PCA-BASED DATA MINING PROBABILISTIC AND FUZZY

APPROACHES WITH APPLICATIONS IN PATTERN

RECOGNITION

Luminita State

Dept. of Computer Science, University of Pitesti, Caderea Bastliei #45, Bucuresti – 1, Romania

Catalina Cocianu

Dept. of Computer Science, Academy of Economic Studies, Calea Dorobantilor #15-17, Bucuresti –1, Romania

Panayiotis Vlamos

Ionian University, Corfu, Greece

Viorica Stefanescu

Dept. of Mathematics, Academy of Economic Studies, Calea Dorobantilor #15-17, Bucuresti –1, Romania

Keywords: Principal component analysis, fuzzy

clustering, supervised learning, cluster analysis, pattern recognition,

data mining.

Abstract: The aim of the paper is to develop a new learning by examples PCA-based algorithm for extracting skeleton

information from data to assure both good recognition performances, and generalization capabilities. Here

the generalization capabilities are viewed twofold, on one hand to identify the right class for new samples

coming from one of the classes taken into account and, on the other hand, to identify the samples coming

from a new class. The classes are represented in the measuremen /feature space by continuous repartitions,

that is the model is given by the family of density functions

t

(

)

f

Hh

h

∈

, where H stands for the finite set of

hypothesis (classes). The basis of the learning process is represented by samples of possible different sizes

coming from the considered classes. The skeleton of each class is given by the principal components

obtained for the corresponding sample.

1 PRINCIPAL COMPONENTS

The starting point for PCA is a n-dimensional

random vector X. There is available a sample X(1),

X(2),… ,X(T) from this random vector. No explicit

assumptions on the probability density of the vectors

are made in PCA, as long as the first – order and the

second –order statistics are known or can be

estimated from the sample. Also, no generative

model is assumed for vector X.

In the PCA transform, the vector X is

first

centered by subtracting its mean, X = X - E(X). In

practice, the mean of the n-dimensional vector X is

estimated from the available sample. In the

following, we assume that the vector X is centered.

Next, X is linearly transformed to another vector Y

with m elements, m<n, so that the redundancy

induced by the correlations is removed. The

transform consists in obtaining a rotated orthogonal

coordinate system such that the elements of X in the

new coordinates become uncorrelated. At the same

time, the variances of the projections of X on the

new coordinates become uncorrelated. At the same

time, the variances of the projections of X on the

new coordinate axes are maximized so that the first

axis corresponds to the maximal variance, the

second axis corresponds to the maximal variance in

the direction orthogonal to the first axis, and so on

(Hyvarinen, 2001).

55

State L., Cocianu C., Vlamos P. and Stefanescu V. (2006).

PCA-BASED DATA MINING PROBABILISTIC AND FUZZY APPROACHES WITH APPLICATIONS IN PATTERN RECOGNITION.

In Proceedings of the First International Conference on Software and Data Technologies, pages 55-60

Copyright

c

 SciTePress

In mathematical terms, consider a linear

combination of the elements of the

vector X, .

n

xxx ,...,,

21

XW

T

n

i

ii

xwy

1

11

==

∑

=

We look for a weight vector maximizing the

PCA criterion,

1

W

() ( )

1

2

1

2

1

EE SWWXW

TT

y =

⎟

⎠

⎞

⎜

⎝

⎛

=

(1)

1

=W

,

where the matrix S is the covariance matrix of X

given for the zero-mean vector X by the correlation

matrix.

It is well known from basic linear algebra that

the solution of the PCA problem is given in terms of

the unit-length eigen vectors of the matrix S,

. The ordering of the eigen vectors is

such that the corresponding eigen values

n

φφφ ,...,,

21

n

λ

,...,,

21

satisfy

n

λ

≥≥≥ ...

21

. The solution

maximizing is given by

and the first principal component of X is

. The PCA criterion can be generalized to

m principal components, . Let

be the m-th principal

component, with the corresponding unit norm

weight vector. The solution maximizing

under the constrain

, for is given by, (Hyvarinen,

2001) and the m-th principal component of

X is .

()

1

2

1

E SWW

T

y =

11

φW =

Xφ

T

y

11

=

nm ≤≤1

XW

T

m

n

i

iimm

xwy ==

∑

=1

m

W

()

m

T

mm

y SWW=

2

E

()

0=

km

yyE mk <≤1

mm

φW =

Xφ

T

mm

y =

2 CLUSTER ANALYSIS

Cluster analysis is a data processing technique

aiming to identify the natural grouping trends

existing in a data collection, producing a set of

overlapping clusters, the elements belonging to the

same cluster sharing similar features. So far, there

have been proposed a relatively small number of

methods for testing the existence/inexistence of a

natural grouping tendency in a data collection, most

of them being based on arguments coming from

mathematical statistics and heuristic graphical

techniques (Panayirci and Dubes,1983, Smith and

Jain,1984,Jain and Dubes,1988, Tukey,1977,

Everitt,1978).

The data are represented by p-dimensional

vectors,

(

)

1

,...,

t

p

Xx x=

, whose components are the

feature values of a specified attributes and the

classification is performed against a certain given

label set. The classification of a data collection

{

}

1

,...,

n

XXℵ=

p

⊂ℜ

corresponds to a labelling

strategy of the objects of

ℵ

.

In the fuzzy approaches, the clusters are

represented as fuzzy sets

()

,

,1

i

uic≤≤

[

]

:0,

i

u ℵ→ 1

)

, where is the

membership degree of to the i-th

cluster,

1

(

ik i k

uuX=

k

X

ic

≤

,

1 kn

≤

. A c-fuzzy partition is

represented by the matrix

ik c n

Uu M

×

=∈

. The

number of labels c has to be selected in advance, the

problem of finding the optimal c is usually referred

as cluster validation.

The main types of label vectors are crisp N

c

,

fuzzy N

p

, and possibilistic N

poz

, defined as follows,

(

){}

{

,1,0,,...,,,|

21

∈=ℜ∈=

ic

c

yyyyyyyN

, (2)

{

c

i

eeeyci ,...,,1,1

21

1

=

⎭

⎬

⎫

=≤≤

∑

=

}

where

()

0,

1,

iij

j

ij

e

ij

δ

≠

⎧

==

⎨

=

⎩

(

)

[]

{

,1,0,,,...,,|

21

∈∀=ℜ∈=

ic

c

p

yiyyyyyN

, (3)

⎭

⎬

⎫

=

∑

=

1

c

i

y

(

)

[]

{

,1,0,,,...,,|

21

∈∀=ℜ∈=

ic

c

poz

yiyyyyyN

}

0,

≠

∃

j

yj

, (4)

Obviously,

p

os p c

NNN

⊃

. If we denote by

[

]

1

,...,

ni

UU U u==

j

a partition of ℵ , then,

according to the types of label vectors, we get the c-

partition types M

pos

, M

p

and M

c

,

[]

{

,,...,,|

1 nncpos

UUUMUUM

=

∈

=

×

(5)

⎭

⎬

⎫

>∀∈∀

∑

=

0,,

1

,

n

k

ikposk

uiNUk

{

}

,,

p

pos k p

M

UU M kU N=∈∀∈

(6)

{

}

,,

cpkc

M

UU M kU N=∈∀∈

(7)

Note that

cppos

M

MM⊂⊂

.

ICSOFT 2006 - INTERNATIONAL CONFERENCE ON SOFTWARE AND DATA TECHNOLOGIES

56

3 C-MEANS MODEL

The most popular classification methods are the c-

means algorithms. The variational problem

corresponding to c-means model is given by

()

() ()

2

,

11 1 1

min , ; 1

cn c n

m

mikiki

UV

ik i k

JUVw uD w u

== = =

⎧⎫

=+−

⎨⎬

⎩⎭

∑∑ ∑ ∑

ik

(8) where

//

c p pos

UMMM∈

,

(

)

1

,...,

ccp

Vv v M

×

=∈

i

v

, is

the centroid of the i-th cluster , is

the penalties vector corresponding to the cluster

system, is the fuzzyfication degree, and

()

1

,..,

T

c

www=

1≥m

2

ik k i

Dxv=−

.

Let us denote by

(

)

VU

ˆ

,

ˆ

a solution of (8). Then,

1. The crisp model:

()

,;0,1

ccpi

UV M M w i c

×

∈× =≤≤

,

⎩

⎨

⎧

≠≤

=

otherwise,0

,,1

ˆ

jiDD

u

ijik

ik

(9)

1

ˆ

n

ik k

k

i

n

ik

k

ux

v

u

=

∑

;

1,

(10)

1ic kn≤≤ ≤ ≤

2. The fuzzy model:

()

,;1,0,1

pcp i

UV M M m w i c

×

∈× > =≤≤

1

2

1

ˆ

m

c

ik

j

jk

D

u

D

−

=

⎡⎤

⎛⎞

⎢

=

⎜⎟

⎢⎥

⎜⎟

⎝⎠

⎢⎥

⎣⎦

∑

⎥

(11)

1

ˆ

n

m

ik k

k

i

n

m

ik

k

ux

v

u

=

∑

;

1,

(12)

1ic kn≤≤ ≤ ≤

3. The possibilistic model:

()

,;

pos c p i

UV M M iw

×

∈× ∀>,0

1

2

1

ˆ

1

m

ik

i

D

u

w

−

⎡⎤

⎛⎞

⎢⎥

=+

⎜⎟

⎢⎥

⎝⎠

⎢⎥

⎣⎦

1

ˆ

n

m

ik k

k

i

n

m

ik

k

ux

v

u

=

∑

;

1,

(13)

1ic kn≤≤ ≤ ≤

The general scheme of a cluster procedure

℘

is ,

()

()( )

1

0

1

,,

tt

t

repeat

tt

UFV

VGU

until t Tor V V

UV U V

ε

℘−

−

←

←+

←

=

−≤

←

where c is the given number of clusters, T is upper

limit on the number of iterations, m is the weight

parameter,

1 m

≤

<∞

, C is the terminal condition, w

is the system of weights ,

,0

i

iw∀>

(

)

0 1,0 ,0

,...,

c

Vv v M

cp

×

=∈

is the initial system of

centroids and , are the updating functions.

℘

F

℘

G

4 PCA-BASED ALGORITHM FOR

EXTRACTING SKELETON

INFORMATION

In the following a new learning by examples PCA-

based algorithm for extracting skeleton information

from data to assure both good recognition

performances, and generalization capabilities, is

developed. Here the generalization capabilities are

viewed twofold, on one hand to identify the right

class for new samples coming from one of the

classes taken into account and, on the other hand, to

identify the samples coming from a new class. The

classes are represented in the measurement/feature

space by continuous repartitions, that is the model is

given by the family of density functions

()

,

where H stands for the finite set of hypothesis

(classes).

Hh

h

f

∈

The basis of the learning process is represented

by samples of possible different sizes coming from

the considered classes. The skeleton of each class is

given by the principal components obtained for the

corresponding sample. The recognition algorithm

identifies the class whose skeleton is the “nearest” to

the tested example, where the closeness degree is

expressed in terms of the amount of disturbance

determined by the decision of allotting it to the

corresponding class. The model is presented as

follows. Let be a series of n-

dimensional vectors coming from a certain class C.

The sample covariance matrix is

N

XXX ,...,,

21

PCA-BASED DATA MINING PROBABILISTIC AND FUZZY APPROACHES WITH APPLICATIONS IN PATTERN

RECOGNITION

57

()(

∑

=

−−

−

=Σ

N

i

T

NiNiN

XX

N

1

μμ

)

, (14)

where

∑

=

N

i

iN

X

N

1

μ

.

We denote by the eigen

values and by the corresponding

orthonormal eigen vectors of .

N

n

NN

λλλ

≥≥≥ ...

21

N

n

N

ψψ

,...,

1

N

Σ

If X

N+1

is a new sample, then, for the series

, we get

121

,,...,,

+NN

XXXX

()()

−−−

+

+Σ=Σ

+++

T

NNNNNN

XX

N

μμ

111

1

N

Σ−

1

(15)

Lemma. In case the eigen values of are

pairwise distinct, the following first order

approximations hold,

N

Σ

() ()

N

iN

T

N

i

N

iN

T

N

i

N

i

N

i

ψΣψψΣψ

1

+

=Δ+=

λλ

(16)

()

∑

≠

=

+

−

Δ

+=

n

ij

j

N

j

N

j

N

i

N

iN

T

j

N

i

N

i

1

ψ

ψΣψ

ψψ

λλ

(17)

Proof Using the perturbation theory, we get,

and, ,

, . Then,

NNN

ΣΣΣ Δ+=

+1

N

i

N

i

N

i

ψψψ Δ+=

+1

N

i

N

i

N

i

λλλ

Δ+=

+1

ni ≤≤1

()()

N

T

NNNNN

N

ΣμXμXΣ

1

11

−−−

+

=Δ

++

(18)

()

(

)

=Δ+Δ+

N

i

N

iNN

ψψΣΣ

(

)

(

)

N

i

N

i

N

i

N

i

ψψ Δ+Δ+=

λλ

(19)

Using first order approximations, from (19) we get,

≅Δ+Δ+

N

iN

N

iN

N

i

N

i

ψΣψΣψ

λ

N

i

N

i

N

i

N

i

N

i

N

i

ψψψ

λλλ

Δ+Δ+≅

(20)

hence,

() ()

≅Δ+Δ

N

iN

T

N

i

N

iN

T

N

i

ψΣψψΣψ

()

2

N

i

N

i

N

i

T

N

i

N

i

ψψψ

λλ

Δ+Δ≅ (21)

Using we obtain ,

()()

N

T

N

i

T

N

i

N

i

Σψψ =

λ

() ()

≅Δ+Δ

N

iN

T

N

i

N

i

T

N

i

N

i

ψΣψψψ

λ

()

N

i

N

i

T

N

i

N

i

λλ

Δ+Δ≅ ψψ

(22)

hence that is,

()

N

iN

T

N

i

N

i

ψΣψ Δ=Δ

λ

() ()

N

iN

T

N

i

N

iN

T

N

i

N

i

N

i

ψΣψψΣψ

1

+

=Δ+=

λλ

(23)

The first order approximations of the orthonormal

eigen vectors of can be derived using the

expansion of each vector in the basis

represented by the orthonormal eigen vectors of

,

1+N

Σ

N

i

ψΔ

N

Σ

∑

=

=Δ

n

j

N

jji

N

i

b

1

,

ψψ , (24)

where

(

)

N

i

T

N

jji

b ψψ Δ=

,

. (25)

Using the orthonormality, we get,

()( )

=Δ+≅Δ+=

N

i

T

N

i

N

i

N

i

N

i

ψψψψψ

21

22

(

)

(

)

N

i

T

N

i

ψψ

Δ+= 21

, (26)

that is

(

)

N

i

T

N

iii

b ψψ Δ=

,

=0 (27)

Using (19), the approximation,

N

i

N

i

N

i

N

i

N

iN

N

iN

ψψψΣψΣ

λλ

Δ+Δ≅Δ+Δ

. (28)

holds for each

ni

≤

1

.

For

nij

≤

≠

≤

1

, from (28) we obtain the following

equations,

(

)

(

)

≅Δ+Δ

N

iN

T

j

N

iN

T

j

N

ψΣψψΣψ

(

)

(

)

N

i

T

j

N

i

N

i

T

j

N

i

ψψψψ

λλ

Δ+Δ≅

(29)

(

)

(

)

≅Δ+Δ

N

iN

T

j

N

iN

T

j

N

ψΣψψΣψ

(

)

N

i

T

j

N

i

ψψ Δ≅

λ

(30)

(

)

(

)

≅Δ+Δ

N

iN

T

j

N

i

T

j

N

j

ψΣψψψ

λ

(

)

N

i

T

j

N

i

ψψ Δ≅

λ

(31)

From (31) we get,

(

)

(

)

(

)

N

iN

T

j

N

i

T

j

N

j

N

i

ψΣψψψ Δ=Δ−

λλ

(32)

()

(

)

N

j

N

i

N

iN

T

j

N

i

T

N

jji

b

λλ

−

Δ

=Δ=

ψΣψ

ψψ

,

(33)

Consequently, the first order approximation of the

eigen vectors of are,

1+N

Σ

(

)

∑

≠

=

−

Δ

+=Δ+

n

ij

j

N

j

N

j

N

i

N

iN

T

j

N

i

N

i

N

i

1

ψ

ψΣψ

ψψψ

λλ

(34)

The skeleton of

C is represented by the set of

estimated principal components .When

the example

X

N

n

N

ψψ

,...,

1

N+1

is included in C, then the new

skeleton is . The skeleton disturbance

induced by the decision that

X

11

1

,...,

++ N

n

N

ψψ

N+1

has to be alloted to

C is measured by

ICSOFT 2006 - INTERNATIONAL CONFERENCE ON SOFTWARE AND DATA TECHNOLOGIES

58

(

∑

=

+

=

n

k

N

k

N

k

d

n

D

1

,

1

ψψ

)

}

(35)

The crisp classification procedure identifies for

each example the closest cluster in terms of the

measure (35). Let . In order to

protect against misclassifications of samples coming

from new classes not belonging to

H, a threshold

T>0 is imposed, that is the example X

{

M

CCCH ,...,,

21

=

N+1

is alloted to

one of

C

j

for which

()

==

∑

=

+

n

k

N

jk

N

jk

d

n

D

1

,,

,

1

ψψ

()

(36)

∑

=

+

≤≤

=

n

k

N

pk

N

pk

Mp

d

n

1

,,

1

,

1

min

ψψ

and

D<T, where the skeleton of C

j

is .

N

jn

N

j ,,1

,...,

ψψ

The classification of samples for which the

resulted value of

D is larger than T is postponed and

the samples are kept in a new possible class CR. The

reclassification of elements of CR is then performed

followed by the decision concerning to either

reconfigure the class system or to add CR as a new

class in

H.

In case of fuzzy classification, the value of the

membership degree of

X

N+1

to each cluster of

is computed as follows. Let

{}

CRHHT ∪=

()

∑

=

+

=

n

k

N

ik

N

ikiN

d

n

CXd

1

,,1

,

1

,

ψψ

, and

Mi ≤≤1

()

⎪

⎩

⎪

⎨

⎧

∞

∅≠

=

∑

=

+

otherwise,

if,,

1

,

1

,,

1

CRd

n

CRXd

n

k

N

Rk

N

Rk

N

ψψ

,

where the skeleton of CR is represented by the set of

estimated principal components .

N

Rn

N

R ,,1

,...,

ψψ

()

S

CXd

X

iN

NC

i

,

1

+

−=

μ

, (37)

Mi ≤≤1

()

⎪

⎩

⎪

⎨

⎧

∞≠−

=

+

otherwise,0

,if,

,

1

CRXd

S

CRXd

X

N

NCR

μ

(38),

where

()

(

)

()( )

⎪

⎩

⎪

⎨

⎧

∞=

∞≠

=

+

∈

+

∈

+

∑

CRXdCXd

S

N

HC

N

HTC

N

,if,,

11

(39)

5 EXPERIMENTAL RESULTS

AND CONCLUDING REMARKS

Several tests were performed on simulated data and

they pointed out very successful performance of the

proposed classification strategy.

A series of tests were performed on 4-

dimensional simulated data coming from 5 clases

each of them having 50 examples. The classes are

represented by normal repartitions ~

()

i

C

ii

Σ

,

μ

,

51

≤

i

, where

1

μ

= [10 11 2 -12]

=

Σ

1

⎥

⎦

⎤

⎢

⎣

⎡

1.8369 0.8010 1.5350 0.6460

0.8010 3.0600 3.1000 1.4720

1.5350 3.1000 3.7500 2.0100

0.6460 1.4720 2.0100 3.5944

2

μ

= [12 -5 8 13]

=

Σ

2

⎥

⎦

⎤

⎢

⎣

⎡

2.3961 1.0050 0.6720 0.3745

1.0050 2.4586 0.9305 0.5230

0.6720 0.9305 1.7766 0.7755

0.3745 0.5230 0.7755 1.5566

3

μ

= [-10 0 9 11]

=

Σ

3

⎥

⎦

⎤

⎢

⎣

⎡

2.3144 0.3250 0.4600 0.6000

0.3250 3.0725 0.0120 0.0200

0.4600 0.0120 2.5144 1.3740

0.6000 0.0200 1.3740 1.7300

4

μ

= [-3 14 3 -11.5]

=

Σ

4

⎥

⎦

⎤

⎢

⎣

⎡

4.1054 0.4250 0.6029 0.9805

0.4250 2.3724 0.9920 0.3814

0.6029 0.9920 2.4038 0.5825

0.9805 0.3814 0.5825 2.3618

5

μ

= [-7 -10.5 -14 11.5]

=

Σ

5

⎥

⎦

⎤

⎢

⎣

⎡

2.9441 0.3454 0.0210 0.6540

0.3454 1.6301 0.0240 0.6330

0.0210 0.0240 2.4436 0.2860

0.6540 0.6330 0.2860 1.8017

The classification criterion is: allote X

N+1

to if

i

j

C

(

∑

=

+

≤≤

=

l

j

ll

l

m

k

Nj

k

j

tl

d

m

D

1

1,

1

,

1

min

ψψ

)

(37)

In order to evaluate the generalization capacities,

100 new examples were generated for each

distribution. The results are presented in Table 1.

PCA-BASED DATA MINING PROBABILISTIC AND FUZZY APPROACHES WITH APPLICATIONS IN PATTERN

RECOGNITION

59

Table 1: Results on new simulated examples.

Class C

1

C

2

C

3

C

4

C

5

Number of

correct

classified

examples

100 100 96 99 100

Number of

missclassified

examples

0 0 4 -

allotted

to C

2

1 –

allotted

to C

1

0

The mean

v

alue of D in

c

ase of

c

orrect

c

lassifications

0.08 0.05 0.75 0.21 0.14

The

maximum

value of D in

case of

correct

classified

examples

0.41

0

.19 1.85 0.55 0.53

The evaluation of the generalization capacities in

case of examples coming from new classes was

performed on 1000 samples generated from

N

(

)

Σ,

μ

, where

=

μ

[0 11 -9 -9.5] and

=Σ

⎥

⎦

⎤

⎢

⎣

⎡

5.2261 1.6410 2.5561 1.3680

1.6410 6.0422 1.8390 1.7925

2.5561 1.8390 6.8986 3.1080

1.3680 1.7925 3.1080 8.2725

.

The admissibility criterion for allotting a sample

to a certain class is given by the maximum value of

D corresponding to correct classifications. The

results showed that about 975 examples were

classified in CR, that is the algorithm managed to

detect the intruded examples.

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