =a * old_value + b where a and b are constants

Thus, the Fahrenheit and Celsius temperature scales differ in terms of where their zero value is and the size of a unit (degree).

Ratio

new_value = a * old_value

Length can be measured in meters or feet.

Discrete and Continuous Attributes

Discrete Attribute
Has only a finite or countably infinite set of values
Examples: zip codes, counts, or the set of words in a collection of documents
Often represented as integer variables.
Note: binary attributes are a special case of discrete attributes

Continuous Attribute
Has real numbers as attribute values
Examples: temperature, height, or weight.
Practically, real values can only be measured and represented using a finite number of digits.
Continuous attributes are typically represented as floating-point variables.

Types of data sets

Record
Data Matrix
Document Data
Transaction Data
Graph
World Wide Web
Molecular Structures
Ordered
Spatial Data
Temporal Data
Sequential Data
Genetic Sequence Data

Important Characteristics of Structured Data

Dimensionality
Curse of Dimensionality

Sparsity
Only presence counts

Resolution
Patterns depend on the scale

Record Data

Data that consists of a collection of records, each of which consists of a fixed set of attributes

Tid	Refund	Marital Status	Taxable Income	Cheat
1	Yes	Single	125K	No
2	No	Married	100K	No
3	No	Single	70K	No
4	Yes	Married	120K	No
5	No	Divorced	95K	Yes
6	No	Married	60K	No
7	Yes	Divorced	220K	No
8	No	Single	85K	Yes
9	No	Married	75K	No
10	No	Single	90K	Yes

Data Matrix

If data objects have the same fixed set of numeric attributes, then the data objects can be thought of as points in a multi-dimensional space, where each dimension represents a distinct attribute

Such data set can be represented by an m by n matrix, where there are m rows, one for each object, and n columns, one for each attribute

Document Data

Each document becomes a `term' vector,
each term is a component (attribute) of the vector,
the value of each component is the number of times the corresponding term occurs in the document.

Document 1�

season�

timeout�

lost�

win�

game�

score�

ball�

play�

coach�

team�

Document 2�

Document 3�

3�

0�

5�

0�

2�

6�

0�

2�

0�

2�

0�

7�

0�

2�

1�

0�

3�

0�

1�

0�

1�

2�

0�

3�

0�

Transaction Data

A special type of record data, where
each record (transaction) involves a set of items.
For example, consider a grocery store. The set of products purchased by a customer during one shopping trip constitute a transaction, while the individual products that were purchased are the items.

TID	Items
1	Bread, Coke, Milk
2	Beer, Bread
3	Beer, Coke, Diaper, Milk
4	Beer, Bread, Diaper, Milk
5	Coke, Diaper, Milk

Graph Data

Examples: Generic graph and HTML Links

Chemical Data

Benzene Molecule: C6H6

Ordered Data

Sequences of transactions

An element of the sequence

Items/Events

Ordered Data

Genomic sequence data

Ordered Data

Spatio-Temporal Data

Average Monthly Temperature of land and ocean

Data Quality

What kinds of data quality problems?
How can we detect problems with the data?
What can we do about these problems?

Examples of data quality problems:
Noise and outliers
missing values
duplicate data

Noise

Noise refers to modification of original values
Examples: distortion of a person’s voice when talking on a poor phone and “snow” on television screen

Two Sine Waves

Two Sine Waves + Noise

Outliers

Outliers are data objects with characteristics that are considerably different than most of the other data objects in the data set

Missing Values

Reasons for missing values
Information is not collected
(e.g., people decline to give their age and weight)
Attributes may not be applicable to all cases
(e.g., annual income is not applicable to children)

Handling missing values
Eliminate Data Objects
Estimate Missing Values
Ignore the Missing Value During Analysis
Replace with all possible values (weighted by their probabilities)

Duplicate Data

Data set may include data objects that are duplicates, or almost duplicates of one another
Major issue when merging data from heterogeous sources

Examples:
Same person with multiple email addresses

Data cleaning
Process of dealing with duplicate data issues

Data Preprocessing

Aggregation
Sampling
Dimensionality Reduction
Feature subset selection
Feature creation
Discretization and Binarization
Attribute Transformation

Aggregation

Combining two or more attributes (or objects) into a single attribute (or object)

Purpose
Data reduction
Reduce the number of attributes or objects
Change of scale
Cities aggregated into regions, states, countries, etc
More “stable” data
Aggregated data tends to have less variability

Aggregation

Standard Deviation of Average Monthly Precipitation

Standard Deviation of Average Yearly Precipitation

Variation of Precipitation in Australia

Sampling

Sampling is the main technique employed for data selection.
It is often used for both the preliminary investigation of the data and the final data analysis.

Statisticians sample because obtaining the entire set of data of interest is too expensive or time consuming.

Sampling is used in data mining because processing the entire set of data of interest is too expensive or time consuming.

Sampling …

The key principle for effective sampling is the following:
using a sample will work almost as well as using the entire data sets, if the sample is representative
A sample is representative if it has approximately the same property (of interest) as the original set of data

Types of Sampling

Simple Random Sampling
There is an equal probability of selecting any particular item

Sampling without replacement
As each item is selected, it is removed from the population

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 up more than once

Stratified sampling
Split the data into several partitions; then draw random samples from each partition

Sample Size

8000 points 2000 Points 500 Points

Sample Size

What sample size is necessary to get at least one object from each of 10 groups.

Curse of Dimensionality

When dimensionality increases, data becomes increasingly sparse in the space that it occupies

Definitions of density and distance between points, which is critical for clustering and outlier detection, become less meaningful

Randomly generate 500 points
Compute difference between max and min distance between any pair of points

Dimensionality Reduction

Purpose:
Avoid curse of dimensionality
Reduce amount of time and memory required by data mining algorithms
Allow data to be more easily visualized
May help to eliminate irrelevant features or reduce noise

Techniques
Principle Component Analysis
Singular Value Decomposition
Others: supervised and non-linear techniques

Dimensionality Reduction: PCA

Goal is to find a projection that captures the largest amount of variation in data

Dimensionality Reduction: PCA

Find the eigenvectors of the covariance matrix
The eigenvectors define the new space

Dimensionality Reduction: ISOMAP

Construct a neighbourhood graph
For each pair of points in the graph, compute the shortest path distances – geodesic distances

By: Tenenbaum, de Silva, Langford (2000)

Dimensionality Reduction: PCA

Feature Subset Selection

Another way to reduce dimensionality of data

Redundant features
duplicate much or all of the information contained in one or more other attributes
Example: purchase price of a product and the amount of sales tax paid

Irrelevant features
contain no information that is useful for the data mining task at hand
Example: students' ID is often irrelevant to the task of predicting students' GPA

Feature Subset Selection

Techniques:
Brute-force approch:
Try all possible feature subsets as input to data mining algorithm
Embedded approaches:
Feature selection occurs naturally as part of the data mining algorithm
Filter approaches:
Features are selected before data mining algorithm is run
Wrapper approaches:
Use the data mining algorithm as a black box to find best subset of attributes

Feature Creation

Create new attributes that can capture the important information in a data set much more efficiently than the original attributes

Three general methodologies:
Feature Extraction
domain-specific
Mapping Data to New Space
Feature Construction
combining features

Mapping Data to a New Space

Two Sine Waves

Two Sine Waves + Noise

Frequency

Fourier transform
Wavelet transform

Discretization Using Class Labels

Entropy based approach

3 categories for both x and y

5 categories for both x and y

Discretization Without Using Class Labels

Data

Equal interval width

Equal frequency

K-means

Attribute Transformation

A function that maps the entire set of values of a given attribute to a new set of replacement values such that each old value can be identified with one of the new values
Simple functions: xk, log(x), ex, |x|
Standardization and Normalization

Similarity and Dissimilarity

Similarity
Numerical measure of how alike two data objects are.
Is higher when objects are more alike.
Often falls in the range [0,1]
Dissimilarity
Numerical measure of how different are two data objects
Lower when objects are more alike
Minimum dissimilarity is often 0
Upper limit varies
Proximity refers to a similarity or dissimilarity

Similarity/Dissimilarity for Simple Attributes

p and q are the attribute values for two data objects.

Euclidean Distance

Euclidean Distance

Where n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q.

Standardization is necessary, if scales differ.

Euclidean Distance

Distance Matrix

Sheet1

point	x	y
	0	2
p2	2	0
p3	3	1
p4	5	1
			point	x	y
			p1	0	2
			p2	2	0
			p3	3	1
			p4	5	1

Sheet2

Sheet3

Sheet1

point	x	y
	0	2
p2	2	0
p3	3	1
p4	5	1
					point	x	y
					p1	0	2
					p2	2	0
					p3	3	1
					p4	5	1
			p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0

Sheet2

Sheet3

Minkowski Distance

Minkowski Distance is a generalization of Euclidean Distance

Where r is a parameter, n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q.

Minkowski Distance: Examples

r = 1. City block (Manhattan, taxicab, L1 norm) distance.
A common example of this is the Hamming distance, which is just the number of bits that are different between two binary vectors

r = 2. Euclidean distance

r  . “supremum” (Lmax norm, L norm) distance.
This is the maximum difference between any component of the vectors

Do not confuse r with n, i.e., all these distances are defined for all numbers of dimensions.

Minkowski Distance

Distance Matrix

Sheet1

point	x	y
	0	2
p2	2	0
p3	3	1
p4	5	1
					point	x	y
					p1	0	2
					p2	2	0
					p3	3	1
					p4	5	1
			p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
		point	x	y
		p1	0	2
		p2	2	0
		p3	3	1
		p4	5	1
		L1	p1	p2	p3	p4
		p1	0	4	4	6
		p2	4	0	2	4
		p3	4	2	0	2
		p4	6	4	2	0
		L2	p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
			p1	p2	p3	p4
		p1	0	2	3	5
		p2	2	0	1	3
		p3	3	1	0	2
		p4	5	3	2	0

Sheet2

Sheet3

Sheet1

point	x	y
	0	2
p2	2	0
p3	3	1
p4	5	1
					point	x	y
					p1	0	2
					p2	2	0
					p3	3	1
					p4	5	1
			p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
		point	x	y
		p1	0	2
		p2	2	0
		p3	3	1
		p4	5	1
		L1	p1	p2	p3	p4
		p1	0	4	4	6
		p2	4	0	2	4
		p3	4	2	0	2
		p4	6	4	2	0
		L2	p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
			p1	p2	p3	p4
		p1	0	2	3	5
		p2	2	0	1	3
		p3	3	1	0	2
		p4	5	3	2	0

Sheet2

Sheet3

Sheet1

point	x	y
	0	2
p2	2	0
p3	3	1
p4	5	1
					point	x	y
					p1	0	2
					p2	2	0
					p3	3	1
					p4	5	1
			p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
		point	x	y
		p1	0	2
		p2	2	0
		p3	3	1
		p4	5	1
		L1	p1	p2	p3	p4
		p1	0	4	4	6
		p2	4	0	2	4
		p3	4	2	0	2
		p4	6	4	2	0
		L2	p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
			p1	p2	p3	p4
		p1	0	2	3	5
		p2	2	0	1	3
		p3	3	1	0	2
		p4	5	3	2	0

Sheet2

Sheet3

Sheet1

point	x	y
	0	2
p2	2	0
p3	3	1
p4	5	1
					point	x	y
					p1	0	2
					p2	2	0
					p3	3	1
					p4	5	1
			p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
		point	x	y
		p1	0	2
		p2	2	0
		p3	3	1
		p4	5	1
		L1	p1	p2	p3	p4
		p1	0	4	4	6
		p2	4	0	2	4
		p3	4	2	0	2
		p4	6	4	2	0
		L2	p1	p2	p3	p4
		p1	0	2.828	3.162	5.099
		p2	2.828	0	1.414	3.162
		p3	3.162	1.414	0	2
		p4	5.099	3.162	2	0
			p1	p2	p3	p4
		p1	0	2	3	5
		p2	2	0	1	3
		p3	3	1	0	2
		p4	5	3	2	0

Sheet2

Sheet3

Mahalanobis Distance

For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6.

 is the covariance matrix of the input data X

Mahalanobis Distance

Covariance Matrix:

A: (0.5, 0.5)

B: (0, 1)

C: (1.5, 1.5)

Mahal(A,B) = 5

Mahal(A,C) = 4

Common Properties of a Distance

Distances, such as the Euclidean distance, have some well known properties.

d(p, q)  0 for all p and q and d(p, q) = 0 only if
p = q. (Positive definiteness)

d(p, q) = d(q, p) for all p and q. (Symmetry)

d(p, r)  d(p, q) + d(q, r) for all points p, q, and r.
(Triangle Inequality)

where d(p, q) is the distance (dissimilarity) between points (data objects), p and q.

A distance that satisfies these properties is a metric

Common Properties of a Similarity

Similarities, also have some well known properties.

s(p, q) = 1 (or maximum similarity) only if p = q.

s(p, q) = s(q, p) for all p and q. (Symmetry)

where s(p, q) is the similarity between points (data objects), p and q.

Similarity Between Binary Vectors

Common situation is that objects, p and q, have only binary attributes

Compute similarities using the following quantities

M01 = the number of attributes where p was 0 and q was 1

M10 = the number of attributes where p was 1 and q was 0

M00 = the number of attributes where p was 0 and q was 0

M11 = the number of attributes where p was 1 and q was 1

Simple Matching and Jaccard Coefficients

SMC = number of matches / number of attributes

= (M11 + M00) / (M01 + M10 + M11 + M00)

J = number of 11 matches / number of not-both-zero attributes values

= (M11) / (M01 + M10 + M11)

SMC versus Jaccard: Example

p = 1 0 0 0 0 0 0 0 0 0

q = 0 0 0 0 0 0 1 0 0 1

M01 = 2 (the number of attributes where p was 0 and q was 1)

M10 = 1 (the number of attributes where p was 1 and q was 0)

M00 = 7 (the number of attributes where p was 0 and q was 0)

M11 = 0 (the number of attributes where p was 1 and q was 1)

SMC = (M11 + M00)/(M01 + M10 + M11 + M00) = (0+7) / (2+1+0+7) = 0.7

J = (M11) / (M01 + M10 + M11) = 0 / (2 + 1 + 0) = 0

Cosine Similarity

If d1 and d2 are two document vectors, then

cos( d1, d2 ) = (d1  d2) / ||d1|| ||d2|| ,

where  indicates vector dot product and || d || is the length of vector d.

Example:

d1 = 3 2 0 5 0 0 0 2 0 0

d2 = 1 0 0 0 0 0 0 1 0 2

d1  d2= 3*1 + 2*0 + 0*0 + 5*0 + 0*0 + 0*0 + 0*0 + 2*1 + 0*0 + 0*2 = 5

||d1|| = (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5 = (42) 0.5 = 6.481

||d2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2) 0.5 = (6) 0.5 = 2.245

cos( d1, d2 ) = .3150

Extended Jaccard Coefficient (Tanimoto)

Variation of Jaccard for continuous or count attributes
Reduces to Jaccard for binary attributes

Correlation

Correlation measures the linear relationship between objects
To compute correlation, we standardize data objects, p and q, and then take their dot product

Visually Evaluating Correlation

Scatter plots showing the similarity from –1 to 1.

General Approach for Combining Similarities

Sometimes attributes are of many different types, but an overall similarity is needed.

Using Weights to Combine Similarities

May not want to treat all attributes the same.
Use weights wk which are between 0 and 1 and sum to 1.

Density

Density-based clustering require a notion of density

Examples:
Euclidean density
Euclidean density = number of points per unit volume

Probability density

Graph-based density

Euclidean Density – Cell-based

Simplest approach is to divide region into a number of rectangular cells of equal volume and define density as # of points the cell contains

Euclidean Density – Center-based

Euclidean density is the number of points within a specified radius of the point

Tid

Refund

Marital

Status

Taxable

Income

Cheat

Yes

Single

125K

Married

100K

Single

70K

Yes

Married

120K

Divorced

95K

Yes

Married

60K

Yes

Divorced

220K

Single

85K

Yes

Married

75K

Singl

90K

Yes

1.1

2.2

16.22

6.25

12.65

1.2

2.7

15.22

5.27

10.23

Thickness

Load

Distance

Projection

of y load

Projection

of x Load

1.1

2.2

16.22

6.25

12.65

1.2

2.7

15.22

5.27

10.23

Thickness

Load

Distance

Projection

of y load

Projection

of x Load

Document 1

Document 2

Document 3

3050260202

702100300

100122030

TID

Items

Bread, Coke, Milk

Beer, Bread

Beer, Coke, Diaper, Milk

Beer, Bread, Diaper, Milk

Coke, Diaper, Milk

Data Mining </a>

<li>

Graph Partitioning </a>

<li>

Parallel Solution of Sparse Linear System of Equations </a>

<li>

N-Body Computation and Dense Linear System Solvers

GGTTCCGCCTTCAGCCCCGCGCC

CGCAGGGCCCGCCCCGCGCCGTC

GAGAAGGGCCCGCCTGGCGGGCG

GGGGGAGGCGGGGCCGCCCGAGC

CCAACCGAGTCCGACCAGGTGCC

CCCTCTGCTCGGCCTAGACCTGA

GCTCATTAGGCGGCAGCGGACAG

GCCAAGTAGAACACGCGAAGCGC

TGGGCTGCCTGCTGCGACCAGGG

Dimensions = 10

Dimensions = 40

Dimensions = 80

Dimensions = 120

Dimensions = 160

Dimensions = 206

dist

)

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pointxy

p102

p220

p331

p451

p1p2p3p4

p102.8283.1625.099

p22.82801.4143.162

p33.1621.41402

p45.0993.16220

dist

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pointxy

p102

p220

p331

p451

L1p1p2p3p4

p10446

p24024

p34202

p46420

L2p1p2p3p4

p102.8283.1625.099

p22.82801.4143.162

p33.1621.41402

p45.0993.16220

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p1p2p3p4

p10235

p22013

p33102

p45320

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