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Lecture 21 Clustering (2). Outline. Similarity (Distance) Measures Distortion Criteria Scattering Criterion Hierarchical Clustering and other clustering methods. Distance Measure. Distance Measure – What does it mean “Similar"? Norm: Mahalanobis distance:

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outline
Outline
  • Similarity (Distance) Measures
  • Distortion Criteria

Scattering Criterion

  • Hierarchical Clustering and other clustering methods

(C) 2001-2003 by Yu Hen Hu

distance measure
Distance Measure
  • Distance Measure – What does it mean “Similar"?
    • Norm:
    • Mahalanobis distance:

d(x,y) = |x – y|TSxy1|x – y|

    • Angle: d(x,y) = xTy/(|x|•|y|)

Binary and symbolic features (x, y contains 0, 1 only):

    • Tanimoto coefficient:

(C) 2001-2003 by Yu Hen Hu

clustering criteria
Clustering Criteria
  • Is the current clustering assignment good enough? Most popular one is the mean-square error distortion measure
  • Other distortion measures can also be used:

(C) 2001-2003 by Yu Hen Hu

scatter matrics
Scatter matrices are defined in the context of analysis of variance in statistics.

They are used in linear discriminant analysis.

However, they can also be used to gauge the fitness of a particular clustering assignment.

Mean vector for i-th cluster:

Total mean vector

Scatter matrix for i-th cluster:

Within-cluster scatter matrix

Between-cluster scatter matrix

Scatter Matrics

(C) 2001-2003 by Yu Hen Hu

scattering criteria
Total scatter matrix:

Note that the total scatter matrix is independent of the assignment I(xk,i). But …

SW and SB both depend on I(xk,i)!

Desired clustering property

SW small

SB large

How to gauge Sw is small or SB is large?

There are several ways.

Tr. Sw (trace of SW): Let

be the eigenvalue decomposition of SW, then

Scattering Criteria

(C) 2001-2003 by Yu Hen Hu

cluster separating measure csm
Similar to scattering criteria.

csm = (mi-mj)/(i+j)

The larger its value, the more separable the two clusters.

Assume underlying data distribution is Gaussian.

Cluster Separating Measure (CSM)

(C) 2001-2003 by Yu Hen Hu

hierarchical clustering
Hierarchical Clustering
  • Merge Method:

Initially, each xk is a cluster. During each iteration, nearest pair of distinct clusters are merged until the number of clusters is reduced to 1.

  • How to measure distance between two clusters:

dmin(C(i), C(j)) = min. d(x,y); x  C(i), y  C(j)

 leads to minimum spanning tree

dmax(C(i), C(j)) = max. d(x,y); x  C(i), y  C(j)

davg(C(i), C(j)) =

dmean(C(i), C(j)) = mi– mj

(C) 2001-2003 by Yu Hen Hu

hierarchical clustering ii
Hierarchical Clustering (II)

Split method:

  • Initially, only one cluster. Iteratively, a cluster is splited into two or more clusters, until the total number of clusters reaches a predefined goal.
  • The scattering criterion can be used to decide how to split a given cluster into two or more clusters.
  • Another way is to perform a m-way clustering, using, say, k-means algorithm to split a cluster into m smaller clusters.

(C) 2001-2003 by Yu Hen Hu

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