Learning

Spatially

Variant

Dissimilarity

-LRB-

Svad

-RRB-

Measures

Clustering

algorithms

typically

operate

on

a

feature

vector

representation

of

the

data

and

find

clusters

that

are

compact

with

respect

to

an

assumed

-LRB-

dis

-RRB-

similarity

measure

between

the

data

points

in

feature

space

This

makes

the

type

of

clusters

identified

highly

dependent

on

the

assumed

similarity

measure

Building

on

recent

work

in

this

area

we

formally

define

a

class

of

spatially

varying

dissimilarity

measures

and

propose

algorithms

to

learn

the

dissimilarity

measure

automatically

from

the

data

The

idea

is

to

identify

clusters

that

are

compact

with

respect

to

the

unknown

spatially

varying

dissimilarity

measure

Our

experiments

show

that

the

proposed

algorithms

are

more

stable

and

achieve

better

accuracy

on

various

textual

data

sets

when

compared

with

similar

algorithms

proposed

in

the

literature

H.

-LSB-

Database

Management

-RSB-

Database

Applications-Data

Mining

Algorithms

Clustering

plays

a

major

role

in

data

mining

as

a

tool

to

discover

structure

in

data

Object

clustering

algorithms

operate

on

a

feature

vector

representation

of

the

data

and

find

clusters

that

are

compact

with

respect

to

an

assumed

-LRB-

dis

-RRB-

similarity

measure

between

the

data

points

in

feature

space

As

a

consequence

the

nature

of

clusters

identified

by

a

clustering

algorithm

is

highly

dependent

on

the

assumed

similarity

measure

The

most

commonly

used

dissimilarity

measure

namely

the

Euclidean

metric

assumes

that

the

dissimilarity

measure

is

isotropic

and

spatially

invariant

and

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KDD

August

Seattle

Washington

USA

Copyright

ACM

it

is

effective

only

when

the

clusters

are

roughly

spherical

and

all

of

them

have

approximately

the

same

size

which

is

rarely

the

case

in

practice

-LSB-

-RSB-

The

problem

of

finding

non-spherical

clusters

is

often

addressed

by

utilizing

a

feature

weighting

technique

These

techniques

discover

a

single

set

of

weights

such

that

relevant

features

are

given

more

importance

than

irrelevant

features

However

in

practice

each

cluster

may

have

a

different

set

of

relevant

features

We

consider

Spatially

Varying

Dissimilarity

-LRB-

SVaD

-RRB-

measures

to

address

this

problem

Diday

et

al.

-LSB-

-RSB-

proposed

the

adaptive

distance

dynamic

clusters

-LRB-

ADDC

-RRB-

algorithm

in

this

vain

A

fuzzified

version

of

ADDC

popularly

known

as

the

Gustafson-Kessel

-LRB-

GK

-RRB-

algorithm

-LSB-

-RSB-

uses

a

dynamically

updated

covariance

matrix

so

that

each

cluster

can

have

its

own

norm

matrix

These

algorithms

can

deal

with

hyperelliposoidal

clusters

of

various

sizes

and

orientations

The

EM

algorithm

-LSB-

-RSB-

with

Gaussian

probability

distributions

can

also

be

used

to

achieve

similar

results

However

the

above

algorithms

are

computationally

expensive

for

high-dimensional

data

since

they

invert

covariance

matrices

in

every

iteration

Moreover

matrix

inversion

can

be

unstable

when

the

data

is

sparse

in

relation

to

the

dimensionality

One

possible

solution

to

the

problems

of

high

computation

and

instability

arising

out

of

using

covariance

matrices

is

to

force

the

matrices

to

be

diagonal

which

amounts

to

weighting

each

feature

differently

in

different

clusters

While

this

restricts

the

dissimilarity

measures

to

have

axis

parallel

isometry

the

weights

also

provide

a

simple

interpretation

of

the

clusters

in

terms

of

relevant

features

which

is

important

in

knowledge

discovery

Examples

of

such

algorithms

are

SCAD

and

Fuzzy-SKWIC

-LSB-

-RSB-

which

perform

fuzzy

clustering

of

data

while

simultaneously

finding

feature

weights

in

individual

clusters

In

this

paper

we

generalize

the

idea

of

the

feature

weighting

approach

to

define

a

class

of

spatially

varying

dissimilarity

measures

and

propose

algorithms

that

learn

the

dissimilarity

measure

automatically

from

the

given

data

while

performing

the

clustering

The

idea

is

to

identify

clusters

inherent

in

the

data

that

are

compact

with

respect

to

the

unknown

spatially

varying

dissimilarity

measure

We

compare

the

proposed

algorithms

with

a

diagonal

version

of

GK

-LRB-

DGK

-RRB-

and

a

crisp

version

of

SCAD

-LRB-

CSCAD

-RRB-

on

a

variety

of

data

sets

Our

algorithms

perform

better

than

DGK

and

CSCAD

and

use

more

stable

update

equations

for

weights

than

CSCAD

The

rest

of

the

paper

is

organized

as

follows

In

the

next

section

we

define

a

general

class

of

dissimilarity

measures

Research

Track

Poster

and

formulate

two

objective

functions

based

on

them

In

Section

we

derive

learning

algorithms

that

optimize

the

objective

functions

We

present

an

experimental

study

of

the

proposed

algorithms

in

Section

We

compare

the

performance

of

the

proposed

algorithms

with

that

of

DGK

and

CSCAD

These

two

algorithms

are

explained

in

Appendix

A.

Finally

we

summarize

our

contributions

and

conclude

with

some

future

directions

in

Section

We

first

define

a

general

class

of

dissimilarity

measures

and

formulate

a

few

objective

functions

in

terms

of

the

given

data

set

Optimization

of

the

objective

functions

would

result

in

learning

the

underlying

dissimilarity

measure

SVaD

Measures

In

the

following

definition

we

generalize

the

concept

of

dissimilarity

measures

in

which

the

weights

associated

with

features

change

over

feature

space

Definition

We

define

the

measure

of

dissimilarity

of

x

from

y

to

be

a

weighted

sum

of

M

dissimilarity

measures

between

x

and

y

where

the

values

of

the

weights

depend

on

the

region

from

which

the

dissimilarity

is

being

measured

Let

P

-LCB-

R

R

K

-RCB-

be

a

collection

of

K

regions

that

partition

the

feature

space

and

w

w

and

w

K

be

the

weights

associated

with

R

R

and

R

K

respectively

Let

g

g

and

g

M

be

M

dissimilarity

measures

Then

each

w

j

j

K

is

an

M

dimensional

vector

where

its

l-th

component

w

jl

is

associated

with

g

l

Let

W

denote

the

K-tuple

-LRB-

w

w

K

-RRB-

and

let

r

be

a

real

number

Then

the

dissimilarity

of

x

from

y

is

given

by

f

W

-LRB-

x

y

-RRB-

M

l

w

r

jl

g

l

-LRB-

x

y

-RRB-

if

y

R

j

-LRB-

-RRB-

We

refer

to

f

W

as

a

Spatially

Variant

Dissimilarity

-LRB-

SVaD

-RRB-

measure

Note

that

f

W

need

not

be

symmetric

even

if

g

i

are

symmetric

Hence

f

W

is

not

a

metric

Moreover

the

behavior

of

f

W

depends

on

the

behavior

of

g

i

There

are

many

ways

to

define

g

i

We

list

two

instances

of

f

W

Example

-LRB-

Minkowski

-RRB-

Let

d

be

the

feature

space

and

M

d.

Let

a

point

x

d

be

represented

as

-LRB-

x

x

d

-RRB-

Then

when

g

i

-LRB-

x

y

-RRB-

x

i

y

i

p

for

i

d

and

p

the

resulting

SVaD

measure

f

M

W

is

called

Minkowski

SVaD

-LRB-

MSVaD

-RRB-

measure

That

is

f

M

W

-LRB-

x

y

-RRB-

d

l

w

r

jl

x

l

y

l

p

if

y

R

j

-LRB-

-RRB-

One

may

note

that

when

w

w

K

and

p

f

M

W

is

the

weighted

Euclidean

distance

When

p

we

call

f

M

W

a

Euclidean

SVaD

-LRB-

ESVaD

-RRB-

measure

and

denote

it

by

f

E

W

We

use

the

phrase

dissimilarity

of

x

from

y

rather

than

dissimilarity

between

x

and

y

because

we

consider

a

general

situation

where

the

dissimilarity

measure

depends

on

the

location

of

y

As

an

example

of

this

situation

in

text

mining

when

the

dissimilarity

is

measured

from

a

document

on

terrorism

to

a

document

x

a

particular

set

of

keywords

may

be

weighted

heavily

whereas

when

the

dissimilarity

is

measured

from

a

document

on

football

to

x

a

different

set

of

keywords

may

be

weighted

heavily

Example

-LRB-

Cosine

-RRB-

Let

the

feature

space

be

the

set

of

points

with

l

norm

equal

to

one

That

is

x

for

all

points

x

in

feature

space

Then

when

g

l

-LRB-

x

y

-RRB-

-LRB-

x

l

y

l

-RRB-

for

l

d

the

resulting

SVaD

measure

f

C

W

is

called

a

Cosine

SVaD

-LRB-

CSVaD

-RRB-

measure

f

C

W

-LRB-

x

y

-RRB-

d

i

w

r

jl

-LRB-

x

l

y

l

-RRB-

if

y

R

j

-LRB-

-RRB-

In

the

formulation

of

the

objective

function

below

we

use

a

set

of

parameters

to

represent

the

regions

R

R

and

R

K

Let

c

c

and

c

K

be

K

points

in

feature

space

Then

y

R

j

iff

f

W

-LRB-

y

c

j

-RRB-

f

W

-LRB-

y

c

i

-RRB-

for

i

j.

-LRB-

-RRB-

In

the

case

of

ties

y

is

assigned

to

the

region

with

the

lowest

index

Thus

the

K-tuple

of

points

C

-LRB-

c

c

c

K

-RRB-

defines

a

partition

in

feature

space

The

partition

induced

by

the

points

in

C

is

similar

in

nature

to

a

Voronoi

tessellation

We

use

the

notation

f

W

C

whenever

we

use

the

set

C

to

parameterize

the

regions

used

in

the

dissimilarity

measure

Objective

Function

for

Clustering

The

goal

of

the

present

work

is

to

identify

the

spatially

varying

dissimilarity

measure

and

the

associated

compact

clusters

simultaneously

It

is

worth

mentioning

here

that

as

in

the

case

of

any

clustering

algorithm

the

underlying

assumption

in

this

paper

is

the

existence

of

such

a

dissimilarity

measure

and

clusters

for

a

given

data

set

Let

x

x

and

x

n

be

n

given

data

points

Let

K

be

a

given

positive

integer

Assuming

that

C

represents

the

cluster

centers

let

us

assign

each

data

point

x

i

to

a

cluster

R

j

with

the

closest

c

j

as

the

cluster

center

i.e.

j

arg

min

l

f

W

C

-LRB-

x

i

c

l

-RRB-

-LRB-

-RRB-

Then

the

within-cluster

dissimilarity

is

given

by

J

-LRB-

W

C

-RRB-

K

j

x

i

R

j

M

l

w

r

jl

g

l

-LRB-

x

i

c

j

-RRB-

-LRB-

-RRB-

J

-LRB-

W

C

-RRB-

represents

the

sum

of

the

dissimilarity

measures

of

all

the

data

points

from

their

closest

centroids

The

objective

is

to

find

W

and

C

that

minimize

J

-LRB-

W

C

-RRB-

To

avoid

the

trivial

solution

to

J

-LRB-

W

C

-RRB-

we

consider

a

normalization

condition

on

w

j

viz.

M

l

w

jl

-LRB-

-RRB-

Note

that

even

with

this

condition

J

-LRB-

W

C

-RRB-

has

a

trivial

solution

w

jp

where

p

arg

min

l

x

i

R

j

g

l

-LRB-

x

i

c

j

-RRB-

and

the

remaining

weights

are

zero

One

way

to

avoid

convergence

of

w

j

to

unit

vectors

is

to

impose

a

regularization

condition

on

w

j

We

consider

the

following

two

regularization

measures

in

this

paper

-LRB-

-RRB-

Entropy

measure

M

l

w

jl

log

-LRB-

w

jl

-RRB-

and

-LRB-

-RRB-

Gini

measure

M

l

w

jl

We

use

P

-LCB-

R

R

R

K

-RCB-

to

represent

the

corresponding

partition

of

the

data

set

as

well

The

intended

interpretation

-LRB-

cluster

or

region

-RRB-

would

be

evident

from

the

context

Research

Track

Poster

The

problem

of

determining

the

optimal

W

and

C

is

similar

to

the

traditional

clustering

problem

that

is

solved

by

the

K-Means

Algorithm

-LRB-

KMA

-RRB-

except

for

the

additional

W

matrix

We

propose

a

class

of

iterative

algorithms

similar

to

KMA

These

algorithms

start

with

a

random

partition

of

the

data

set

and

iteratively

update

C

W

and

P

so

that

J

-LRB-

W

C

-RRB-

is

minimized

These

iterative

algorithms

are

instances

of

Alternating

Optimization

-LRB-

AO

-RRB-

algorithms

In

-LSB-

-RSB-

it

is

shown

that

AO

algorithms

converge

to

a

local

optimum

under

some

conditions

We

outline

the

algorithm

below

before

actually

describing

how

to

update

C

W

and

P

in

every

iteration

Randomly

assign

the

data

points

to

K

clusters

REPEAT

Update

C

Compute

the

centroid

of

each

cluster

c

j

Update

W

Compute

the

w

jl

j

l.

Update

P

Reassign

the

data

points

to

the

clusters

UNTIL

-LRB-

termination

condition

is

reached

-RRB-

In

the

above

algorithm

the

update

of

C

depends

on

the

definition

of

g

i

and

the

update

of

W

on

the

regularization

terms

The

update

of

P

is

done

by

reassigning

the

data

points

according

to

-LRB-

-RRB-

Before

explaining

the

computation

of

C

in

every

iteration

for

various

g

i

we

first

derive

update

equations

for

W

for

various

regularization

measures

Update

of

Weights

While

updating

weights

we

need

to

find

the

values

of

weights

that

minimize

the

objective

function

for

a

given

C

and

P

As

mentioned

above

we

consider

the

two

regularization

measures

for

w

jl

and

derive

update

equations

If

we

consider

the

entropy

regularization

with

r

the

objective

function

becomes

J

EN

T

-LRB-

W

C

-RRB-

K

j

x

i

R

j

M

l

w

jl

g

l

-LRB-

x

i

c

j

-RRB-

K

j

j

M

l

w

jl

log

-LRB-

w

jl

-RRB-

K

j

j

M

l

w

jl

-LRB-

-RRB-

Note

that

j

are

the

Lagrange

multipliers

corresponding

to

the

normalization

constraints

in

-LRB-

-RRB-

and

j

represent

the

relative

importance

given

to

the

regularization

term

relative

to

the

within-cluster

dissimilarity

Differentiating

J

EN

T

-LRB-

W

C

-RRB-

with

respect

to

w

jl

and

equating

it

to

zero

we

obtain

w

jl

exp

-LRB-

j

x

i

Rj

g

l

-LRB-

x

i

c

j

-RRB-

-RRB-

j

Solving

for

j

by

substituting

the

above

value

of

w

jl

in

-LRB-

-RRB-

and

substituting

the

value

of

j

back

in

the

above

equation

we

obtain

w

jl

exp

x

i

R

j

g

l

-LRB-

x

i

c

j

-RRB-

j

M

n

exp

x

i

R

j

g

n

-LRB-

x

i

c

j

-RRB-

j

-LRB-

-RRB-

If

we

consider

the

Gini

measure

for

regularization

with

r

the

corresponding

w

jl

that

minimizes

the

objective

function

can

be

shown

to

be

w

jl

-LRB-

j

x

i

R

j

g

l

-LRB-

x

i

c

j

-RRB-

-RRB-

M

n

-LRB-

-LRB-

j

x

i

R

j

g

n

-LRB-

x

i

c

j

-RRB-

-RRB-

-RRB-

-LRB-

-RRB-

In

both

cases

the

updated

value

of

w

jl

is

inversely

related

Algorithm

Update

Equations

Acronyms

P

C

W

EEnt

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

EsGini

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

CEnt

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

CsGini

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Table

Summary

of

algorithms

to

x

i

R

j

g

l

-LRB-

x

i

c

j

-RRB-

This

has

various

interpretations

based

on

the

nature

of

g

l

For

example

when

we

consider

the

ESVaD

measure

w

jl

is

inversely

related

to

the

variance

of

l-th

element

of

the

data

vectors

in

the

j-th

cluster

In

other

words

when

the

variance

along

a

particular

dimension

is

high

in

a

cluster

then

the

dimension

is

less

important

to

the

cluster

This

popular

heuristic

has

been

used

in

various

contexts

-LRB-

such

as

relevance

feedback

-RRB-

in

the

literature

-LSB-

-RSB-

Similarly

when

we

consider

the

CSVaD

measure

w

jl

is

directly

proportional

to

the

correlation

of

the

j-th

dimension

in

the

l-th

cluster

Update

of

Centroids

Learning

ESVaD

Measures

Substituting

the

ESVaD

measure

in

the

objective

function

and

solving

the

first

order

necessary

conditions

we

observe

that

c

jl

R

j

x

i

R

j

x

il

-LRB-

-RRB-

minimizes

J

ESV

AD

-LRB-

W

C

-RRB-

Learning

CSVaD

Measures

Let

x

il

w

jl

x

il

then

using

the

Cauchy-Swartz

inequality

it

can

be

shown

that

c

jl

R

j

x

i

R

j

x

il

-LRB-

-RRB-

maximizes

x

i

R

j

d

l

w

jl

x

il

c

jl

Hence

-LRB-

-RRB-

also

minimizes

the

objective

function

when

CSVaD

is

used

as

the

dissimilarity

measure

Table

summarizes

the

update

equations

used

in

various

algorithms

We

refer

to

this

set

of

algorithms

as

SVaD

learning

algorithms

In

this

section

we

present

an

experimental

study

of

the

algorithms

described

in

the

previous

sections

We

applied

the

proposed

algorithms

on

various

text

data

sets

and

compared

the

performance

of

EEnt

and

EsGini

with

that

of

K-Means

CSCAD

and

DGK

algorithms

The

reason

for

choosing

the

K-Means

algorithm

-LRB-

KMA

-RRB-

apart

from

CSCAD

and

DGK

is

that

it

provides

a

baseline

for

assessing

the

advantages

of

feature

weighting

KMA

is

also

a

popular

algorithm

for

text

clustering

We

have

included

a

brief

description

of

CSCAD

and

DGK

algorithms

in

Appendix

A.

Text

data

sets

are

sparse

and

high

dimensional

We

consider

standard

labeled

document

collections

and

test

the

proposed

algorithms

for

their

ability

to

discover

dissimilarity

measures

that

distinguish

one

class

from

another

without

actually

considering

the

class

labels

of

the

documents

We

measure

the

success

of

the

algorithms

by

the

purity

of

the

regions

that

they

discover

Research

Track

Poster

Data

Sets

We

performed

our

experiments

on

three

standard

data

sets

News

Group

Yahoo

and

Classic

These

data

sets

are

described

below

News

Group

We

considered

different

subsets

of

News

Group

data

that

are

known

to

contain

clusters

of

varying

degrees

of

separation

-LSB-

-RSB-

As

in

-LSB-

-RSB-

we

considered

three

random

samples

of

three

subsets

of

the

News

Group

data

The

subsets

denoted

by

Binary

has

documents

each

from

talk.politics.mideast

and

talk.politics.misc

has

documents

each

from

comp.graphics

rec.motorcycles

rec.sport.baseball

sci.space

and

talk.politics.mideast

Finally

has

documents

each

from

alt.atheism

comp

sys.mac.hardware

misc.forsale

rec.autos

rec.sport.hockey

sci.crypt

sci.electronics

sci.med

sci.space

and

talk.politics

gun

It

may

be

noted

that

Binary

data

sets

have

two

highly

overlapping

classes

Each

of

data

sets

has

samples

from

distinct

classes

whereas

data

sets

have

only

a

few

samples

from

different

classes

The

size

of

the

vocabulary

used

to

represent

the

documents

in

Binary

data

set

is

about

about

and

about

We

observed

that

the

relative

performance

of

the

algorithms

on

various

samples

of

Binary

and

data

sets

was

similar

Hence

we

report

results

on

only

one

of

them

Yahoo

This

data

set

contains

Reuters

news

articles

downloaded

from

Yahoo

in

There

are

from

Health

from

Entertainment

from

Sports

from

Politics

from

Technology

and

from

Business

After

preprocessing

the

documents

from

this

data

set

are

represented

using

words

Note

that

this

data

set

has

samples

from

different

classes

Here

the

distribution

of

data

points

across

the

class

is

uneven

ranging

from

to

Classic

Classic

data

set

contains

aerospace

systems

abstracts

from

the

Cranfield

collection

medical

abstracts

from

the

Medline

collection

and

information

retrieval

abstracts

from

the

Cisi

collection

making

up

documents

in

all

After

preprocessing

this

data

set

has

words

The

points

are

almost

equally

distributed

among

the

three

distinct

classes

The

data

sets

were

preprocessed

using

two

major

steps

First

a

set

of

words

-LRB-

vocabulary

-RRB-

is

extracted

and

then

each

document

is

represented

with

respect

to

this

vocabulary

Finding

the

vocabulary

includes

-LRB-

-RRB-

elimination

of

the

standard

list

of

stop

words

from

the

documents

-LRB-

-RRB-

application

of

Porter

stemming

for

term

normalization

and

-LRB-

-RRB-

keeping

only

the

words

which

appear

in

at

least

documents

We

represent

each

document

by

the

unitized

frequency

vector

Evaluation

of

Algorithms

We

use

the

accuracy

measure

to

compare

the

performance

of

various

algorithms

Let

a

ij

represent

the

number

of

data

points

from

class

i

that

are

in

cluster

j

Then

the

accuracy

of

the

partition

is

given

by

j

max

i

a

ij

n

where

n

is

the

total

number

of

data

points

It

is

to

be

noted

that

points

coming

from

a

single

class

need

not

form

a

single

cluster

There

could

be

multiple

tar.gz

Iteration

J

-LRB-

W

C

-RRB-

Accuracy

Table

Evolution

of

J

-LRB-

W

C

-RRB-

and

Accuracies

with

iterations

when

EEnt

applied

on

a

data

clusters

in

a

class

that

represent

sub-classes

We

study

the

performance

of

SVaD

learning

algorithms

for

various

values

of

K

i.e.

the

number

of

clusters

Experimental

Setup

In

our

implementations

we

have

observed

that

the

proposed

algorithms

if

applied

on

randomly

initialized

centroids

show

unstable

behavior

One

reason

for

this

behavior

is

that

the

number

of

parameters

that

are

estimated

in

feature-weighting

clustering

algorithms

is

twice

as

large

as

that

estimated

by

the

traditional

KMA

We

therefore

first

estimate

the

cluster

centers

giving

equal

weights

to

all

the

dimensions

using

KMA

and

then

fine-tune

the

cluster

centers

and

the

weights

using

the

feature-weighting

clustering

algorithms

In

every

iteration

the

new

sets

of

weights

are

updated

as

follows

Let

w

n

-LRB-

t

-RRB-

represent

the

weights

com-puted

using

one

of

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

or

-LRB-

-RRB-

in

iteration

-LRB-

t

-RRB-

and

w

-LRB-

t

-RRB-

the

weights

in

iteration

t.

Then

the

weights

in

iteration

-LRB-

t

-RRB-

are

w

-LRB-

t

-RRB-

-LRB-

-LRB-

t

-RRB-

-RRB-

w

-LRB-

t

-RRB-

-LRB-

t

-RRB-

w

n

-LRB-

t

-RRB-

-LRB-

-RRB-

where

-LRB-

t

-RRB-

-LSB-

-RSB-

decreases

with

t

That

is

-LRB-

t

-RRB-

-LRB-

t

-RRB-

for

a

given

constant

-LSB-

-RSB-

In

our

experiments

we

observed

that

the

variance

of

purity

values

for

different

initial

values

of

-LRB-

-RRB-

and

above

is

very

small

Hence

we

report

the

results

for

-LRB-

-RRB-

and

We

set

the

value

of

j

It

may

be

noted

that

when

the

documents

are

represented

as

unit

vectors

KMA

with

the

cosine

dissimilarity

measure

and

Euclidean

distance

measure

would

yield

the

same

clusters

This

is

essentially

the

same

as

Spherical

K-Means

algorithms

described

in

-LSB-

-RSB-

Therefore

we

consider

only

the

weighted

Euclidean

measure

and

restrict

our

comparisons

to

EEnt

and

EsGini

in

the

experiments

Since

the

clusters

obtained

by

KMA

are

used

to

initialize

all

other

algorithms

considered

here

and

since

the

results

of

KMA

are

sensitive

to

initialization

the

accuracy

numbers

reported

in

this

section

are

averages

over

random

initializations

of

KMA

Results

and

Observations

Effect

of

SVaD

Measures

on

Accuracies

In

Table

we

show

a

sample

run

of

EEnt

algorithm

on

one

of

the

data

sets

This

table

shows

the

evolution

of

J

-LRB-

W

C

-RRB-

and

the

corresponding

accuracies

of

the

clusters

with

the

iterations

The

accuracy

shown

at

iteration

is

that

of

the

clusters

obtained

by

KMA

The

purity

of

clusters

increases

with

decrease

in

the

value

of

the

objective

function

defined

using

SVaD

measures

We

have

observed

a

similar

behavior

of

EEnt

and

EsGini

on

other

data

sets

also

This

validates

our

hypothesis

that

SVaD

measures

capture

the

underlying

structure

in

the

data

sets

more

accurately

Research

Track

Poster

Comparison

with

Other

Algorithms

Figure

to

Figure

show

average

accuracies

of

various

algorithms

on

the

data

sets

for

various

number

of

clusters

The

accuracies

of

KMA

and

DGK

are

very

close

to

each

other

and

hence

in

the

figures

the

lines

corresponding

to

these

algorithms

are

indistinguishable

The

lines

corresponding

to

CSCAD

are

also

close

to

that

of

KMA

in

all

the

cases

except

Class

General

observations

The

accuracies

of

SVaD

algorithms

follow

the

trend

of

the

accuracies

of

other

algorithms

In

all

our

experiments

both

SVaD

learning

algorithms

improve

the

accuracies

of

clusters

obtained

by

KMA

It

is

observed

in

our

experiments

that

the

improvement

could

be

as

large

as

in

some

instances

EEnt

and

EsGini

consis-tently

perform

better

than

DGK

on

all

data

sets

and

for

all

values

of

K.

EEnt

and

EsGini

perform

better

than

CSCAD

on

all

data

sets

excepts

in

the

case

of

Classic

and

for

a

few

values

of

K.

Note

that

the

weight

update

equation

of

CSCAD

-LRB-

-RRB-

may

result

in

negative

values

of

w

jl

Our

experience

with

CSCAD

shows

that

it

is

quite

sensitive

to

initialization

and

it

may

have

convergence

problems

In

contrast

it

may

be

observed

that

w

jl

in

-LRB-

-RRB-

and

-LRB-

-RRB-

are

always

positive

Moreover

in

our

experience

these

two

versions

are

much

less

sensitive

to

the

choice

of

j

Data

specific

observations

When

K

EEnt

and

EsGini

could

not

further

improve

the

results

of

KMA

on

the

Binary

data

set

The

reason

is

that

the

data

set

contains

two

highly

overlapping

classes

However

for

other

values

of

K

they

marginally

improve

the

accuracies

In

the

case

of

the

accuracies

of

the

algorithms

are

non-monotonic

with

K

The

improvement

of

accuracies

is

large

for

intermediate

values

of

K

and

small

for

extreme

values

of

K

When

K

KMA

finds

relatively

stable

clusters

Hence

SVaD

algorithms

are

unable

to

improve

the

accuracies

as

much

as

they

did

for

intermediate

values

of

K.

For

larger

values

of

K

the

clusters

are

closely

spaced

and

hence

there

is

little

scope

for

improvement

by

the

SVaD

algorithms

data

sets

are

the

toughest

to

cluster

because

of

the

large

number

of

classes

present

in

the

data

In

this

case

the

accuracies

of

the

algorithms

are

monotonically

increasing

with

the

number

of

clusters

The

extent

of

improvement

of

accuracies

of

SVaD

algorithms

over

KMA

is

almost

constant

over

the

entire

range

of

K

This

reflects

the

fact

that

the

documents

in

data

set

are

uniformly

distributed

over

feature

space

The

distribution

of

documents

in

Yahoo

data

set

is

highly

skewed

The

extent

of

improvements

that

the

SVaD

algorithms

could

achieve

decrease

with

K.

For

higher

values

of

K

KMA

is

able

to

find

almost

pure

sub-clusters

resulting

in

accuracies

of

about

This

leaves

little

scope

for

improvement

The

performance

of

CSCAD

differs

noticeably

in

the

case

of

Classic

It

performs

better

than

the

SVaD

algorithms

for

K

and

better

than

EEnt

for

K

However

for

larger

values

of

K

the

SVaD

algorithms

perform

better

than

the

rest

As

in

the

case

of

the

improvements

of

SVaD

algorithms

over

others

are

significant

and

consistent

One

may

recall

that

and

Classic

consist

of

documents

from

distinct

classes

Therefore

this

observation

implies

that

when

there

are

distinct

clusters

in

the

data

set

KMA

yields

confusing

clusters

when

the

number

of

clusters

is

over-Figure

Accuracy

results

on

Binary

data

Figure

Accuracy

results

on

data

specified

In

this

scenario

EEnt

and

EsGini

can

fine-tune

the

clusters

to

improve

their

purity

We

have

defined

a

general

class

of

spatially

variant

dissimilarity

measures

and

proposed

algorithms

to

learn

the

measure

underlying

a

given

data

set

in

an

unsupervised

learning

framework

Through

our

experiments

on

various

textual

data

sets

we

have

shown

that

such

a

formulation

of

dissimilarity

measure

can

more

accurately

capture

the

hidden

structure

in

the

data

than

a

standard

Euclidean

measure

that

does

not

vary

over

feature

space

We

have

also

shown

that

the

proposed

learning

algorithms

perform

better

than

other

similar

algorithms

in

the

literature

and

have

better

stability

properties

Even

though

we

have

applied

these

algorithms

only

to

text

data

sets

the

algorithms

derived

here

do

not

assume

any

specific

characteristics

of

textual

data

sets

Hence

they

Figure

Accuracy

results

on

data

Research

Track

Poster

Figure

Accuracy

results

on

Yahoo

data

Figure

Accuracy

results

on

Classic

data

are

applicable

to

general

data

sets

Since

the

algorithms

perform

better

for

larger

K

it

would

be

interesting

to

investigate

whether

they

can

be

used

to

find

subtopics

of

a

topic

Finally

it

will

be

interesting

to

learn

SVaD

measures

for

labeled

data

sets

-LSB-

-RSB-

J.

C.

Bezdek

and

R.

J.

Hathaway

Some

notes

on

alternating

optimization

In

Proceedings

of

the

AFSS

International

Conference

on

Fuzzy

Systems

Calcutta

pages

Springer-Verlag

-LSB-

-RSB-

A.

P.

Dempster

N.

M.

Laird

and

Rubin

Maximum

likelihood

from

incomplete

data

via

the

EM

algorithm

Journal

Royal

Statistical

Society

B

-LRB-

-RRB-

-LSB-

-RSB-

I.

S.

Dhillon

and

D.

S.

Modha

Concept

decompositions

for

large

sparse

text

data

using

clustering

Machine

Learning

-LRB-

-RRB-

January

-LSB-

-RSB-

E.

Diday

and

J.

C.

Simon

Cluster

analysis

In

K.

S.

Fu

editor

Pattern

Recognition

pages

Springer-Verlag

-LSB-

-RSB-

H.

Frigui

and

O.

Nasraoui

Simultaneous

clustering

and

attribute

discrimination

In

Proceedings

of

FUZZIEEE

pages

San

Antonio

-LSB-

-RSB-

H.

Frigui

and

O.

Nasraoui

Simultaneous

categorization

of

text

documents

and

identification

of

cluster-dependent

keywords

In

Proceedings

of

FUZZIEEE

pages

Honolulu

Hawaii

-LSB-

-RSB-

D.

E.

Gustafson

and

W.

C.

Kessel

Fuzzy

clustering

with

the

fuzzy

covariance

matrix

In

Proccedings

of

IEEE

CDC

pages

San

Diego

California

-LSB-

-RSB-

R.

Krishnapuram

and

J.

Kim

A

note

on

fuzzy

clustering

algorithms

for

Gaussian

clusters

IEEE

Transactions

on

Fuzzy

Systems

-LRB-

-RRB-

Aug

-LSB-

-RSB-

Y.

Rui

T.

S.

Huang

and

S.

Mehrotra

Relevance

feedback

techniques

in

interactive

content-based

image

retrieval

In

Storage

and

Retrieval

for

Image

and

Video

Databases

-LRB-

SPIE

-RRB-

pages

-LSB-

-RSB-

N.

Slonim

and

N.

Tishby

Document

clustering

using

word

clusters

via

the

information

bottleneck

method

In

Proceedings

of

SIGIR

pages

APPENDIX

A

OTHER

FEATURE

WEIGHTING

CLUSTERING

TECHNIQUES

A.

Diagonal

Gustafson-Kessel

-LRB-

DGK

-RRB-

Gustafson

and

Kessel

-LSB-

-RSB-

associate

each

cluster

with

a

different

norm

matrix

Let

A

-LRB-

A

A

k

-RRB-

be

the

set

of

k

norm

matrices

associated

with

k

clusters

Let

u

ji

is

the

fuzzy

membership

of

x

i

in

cluster

j

and

U

-LSB-

u

ji

-RSB-

By

restricting

A

j

s

to

be

diagonal

and

u

ji

-LCB-

-RCB-

we

can

reformulate

the

original

optimization

problem

in

terms

of

SVaD

measures

as

follows

min

C

W

J

DGK

-LRB-

C

W

-RRB-

k

j

x

i

R

j

M

l

w

jl

g

l

-LRB-

x

i

c

j

-RRB-

subject

to

l

w

jl

j

Note

that

this

problem

can

be

solved

using

the

same

AO

algorithms

described

in

Section

Here

the

update

for

C

and

P

would

remain

the

same

as

that

discussed

in

Section

It

can

be

easily

shown

that

when

j

j

w

jl

M

m

x

i

R

j

g

m

-LRB-

x

i

c

j

-RRB-

x

i

R

j

g

l

-LRB-

x

i

c

j

-RRB-

-LRB-

-RRB-

minimize

J

DGK

for

a

given

C.

A.

Crisp

Simultaneous

Clustering

and

Attribute

Discrimination

-LRB-

CSCAD

-RRB-

Frigui

et

al.

in

-LSB-

-RSB-

considered

a

fuzzy

version

of

the

feature-weighting

based

clustering

problem

-LRB-

SCAD

-RRB-

To

make

a

fair

comparison

of

our

algorithms

with

SCAD

we

derive

its

crisp

version

and

refer

to

it

as

Crisp

SCAD

-LRB-

CSCAD

-RRB-

In

-LSB-

-RSB-

the

Gini

measure

is

used

for

regularization

If

the

Gini

measure

is

considered

with

r

the

weights

w

jl

that

minimize

the

corresponding

objective

function

for

a

given

C

and

P

are

given

by

w

jl

M

j

M

M

n

x

i

R

j

g

n

-LRB-

x

i

c

j

-RRB-

x

i

R

j

g

l

-LRB-

x

i

c

j

-RRB-

-LRB-

-RRB-

Since

SCAD

uses

the

weighted

Euclidean

measure

the

update

equations

of

centroids

in

CSCAD

remain

the

same

as

in

-LRB-

-RRB-

The

update

equation

for

w

jl

in

SCAD

is

quite

similar

to

-LRB-

-RRB-

One

may

note

that

in

-LRB-

-RRB-

the

value

of

w

jl

can

become

negative

In

-LSB-

-RSB-

a

heuristic

is

used

to

estimate

the

value

j

in

every

iteration

and

set

the

negative

values

of

w

jl

to

zero

before

normalizing

the

weights

Research

Track

Poster

O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

dissimilarity measure
clustering
feature weighting

spatially varying dissimilarity (svad)
learning dissimilarity measures