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Pattern Recognition: Statistical and Neural

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Nanjing University of Science & Technology

Pattern Recognition:Statistical and Neural

Lonnie C. Ludeman

Lecture 27

Nov 9, 2005

Lecture 27 Topics

- K-Means Clustering Algorithm Details
- K-Means Step by Step Example
- ISODATA Algorithm -Overview
- 4. Agglomerative Hierarchical Clustering Algorithm Description

K-Means Clustering Algorithm:

Basic Procedure

Randomly Select K cluster centers from Pattern Space

Distribute set of patterns to the cluster center using minimum distance

Compute new Cluster centers for each cluster

Continue this process until the cluster centers do not change.

Flow Diagram for K-Means Algorithm

Step 1 Initialization

Choose K initial Cluster centers

M1(1), M2(1), ... , MK(1)

Method 1 – First K samples Method 2 – K data samples selected randomly Method 3 – K random vectors

Set m = 1 and Go To Step 2

Step 2 Determine New Clusters

Using Cluster centers Distribute pattern vectors using minimum distance.

Method 1 – Use Euclidean distance Method 2 – Use other distance measures

Assign sample xjto class Ck if

Go to Step 3

Step 3 Compute New Cluster Centers

Using the new Cluster assignment

Clk(m) m = 1, 2, ... , K

Compute new cluster centers

Mk(m+1) m = 1, 2, ... , K

using

where Nk, k = 1, 2, ... , K

is the number of pattern vectors in Clk(m)

Go to Step 4

Step 4 Check for Convergence

Using Cluster centers from step 3 check for convergence

Convergence occurs if the means do not change

If Convergence occurs Clustering is complete and the results given.

If No Convergence then Go to Step 5

Step 5 Check for Maximum Number of Iterations

Define MAXIT as the maximum number of iterations that is acceptable.

If m = MAXIT Then display no convergence

and Stop.

If m < MAXITThen m=m+1 (increment m)

and Return to Step 2

Example:K-Means cluster algorithm

Given the following set of pattern vectors

Plot of Data points in Given set of samples

Do the following

(a) Solution – 2-class case

Initial Cluster centers

Plot of Data points in Given set of samples

Initial Cluster Centers

Distances from all Samples to cluster centers

Cl2

Cl1

Cl2

Cl1

Cl2

Cl2

Cl2

With tie select randomly

First Cluster assignment

Closest to x2

Closest to x1

Plot of Data points in Given set of samples

First Cluster Assignment

Compute New Cluster centers

New Cluster centers

Plot of Data points in Given set of samples

Distances from all Samples to cluster centers

2

2

Cl2

Cl2

Cl1

Cl1

Cl2

Cl2

Cl1

Second Cluster assignment

Old Cluster Center

M2(2)

New Clusters

M1(2)

Old Cluster Center

Plot of Data points in Given set of samples

Compute New Cluster Centers

ClusterCenters

M2(3)

New Clusters

M1(3)

Plot of Data points in Given set of samples

Distances from all Samples to cluster centers

3

3

Cl1

Cl1

Cl1

Cl2

Cl2

Cl2

Cl2

Compute New Cluster centers

(b) Solution: 3-Class case

Select Initial Cluster Centers

First Cluster assignment using distances from pattern vectors to initial cluster centers

Compute New Cluster centers

Second Cluster assignment using distances from pattern vectors to cluster centers

At the next step we have convergence as the cluster centers do not change thus the Final Cluster Assignment becomes

Final 3-Class Clusters

Cl3

Cl2

Final Cluster Centers

Cl1

Plot of Data points in Given set of samples

Iterative Self Organizing Data Analysis Technique A

ISODATA Algorithm

Performs Clustering of unclassified quantitative data with an unknown number of clusters

Similar to K-Means but with ablity to merge and split clusters thus giving flexibility in number of clusters

ISODATA Parameters that need to be specified

merged at each step

Requires more specified information than for the K-Means Algorithm

ISODATA Algorithm

Final Clustering

Hierarchical Clustering

Approach 1 Agglomerative

Combines groups at each level

Approach 2 Devisive

Combines groups at each level

Will present only Agglomerative Hierarchical Clustering as it is most used.

Agglomerative Hierarchical Clustering

Consider a set S of patterns to be clustered

S = { x1, x2, ... , xk, ... , xN}

Define Level N by

S1(N)= { x1}

Clusters at level N are the individual pattern vectors

S2(N)= { x2}

...

SN(N)= { xN}

Define Level N -1 to be N – 1 Clusters formed by merging two of the Level N clusters by the following process.

Compute the distances between all the clusters at level N and merge the two with the smallest distance (resolve ties randomly) to give the Level N-1 clusters as

S1(N-1)

Clusters at level N -1 result from this merging

S2(N-1)

...

SN-1(N-1)

The process of merging two clusters at each step is performed sequentially until Level 1 is reached. Level one is a single cluster containing all samples

S1(1)= { x1, x2, ... , xk, ... , xN}

Thus Hierarchical clustering provides cluster assignments for all numbers of clusters from N to 1.

Definition:

A Dendrogram is a tree like structure that illustrates the mergings of clusters at each step of the Hierarchical Approach.

A typical dendrogram appears on the next slide

Typical Dendrogram

Summary Lecture 27

- Presented the K-Means Clustering Algorithm Details
- Showed Example of Clustering using the K-Means Algorithm (Step by Step)
- Briefly discussed the ISODATA Algorithm
- 4. Introduced the Agglomerative Hierarchical Clustering Algorithm

End of Lecture 27