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Data Mining�Cluster Analysis: Basic Concepts �and Algorithms

Module 5

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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?

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

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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 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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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

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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 (more similar) to the “center” of a cluster, than to the center of any other cluster
The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster

4 center-based clusters

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Types of Clusters: Contiguity-Based

Contiguous Cluster (Nearest neighbor or Transitive)

A cluster is a set of points such that 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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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

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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

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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
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

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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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Two different K-means Clusterings

Sub-optimal Clustering

Optimal Clustering

Original Points

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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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 C_iand m_i is the representative point for cluster C_i

can show that m_icorresponds 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

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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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10 Clusters Example

Starting with some pairs of clusters having three initial centroids, while other have only one.

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10 Clusters Example

Starting with some pairs of clusters having three initial centroids, while other have only one.

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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

Postprocessing
Bisecting K-means

Not as susceptible to initialization issues

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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.

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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

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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

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Bisecting K-means

Bisecting K-means algorithm

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

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Bisecting K-means Example

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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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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 dendogram at the proper level

They may correspond to meaningful taxonomies

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

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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 a 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

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

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Starting Situation

Start with clusters of individual points and a proximity matrix

p1

p3

p5

p4

p2

p1

p2

p3

p4

p5

. . .

.

Proximity Matrix

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Intermediate Situation

After some merging steps, we have some clusters

C1

C4

C2

C5

C3

C2

C1

C3

C5

C4

C2

C3

C4

C5

Proximity Matrix

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Intermediate Situation

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

C1

C4

C2

C5

C3

C2

C1

C3

C5

C4

C2

C3

C4

C5

Proximity Matrix

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After Merging

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

C1

C4

C2 U C5

C3

? ? ? ?

?

C2 U C5

C1

C3

C4

C2 U C5

C3

C4

Proximity Matrix

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How to Define Inter-Cluster Similarity

p1

p3

p5

p4

p2

p1

p2

p3

p4

p5

. . .

.

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

p1

p3

p5

p4

p2

p1

p2

p3

p4

p5

. . .

.

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

p1

p3

p5

p4

p2

p1

p2

p3

p4

p5

. . .

.

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

p1

p3

p5

p4

p2

p1

p2

p3

p4

p5

. . .

.

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

p1

p3

p5

p4

p2

p1

p2

p3

p4

p5

. . .

.

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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Cluster Similarity: MIN or Single Link

Similarity of two clusters is based on the two most similar (closest) points in the different clusters

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

1

2

3

4

5

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Hierarchical Clustering: MIN

Nested Clusters

Dendrogram

1

2

3

4

5

6

1

2

3

4

5

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Strength of MIN

Original Points

Two Clusters

Can handle non-elliptical shapes

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Limitations of MIN

Original Points

Two Clusters

Sensitive to noise and outliers

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Cluster Similarity: MAX or Complete Linkage

Similarity of two clusters is based on the two least similar (most distant) points in the different clusters

Determined by all pairs of points in the two clusters

1

2

3

4

5

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Hierarchical Clustering: MAX

Nested Clusters

Dendrogram

1

2

3

4

5

6

1

2

5

3

4

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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
Biased towards globular clusters

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Cluster Similarity: 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

1

2

3

4

5

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Hierarchical Clustering: Group Average

Nested Clusters

Dendrogram

1

2

3

4

5

6

1

2

5

3

4

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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

1

2

3

4

5

6

1

2

5

3

4

MIN

MAX

1

2

3

4

5

6

1

2

5

3

4

1

2

3

4

5

6

1

2

5

3

4

1

2

3

4

5

6

1

2

3

4

5

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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 for some approaches

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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

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MST: Divisive Hierarchical Clustering

Build MST (Minimum Spanning Tree)

Start with a tree that consists of any point
In successive steps, look for the closest pair of points (p, q) such that one point (p) is in the current tree but the other (q) is not
Add q to the tree and put an edge between p and q

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MST: Divisive Hierarchical Clustering

Use MST for constructing hierarchy of clusters

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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.

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DBSCAN: Core, Border, and Noise Points

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DBSCAN Algorithm

Eliminate noise points
Perform clustering on the remaining points

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DBSCAN: Core, Border and Noise Points

Original Points

Point types: core, border and noise

Eps = 10, MinPts = 4

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When DBSCAN Works Well

Original Points

Clusters

Resistant to Noise
Can handle clusters of different shapes and sizes

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When DBSCAN Does NOT Work Well

Original Points

(MinPts=4, Eps=9.75).

(MinPts=4, Eps=9.92)

Varying densities
High-dimensional data

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DBSCAN: Determining EPS and MinPts

Idea is that for points in a cluster, their k^th nearest neighbors are at roughly the same distance
Noise points have the k^th nearest neighbor at farther distance
So, plot sorted distance of every point to its k^th nearest neighbor

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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

DBSCAN

Complete Link

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Different Aspects of Cluster Validation

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.

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Measures of Cluster Validity

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.

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Measuring Cluster Validity Via Correlation

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.

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Measuring Cluster Validity Via Correlation

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

Corr = -0.9235

Corr = -0.5810

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