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Dynamic Clustering(some unfinished business)

Keith van Rijsbergen

Glasgow

October, 2002

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OUTLINE• Introduction

• Scales

• Dissimilarity/Similarity

• Information-theoretic approach

• Static Clustering

• Dynamic Clustering

• Application

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Introduction

•Theory…..where did it come from (Sneath and Sokal)•Implementation….not yet•Experimentation…..none

Sneath and Sokal, Numerical Taxonomy (1973)Jardine and Sibson, Mathematical Taxonomy (1971)Van Rijsbergen, Information Retrieval (1979)

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Scales

scale operation group statistics

nominal equality permutation

1:1

mode

ordinal greater/less isotonic

monotone

median

interval equality/diff

of intervals

linear

x’=ax+b

mean

ratio equality of

ratios

similarity

x’=ax

coeff of

variation

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Dissimilarity/Similarity

d(x,y) 0 for all x,y

d(x,x) = 0 for all x

d(x,y) = d(y,x)

d(x,y) d(x,z) +d(z,y)

{d(x,y) max [d(x,z), d(z,y)]}

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Information-theoretic approach I

1:2I2:1I2,1J

dxxf

fxf2:1I

HPHP

logxHPxHP

logxfxf

log

2

x1

1

2

1

2

1

2

1

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2121

21

2211

21222

2211

21111

21

21

cw,cw2,1Kw,w2,1K

1,2K2,1K

02,1K

0wor 0wif 0)2,1(K

dxfwfw

wwflogfwfwfw

wwflogfw

ww1

w,w2,1K

Information-theoretic approach II

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Navigation - Browsing

T-space

D-space

Duality is the key.

Class definition!

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Static Clustering

1. dependence on rank-ordering of dissimilarity2. insensitive to small errors in DC3. preservation of well marked clusters4. stable under growth5. labelling independence6. invariance of ultrametric7. subject to 3 minimises distortion

DCBA

1.4.4.4.E

3.3.3.D

2.4.C

1.B

DCBA

2.2.3.3.E

2.3.3.D

3.3.C

1.BT

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Dendrogram

.3

.2

.1

Spanning tree?

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Dynamic Clustering

Hilbert-Schmidt: (A,B) = trace(A’B)

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Applications

• Image Retrieval• Web Retrieval

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Ostension

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Conclusions

?