Data Mining

Cluster Analysis: Basic Concepts

and Algorithms

Lecture Notes for Chapter 7

Introduction to Data Mining

by

Tan, Steinbach, Kumar

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 *

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

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

What is not Cluster Analysis?

Supervised classification

Have class label information

Simple segmentation

Dividing students into different registration groups alphabetically, by last name

Results of a query

Groupings are a result of an external specification

Graph partitioning

Some mutual relevance and synergy, but areas are not identical

Types of Clusterings

A clustering is a set of clusters

Important distinction between hierarchical and partitional sets of clusters

Partitional Clustering

A division 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

Other Distinctions Between Sets of Clusters

Exclusive versus non-exclusive

In non-exclusive clusterings, points may belong to multiple clusters.

Can represent multiple classes or border points

Fuzzy versus non-fuzzy

In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1

Weights must sum to 1

Probabilistic clustering has similar characteristics

Partial versus complete

In some cases, we only want to cluster some of the data

Heterogeneous versus homogeneous

Cluster of widely different sizes, shapes, and densities

Types of Clusters

Well-separated clusters

Center-based clusters

Contiguous clusters

Density-based clusters

Property or Conceptual

Described by an Objective Function

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.

Types of Clusters: Objective Function

Map the clustering problem to a different domain and solve a related problem in that domain

Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points

Clustering is equivalent to breaking the graph into connected components, one for each cluster.

Want to minimize the edge weight between clusters and maximize the edge weight within clusters

Characteristics of the Input Data Are Important

Type of proximity or density measure

This is a derived measure, but central to clustering

Sparseness

Dictates type of similarity

Adds to efficiency

Attribute type

Dictates type of similarity

Type of Data

Dictates type of similarity

Other characteristics, e.g., autocorrelation

Dimensionality

Noise and Outliers

Type of Distribution

Clustering Algorithms

K-means and its variants

Hierarchical clustering

Density-based clustering

K-means Clustering

Partitional clustering approach

Each cluster is associated with a centroid (center point)

Each point is assigned to the cluster with the closest centroid

Number of clusters, K, must be specified

The basic algorithm is very simple

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

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 clusters, we can choose 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

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 dont

Consider an example of five pairs of clusters

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

Postprocessing

Bisecting K-means

Not as susceptible to initialization issues

Handling Empty Clusters

Basic K-means algorithm can yield empty clusters

Several strategies

Choose the point that contributes most to SSE

Choose a point from the cluster with the highest SSE

If there are several empty clusters, the above can be repeated several times.

Updating Centers Incrementally

In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid

An alternative is to update the centroids after each assignment (incremental approach)

Each assignment updates zero or two centroids

More expensive

Introduces an order dependency

Never get an empty cluster

Can use “weights” to change the impact

Pre-processing and Post-processing

Pre-processing

Normalize the data

Eliminate outliers

Post-processing

Eliminate small clusters that may represent outliers

Split loose clusters, i.e., clusters with relatively high SSE

Merge clusters that are close and that have relatively low SSE

Can use these steps during the clustering process

ISODATA

Bisecting K-means

Bisecting K-means algorithm

Variant of K-means that can produce a partitional or a hierarchical clustering

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.

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 dendogram at the proper level

They may correspond to meaningful taxonomies

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

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 a point (or there are k clusters)

Traditional hierarchical algorithms use a similarity or distance matrix

Merge or split one cluster at a time

Agglomerative Clustering Algorithm

More 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

Hierarchical Clustering: Group Average

Compromise between Single and Complete Link

Strengths

Less susceptible to noise and outliers

Limitations

Biased towards globular clusters

Cluster Similarity: Wards 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

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 for some approaches

Hierarchical Clustering: Problems and Limitations

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

No objective function is directly minimized

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

Sensitivity to noise and outliers

Difficulty handling different sized clusters and convex shapes

Breaking large clusters

DBSCAN

DBSCAN is a density-based algorithm.

Density = number of points within a specified radius (Eps)

A point is a core point if it has more than a specified number of points (MinPts) within Eps

These are points that are at the interior of a cluster

A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point

A noise point is any point that is not a core point or a border point.

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

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

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

Two matrices

Proximity Matrix

“Incidence” 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

Need a framework to interpret any measure.

For example, if our measure of evaluation has the value, 10, is that good, fair, or poor?

Statistics provide a framework for cluster validity

The more “atypical” a clustering result is, the more likely it represents valid structure in the data

Can compare the values of an index that result from random data or clusterings to those of a clustering result.

If the value of the index is unlikely, then the cluster results are valid

These approaches are more complicated and harder to understand.

For comparing the results of two different sets of cluster analyses, a framework is less necessary.

However, there is the question of whether the difference between two index values is significant

Framework for Cluster Validity

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

“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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