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Chapter 4: Clustering

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What is Cluster Analysis?

Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups

Inter-cluster distances are maximized

Intra-cluster distances are minimized

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Applications of Cluster Analysis

Understanding

Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations

Summarization

Reduce the size of large data sets

Clustering precipitation in Australia

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What is not Cluster Analysis?

Simple segmentation

Dividing students into different registration groups alphabetically, by last name

Results of a query

Groupings are a result of an external specification

Clustering is a grouping of objects based on the data

Supervised classification

Have class label information

Association Analysis

Local vs. global connections

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Notion of a Cluster can be Ambiguous

How many clusters?

Four Clusters

Two Clusters

Six Clusters

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Types of Clusterings

A clustering is a set of clusters

Important distinction between hierarchical and partitional sets of clusters

Partitional Clustering

A division of data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset

Hierarchical clustering

A set of nested clusters organized as a hierarchical tree

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Partitional Clustering

Original Points

A Partitional Clustering

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Hierarchical Clustering

Traditional Hierarchical Clustering

Non-traditional Hierarchical Clustering

Non-traditional Dendrogram

Traditional Dendrogram

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Types of Clusters

Well-separated clusters

Center-based clusters

Contiguous clusters

Density-based clusters

Property or Conceptual

Described by an Objective Function

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Types of Clusters: Well-Separated

Well-Separated Clusters:

A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster.

3 well-separated clusters

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Types of Clusters: Center-Based

Center-based

A cluster is a set of objects such that an object in a cluster is closer to the “center” of a cluster, than to the center of any other cluster

The center of a cluster is a centroid.

The average of all the points in the cluster is a medoid

4 center-based clusters

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Types of Clusters: Contiguity-Based

Contiguous Cluster (Nearest neighbor or Transitive)

A point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster.

8 contiguous clusters

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Types of Clusters: Density-Based

Density-based

A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density.

Used when the clusters are irregular or intertwined, and when noise and outliers are present.

6 density-based clusters

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Types of Clusters: Conceptual Clusters

Shared Property or Conceptual Clusters

Finds clusters that share some common property or represent a particular concept.

2 Overlapping Circles

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Types of Clusters: Objective Function

Clusters Defined by an Objective Function

Finds clusters that minimize or maximize an objective function.

Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard)

Can have global or local objectives.

Hierarchical clustering algorithms typically have local objectives

Partitional algorithms typically have global objectives

A variation of the global objective function approach is to fit the data to a parameterized model.

Parameters for the model are determined from the data.

Mixture models assume that the data is a ‘mixture' of a number of statistical distributions.

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Characteristics of the Input Data Are Important

Type of proximity or density measure

Central to clustering

Depends on data and application

Data characteristics that affect proximity and/or density are

Dimensionality

Sparseness

Attribute type

Special relationships in the data

For example, autocorrelation

Distribution of the data

Noise and Outliers

Often interfere with the operation of the clustering algorithm

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Clustering Algorithms

K-means and its variants

Hierarchical clustering

Density-based clustering

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K-means Clustering

Partitional clustering approach

Number of clusters, K, must be specified

Each cluster is associated with a centroid (center point)

Each point is assigned to the cluster with the closest centroid

The basic algorithm is very simple

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Example of K-means Clustering

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Example of K-means Clustering

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K-means Clustering – Details

Initial centroids are often chosen randomly.

Clusters produced vary from one run to another.

The centroid is (typically) the mean of the points in the cluster.

‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc.

K-means will converge for common similarity measures mentioned above.

Most of the convergence happens in the first few iterations.

Often the stopping condition is changed to ‘Until relatively few points change clusters’

Complexity is O( n * K * I * d )

n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

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Evaluating K-means Clusters

Most common measure is Sum of Squared Error (SSE)

For each point, the error is the distance to the nearest cluster

To get SSE, we square these errors and sum them.

x is a data point in cluster Ci and mi is the representative point for cluster Ci

can show that mi corresponds to the center (mean) of the cluster

Given two sets of clusters, we prefer the one with the smallest error

One easy way to reduce SSE is to increase K, the number of clusters

A good clustering with smaller K can have a lower SSE than a poor clustering with higher K

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Two different K-means Clusterings

Sub-optimal Clustering

Optimal Clustering

Original Points

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Limitations of K-means

K-means has problems when clusters are of differing

Sizes

Densities

Non-globular shapes

K-means has problems when the data contains outliers.

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Limitations of K-means: Differing Sizes

Original Points

K-means (3 Clusters)

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Limitations of K-means: Differing Density

Original Points

K-means (3 Clusters)

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Limitations of K-means: Non-globular Shapes

Original Points

K-means (2 Clusters)

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Overcoming K-means Limitations

Original Points K-means Clusters

One solution is to use many clusters.

Find parts of clusters, but need to put together.

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Overcoming K-means Limitations

Original Points K-means Clusters

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Overcoming K-means Limitations

Original Points K-means Clusters

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids …

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Importance of Choosing Initial Centroids …

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Problems with Selecting Initial Points

If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small.

Chance is relatively small when K is large

If clusters are the same size, n, then

For example, if K = 10, then probability = 10!/1010 = 0.00036

Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t

Consider an example of five pairs of clusters

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10 Clusters Example

Starting with two initial centroids in one cluster of each pair of clusters

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10 Clusters Example

Starting with two initial centroids in one cluster of each pair of clusters

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Solutions to Initial Centroids Problem

Multiple runs

Helps, but probability is not on your side

Sample and use hierarchical clustering to determine initial centroids

Select more than k initial centroids and then select among these initial centroids

Select most widely separated

Generate a larger number of clusters and then perform a hierarchical clustering

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Hierarchical Clustering

Produces a set of nested clusters organized as a hierarchical tree

Can be visualized as a dendrogram

A tree like diagram that records the sequences of merges or splits

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Strengths of Hierarchical Clustering

Do not have to assume any particular number of clusters

Any desired number of clusters can be obtained by ‘cutting’ the dendrogram at the proper level

They may correspond to meaningful taxonomies

Example in biological sciences (e.g., animal kingdom)

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Hierarchical Clustering

Two main types of hierarchical clustering

Agglomerative:

Start with the points as individual clusters

At each step, merge the closest pair of clusters until only one cluster (or k clusters) left

Divisive:

Start with one, all-inclusive cluster

At each step, split a cluster until each cluster contains an individual point (or there are k clusters)

Traditional hierarchical algorithms use a similarity or distance matrix

Merge or split one cluster at a time

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Agglomerative Clustering Algorithm

Most popular hierarchical clustering technique

Basic algorithm is straightforward

Compute the proximity matrix

Let each data point be a cluster

Repeat

Merge the two closest clusters

Update the proximity matrix

Until only a single cluster remains

Key operation is the computation of the proximity of two clusters

Different approaches to defining the distance between clusters distinguish the different algorithms

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Proximity Matrix

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Starting Situation

Start with clusters of individual points and a proximity matrix

. . .

Proximity Matrix

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Intermediate Situation

After some merging steps, we have some clusters

Proximity Matrix

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Intermediate Situation

We want to merge the two closest clusters (C2 and C5) and update the proximity matrix.

Proximity Matrix

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After Merging

The question is “How do we update the proximity matrix?”

C2 U C5

? ? ? ?

C2 U C5

Proximity Matrix

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How to Define Inter-Cluster Distance

. . .

Similarity?

MIN

MAX

Group Average

Distance Between Centroids

Other methods driven by an objective function

Ward’s Method uses squared error

Proximity Matrix

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How to Define Inter-Cluster Similarity

. . .

Proximity Matrix

MIN

MAX

Group Average

Distance Between Centroids

Other methods driven by an objective function

Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity

. . .

Proximity Matrix

MIN

MAX

Group Average

Distance Between Centroids

Other methods driven by an objective function

Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity

. . .

Proximity Matrix

MIN

MAX

Group Average

Distance Between Centroids

Other methods driven by an objective function

Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity

. . .

Proximity Matrix

MIN

MAX

Group Average

Distance Between Centroids

Other methods driven by an objective function

Ward’s Method uses squared error



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MIN or Single Link

Proximity of two clusters is based on the two closest points in the different clusters

Determined by one pair of points, i.e., by one link in the proximity graph

Example:

Distance Matrix:

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Hierarchical Clustering: MIN

Nested Clusters

Dendrogram

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Strength of MIN

Original Points

Six Clusters

Can handle non-elliptical shapes

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Limitations of MIN

Original Points

Two Clusters

Sensitive to noise and outliers

Three Clusters

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MAX or Complete Linkage

Proximity of two clusters is based on the two most distant points in the different clusters

Determined by all pairs of points in the two clusters

Distance Matrix:

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Hierarchical Clustering: MAX

Nested Clusters

Dendrogram

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Strength of MAX

Original Points

Two Clusters

Less susceptible to noise and outliers

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Limitations of MAX

Original Points

Two Clusters

Tends to break large clusters

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Group Average

Proximity of two clusters is the average of pairwise proximity between points in the two clusters.

Need to use average connectivity for scalability since total proximity favors large clusters

Distance Matrix:

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Hierarchical Clustering: Group Average

Nested Clusters

Dendrogram

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Hierarchical Clustering: Group Average

Compromise between Single and Complete Link

Strengths

Less susceptible to noise and outliers

Limitations

Biased towards globular clusters

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Cluster Similarity: Ward’s Method

Similarity of two clusters is based on the increase in squared error when two clusters are merged

Similar to group average if distance between points is distance squared

Less susceptible to noise and outliers

Biased towards globular clusters

Hierarchical analogue of K-means

Can be used to initialize K-means

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Hierarchical Clustering: Comparison

Group Average

Ward’s Method

MIN

MAX

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Hierarchical Clustering: Time and Space requirements

O(N2) space since it uses the proximity matrix.

N is the number of points.

O(N3) time in many cases

There are N steps and at each step the size, N2, proximity matrix must be updated and searched

Complexity can be reduced to O(N2 log(N) ) time with some cleverness

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Hierarchical Clustering: Problems and Limitations

Once a decision is made to combine two clusters, it cannot be undone

No global objective function is directly minimized

Different schemes have problems with one or more of the following:

Sensitivity to noise and outliers

Difficulty handling clusters of different sizes and non-globular shapes

Breaking large clusters

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Cluster Validity

For supervised classification we have a variety of measures to evaluate how good our model is

Accuracy, precision, recall

For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters?

But “clusters are in the eye of the beholder”!

Then why do we want to evaluate them?

To avoid finding patterns in noise

To compare clustering algorithms

To compare two sets of clusters

To compare two clusters

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Clusters found in Random Data

Random Points

K-means

Complete Link

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Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data.

Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels.

Evaluating how well the results of a cluster analysis fit the data without reference to external information.

- Use only the data

Comparing the results of two different sets of cluster analyses to determine which is better.

Determining the ‘correct’ number of clusters.

For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.

Different Aspects of Cluster Validation

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Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types.

External Index: Used to measure the extent to which cluster labels match externally supplied class labels.

Entropy

Internal Index: Used to measure the goodness of a clustering structure without respect to external information.

Sum of Squared Error (SSE)

Relative Index: Used to compare two different clusterings or clusters.

Often an external or internal index is used for this function, e.g., SSE or entropy

Sometimes these are referred to as criteria instead of indices

However, sometimes criterion is the general strategy and index is the numerical measure that implements the criterion.

Measures of Cluster Validity

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Two matrices

Proximity Matrix

Ideal Similarity Matrix

One row and one column for each data point

An entry is 1 if the associated pair of points belong to the same cluster

An entry is 0 if the associated pair of points belongs to different clusters

Compute the correlation between the two matrices

Since the matrices are symmetric, only the correlation between n(n-1) / 2 entries needs to be calculated.

High correlation indicates that points that belong to the same cluster are close to each other.

Not a good measure for some density or contiguity based clusters.

Measuring Cluster Validity Via Correlation

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Measuring Cluster Validity Via Correlation

Correlation of ideal similarity and proximity matrices for the K-means clusterings of the following two data sets.

Corr = -0.9235

Corr = -0.5810

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Order the similarity matrix with respect to cluster labels and inspect visually.

Using Similarity Matrix for Cluster Validation

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Using Similarity Matrix for Cluster Validation

Clusters in random data are not so crisp

K-means

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Using Similarity Matrix for Cluster Validation

Clusters in random data are not so crisp

Complete Link

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Clusters in more complicated figures aren’t well separated

Internal Index: Used to measure the goodness of a clustering structure without respect to external information

SSE

SSE is good for comparing two clusterings or two clusters (average SSE).

Can also be used to estimate the number of clusters

Internal Measures: SSE

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Internal Measures: SSE

SSE curve for a more complicated data set

SSE of clusters found using K-means

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Correlation of ideal similarity and proximity matrices for the K-means clusterings of the following two data sets.

Statistical Framework for Correlation

Corr = -0.9235

Corr = -0.5810

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Cluster Cohesion: Measures how closely related are objects in a cluster

Example: SSE

Cluster Separation: Measure how distinct or well-separated a cluster is from other clusters

Example: Squared Error

Cohesion is measured by the within cluster sum of squares (SSE)

Separation is measured by the between cluster sum of squares

Where |Ci| is the size of cluster i

Internal Measures: Cohesion and Separation

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Internal Measures: Cohesion and Separation

Example: SSE

BSS + WSS = constant



K=2 clusters:

K=1 cluster:

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A proximity graph based approach can also be used for cohesion and separation.

Cluster cohesion is the sum of the weight of all links within a cluster.

Cluster separation is the sum of the weights between nodes in the cluster and nodes outside the cluster.

Internal Measures: Cohesion and Separation

cohesion

separation

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“The validation of clustering structures is the most difficult and frustrating part of cluster analysis.

Without a strong effort in this direction, cluster analysis will remain a black art accessible only to those true believers who have experience and great courage.”

Algorithms for Clustering Data, Jain and Dubes

Final Comment on Cluster Validity

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Discovered Clusters Industry Group

Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN,

Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN,

DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN,

Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down,

Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN,

Sun-DOWN

Technology1-DOWN

Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN,

ADV-Micro-Device-DOWN,Andrew-Corp-DOWN,

Computer-Assoc-DOWN,Circuit-City-DOWN,

Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN,

Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN

Technology2-DOWN

Fannie-Mae-DOWN,Fed-Home-Loan-DOWN,

MBNA-Corp-DOWN,Morgan-Stanley-DOWN

Financial-DOWN

Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP,

Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP,

Schlumberger-UP

Oil-UP

	Discovered Clusters	*Industry Group*
1	Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN, Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN, DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN, Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down, Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN, Sun-DOWN	Technology1-DOWN
2	Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN, ADV-Micro-Device-DOWN,Andrew-Corp-DOWN, Computer-Assoc-DOWN,Circuit-City-DOWN, Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN, Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN	Technology2-DOWN
3	Fannie-Mae-DOWN,Fed-Home-Loan-DOWN, MBNA-Corp-DOWN,Morgan-Stanley-DOWN	Financial-DOWN
4	Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP, Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP, Schlumberger-UP	Oil-UP

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0.6

0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Points

100

Similarity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Points

100

Similarity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SSE

-6

-4

-2

WSS

SSE

)

(

BSS

)

(

)

(

)

(

)

(

)

(

)

(

)

(

Total

BSS

WSS

SSE

)

(

)

(

)

(

)

(

)

(

Total

BSS

WSS

SSE