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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

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

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.

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

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

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

Distances from all Samples to cluster centers

Cl2

Cl1

Cl2

Cl1

Cl2

Cl2

Cl2

With tie select randomly

First Cluster assignment

Compute 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

M2(2)

New Clusters

M1(2)

Old Cluster Center

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

Select Initial Cluster Centers

First Cluster assignment using distances from pattern vectors to initial 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 do not change thus the Final Cluster Assignment becomes

Cl3

Cl2

Final Cluster Centers

Cl1

Plot of Data points in Given set of samples

I do not change thus the Final Cluster Assignment becomesterative 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 do not change thus the Final Cluster Assignment becomesParameters that need to be specified

merged at each step

Requires more specified information than for the K-Means Algorithm

ISODATA Algorithm do not change thus the Final Cluster Assignment becomes

Final Clustering

Hierarchical Clustering do not change thus the Final Cluster Assignment becomes

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 do not change thus the Final Cluster Assignment becomes

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 do not change thus the Final Cluster Assignment becomes 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 do not change thus the Final Cluster Assignment becomesprocess 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: do not change thus the Final Cluster Assignment becomes

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 do not change thus the Final Cluster Assignment becomes

Summary Lecture 27 do not change thus the Final Cluster Assignment becomes

- 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 do not change thus the Final Cluster Assignment becomes

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