DSpace at VNU: Some context fuzzy clustering methods for classification problems

SOME CONTEXT FUZZY CLUSTERING METHODS FOR CLASSIFICATION PROBLEMS Bui Cong Cuong Institute of Mathematics, VAST 18 Hoang Quoc Viet, HaNoi, Vietnam ccuong@inbox.com Le Hoang Son Ha

SOME CONTEXT FUZZY CLUSTERING METHODS FOR

CLASSIFICATION PROBLEMS

Bui Cong Cuong

Institute of Mathematics, VAST

18 Hoang Quoc Viet, HaNoi, Vietnam

ccuong@inbox.com

Le Hoang Son

Ha Noi University of Science, VNU

334 Nguyen Trai, Ha Noi, Viet Nam sonlh@vnu.edu.vn

Hoang Thi Minh Chau University of Economic and Technical Industries

456 Minh Khai, Ha Noi, Viet Nam htmchau.uneti@moet.edu.vn

ABSTRACT

In this paper, we will propose a two-context fuzzy clustering

algorithm (2C-FCM) and its parallel solution so called P2C-FCM

for the classification problems Some initial experiments show

the effectiveness of P2C-FCM and 2C-FCM when comparing

with traditional Context FCM The applications of P2C-FCM

and 2C-FCM are the basis to generate fuzzy rules for classifying

member countries of United Nation Organization (UNO)

according to the Human Development Index based on the

statistics of UNO in 2005

Categories and Subject Descriptors

I.5.3 [Pattern Recognition]: Clustering – algorithms

General Terms

Algorithms, Experimentation, Theory

Keywords

Parallel Fuzzy Clustering, Fuzzy Rules, Classification

1 INTRODUCTION

The problem of classification and data clustering were

studied long time ago Recent striking approaches have

concentrated on fuzzy clustering method (FCM) whose

applications range from data analysis, pattern recognition, image

segmentation, group-positioning analysis, satellite images,

financial analysis, With the growing demands for the

exploitation of intelligent and highly autonomous systems, it

would be beneficial to combine robust learning capabilities with

a high level of knowledge interpretability Fuzzy neuro

computation supports a new paradigm of intelligent information

processing [1, 2, 5-8], in which we are able to achieve this

powerful combination Nowadays, W.Pedrycz et al presented

some knowledge-based clustering methods, including context

fuzzy C-means method (CFCM) It is also considered as a strong

aid of rule extraction and data mining from a set of data [1, 2,

4-8], in which fuzzy factors are really common and rise up various

trends to work on

H.M Berenji in [3] considered FCM as a method for tuning fuzzy rules and in [2] G Bortolan, W Pedrycz used CFCM in the design of fuzzy neural network In the section 5 of [2] the contexts and the resulting prototypes in the input space are directly used towards the construction of the fuzzy neural networks Obviously, FCM and CFCM are useful techniques in computing methodologies nowadays

1.1 Previous work

The first context-based clustering approach was proposed by

W.Pedrycz [2] namely as context fuzzy C-means method (CFCM)

In this study, they defined a context variable in order to narrow the origin dataset under some conditions of certain dimensions Because only a subset of origin dataset which has considerable meaning to the context is invoked, the velocity and effeciency of classification can be improved considerably, and the result focuses on the area that really has many relevant points In a specific case, the context-sensitive FCM allows us to concentrate the classification into a subspace due to conditions of some dimensions showed in defined context The convergence conditions of CFCM algorithm is analogous to standard FCM as shown in [3] The speed of CFCM is relatively faster than FCM

in case of little context variables CFCM has been being the state-of-the-art algorithm in context-based clustering

However, sometimes we need to concentrate to 2 more important variables of the set of antecedent attributes of fuzzy rules For example, we need to look for countries which have

‘high’ GDP Per capita (GDPPC) and ‘high’ Education Index (EI)

in the statistics of the United Nation Organization (UNO) Traditional CFCM does not allow us to do this analysis due to the use of a context variable in its definition only Consequently,

we need another method to deal with this kind of requests

1.2 This work

In this paper, we will present a 2-context fuzzy clustering

method (2C-FCM) and a parallel solution for this algorithm so

called Parallel Two-Context Fuzzy Clustering Method

(P2C-FCM) These methods are applied to generate fuzzy rules in

contextual situations for the classification problem The experiment on the statistics of UNO in 2005 shows the effectiveness of the proposed algorithms

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The paper is organized as follows: The second section

reviews the CFCM method Section 3 devotes to a concrete

2C-FCM algorithm which could be applied in the parameter training

phase of the process and to calculation the partition matrix The

parallel solution P2C-FCM will be presented in Section 4 In

Section 5, we will present some evaluations and the application

of these clustering methods for classifying member countries of

United Nation Organization (UNO) according to the Human

Development Index based on the statistics of UNO 2005 will be

presented in Section 6 Finally, we make conclusion and future

works in the last section

2 CONTEXT FUZZY C-MEANS METHOD

Given a dataset of n attributes:X   x1, , xN, supposed

that missing data have been processed, our purpose is to classify

them into C clusters We will work in n-dimension space

(x  R n) with xk is the kth point and vi is the center of ith cluster

Then, we define a context variable in Y  Xwhose definition is

stated through the map:

A: Y → [0,1]

yk  fk = A(yk)

The value fk can be understood as the representation for the

level of relation of the kth point to the supposed context Y These

are some ways to define the relation between fk and the

membership of kth point to the ith cluster, for instance, using the

sum operator (1) or maximum operator (2)







c

i

k

ik f

u

1

(1)

k

ik

c

1

max (2)

with k=1, , N The basic objective function is:

2

1 1

m

 

where m is a coefficient of fuzziness and uik is an element of

partition matrix U defined as following:

 

































C i N u N k f u u

f

U

N

k ik c

i k ik

)

(

1 1

where uik is the membership value of the kth point to the ith

cluster

The algorithm CFCM

The context fuzzy c-means algorithm has four steps:

1 Initiate the matrix U(t) with t=0

2 Re-calculate centers of each clusters according to :

1

1

N m

ik k k

m ik k

u x u









for i = 1,…, C

3 Re- calculate matrix U(t+1) as follow:

2 1

1

k ik

m C

f u

x x













for i = 1, , C ; k=1, , N

4 If the error of the partition matrix ||U(t+1) - U(t)||, defined through some analysis normal, is less than given threshold  then the algorithm stops, else return step 2

We arrive at the formula in step 3 by transforming the condition: U  U ( f) to a standard unconstrained optimization

by making use of Lagrange multipliers and determining a critical point of the resulting function That means we only need to change the total membership of each point to all the groups That sum is not necessary equal to 1, but it can vary from 0 to 1 It is obvious from those formulae that, if a point has no meanings in a certain context, its contextual value fk will be equal to 0, and it plays no role in re-manipulating the positions of centers and the membership measures The target function of the algorithm remains unchanged

We have known that standard FCM is a direction-free construction, which means it is regardless if a dimension is an input or an output variable This may lead to a not very reasonable distribution of prototypes or centers of groups in that the algorithm sweeps over even unrelated areas in data space In

a specific case, the context-sensitive FCM (CFCM) allows us to concentrate the classification into a subspace due to conditions of some dimensions showed in defined context Only a subset of original dataset which has considerable meaning to the context is helpful in the algorithm

3 2-CONTEXT FUZZY CLUSTERING

Sometimes we need to concentrate to 2 more important variables of the set of antecedent attributes of fuzzy rules The following is a 2-context fuzzy clustering algorithm (2C-FCM) for classification the data set  n

k

k x R

x :  Choose 2 context variables, then two maps A and B are defined on subspaces Y and Z as follow:

A: Y → [0,1]

yk  f1k = A(yk) (3) and B: Z → [0,1]

zk  f2k = B(zk) (4) The objective function is:

2

1 1

m

 

The 2-context fuzzy clustering algorithm (2C-FCM)

The algorithm consists of 5 steps:

1 Use FCM with the first choosen context variblable to

classify into C 1 clusters with the objective function:

1 1

m

 

 

where xk R with k=1, , N are data values of the

first context variable The first resulting partition matrix is:

1

1

{ (0,1) : 1, 1, , }

C

j



    

The output of this step are the matrix U and C 1 cluster

centers in R

2 Calculate f k as

kj

j

k u

f )

1 , k=1, ,N and j=1, C 1 (5)

3 For each context value )

1

j k

f with j=1, C 1 , use CFCM

to classify the second context choosen variable into C 2

clusters according to the objective function:

1 1

m

 

 

Where xk R with k=1, , N are data values of the

second context variable The second resulting partition matrix

is:

( )

1

N

j

j



4 Define the context values for the choosen second

attribute:

( )

2

j

for k=1, ,N and j=1, ,C2 (6)

2

1 C

1 2 1, ,

For each context values f2l k), use CFCM to classify

the remains data set into C groups with

2

1 1

m

 

 n

k R

x }, which are the data values according to

n-2 corresponding attributes The partition matrix is:

( ) 2 1

C

l

s



In the final step we receive C 1 *C 2 *C clusters according to

C 1 *C 2 contexts from 2 choosen context variables The final partition matrix is:

1 2

1

{ (0,1) : 1, 1, , }

j

 



     

As we can see, the level of details in knowledge-based clustering is enhanced by making use of 2 context variables For example, some countries having ‘high’ GDPPC and ‘high’ EI will be listed in 2C-FCM algorithms whilist the information of

‘high’ GDPPC or ‘high’ EI is shown in CFCM Therefore, CFCM is sometimes called one-context FCM (1C-FCM) Although the remains data set except 2 context data values contribute less importance, however, in any case, we have to count these data with premise that every data have some relationships with given 2 context variables

4 PARALLEL 2-CONTEXT FCM

The Two-Context Fuzzy Clustering Method (2C-FCM) increase the level of details in comparison with traditional CFCM However, the computation time of this algorithm also increase as a result, due to dealing with one more variable context Basically, each step in 2C-FCM uses CFCM or FCM as

a tool to classify specific data values Assume that the complexity

of CFCM and FCM in this scene are the same Therefore, we have total evaluation of 2C-FCM algorithms:

Step 1: one time uses FCM Step 3: C1 times use CFCM Step 5: C 1 C2 times use CFCM The complexity of general FCM (CFCM) is O(n4) Consequently, the complexity of 2C-FCM is equivalent to O(n6) Although we take advantage of knowledge-detail, however, the velocity of 2C-FCM is a big ostacle when the size

of data set is relative large For example, in stock market where there are a lot of shareholders and transactions in a short time; the classification is really difficult! Until now, we have not found

an optimal solution for this case

Thanks to the invention of supercomputer and especially parallel computation, the answer for this question has been solved Here is the parallel solution for 2-context fuzzy clustering method

The parallel 2-context fuzzy clustering algorithm (P2C-FCM)

The algorithm consists of 4 steps:

1 Use FCM with the first choosen context variblable to classify into C 1 clusters with the objective function:

1 1

m

 

 

where xk R with k=1, , N are data values of the

first context variable The output of this step are the matrix

U 1 and C 1 cluster centers in R:





















N k u u

U

C

j kj

kj 0 , 1 : 1 , 1 , ,

1

1 1

) , , 1



2 Supposed that the number of processors are h, split the

matrix U 1 and C 1 cluster centers according to this

figure Indeed, the number of context values and cluster

centers per processor is equivalent to the quotient of C 1

and h Moreover, some first processors have to

undertake extra context values and cluster centers

depending on the surplus of this number This procedure

can be exemplified by the following pseudo-codes:

int NumRows = C 1 / h;

int Surpluses = C 1 % h;

int pos = 1;

For each processor ID:

- Calculate the number of data which will be sent

to processor ID: int NumData = (ID <=

Surpluses)?NumRows+1: NumRows;

- Send to processor ID the position of rows in

original matrix as well as some parts of matrix

U 1 and C 1 cluster centers according to NumData

- Update the position of rows by adding NumData

to it

3 For each processor, we perform the same steps of

2C-FCM method from step 3 to 5 with data assigned to it

Finally, in this processor, we will receive L*C 2 *C

clusters according to L*C 2 contexts from 2 choosen

context variables with L is the number of context values

and cluster centers in this processor

4 After all processors successfully compute, gather and

connect results into single output

- Each processor sends its ID and the position of

rows in original matrix (pos)

- The Master process relies on these above numbers

to connect clusters on all processors

- The final output is a file containing some cluster

centers and partition matrices

Obviously, the velocity of the algorithm increase due to

simultaneous actions from different processes The accuracy of

results is similar to those in 2C-FCM because all processes

perform exactly same routines with origin 2C-FCM, but for a

smaller data set This procedure is useful in large data set where

the size can be thousands or even millions

5 EVALUATION

The 2C-FCM and P2C-FCM methods are implemented in C and MPI/C respectively and executed on a Linux Cluster 1350 with eight computing nodes of 51.2GFlops Each node contains two Intel Xeon dual core 3.2GHz, 2GB Ram We begin the experiment on the dataset of UNO in 2005 This scenario consists of 3 parts: 1) Does 2C-FCM really bring more information than CFCM? 2) What is the optimal number of processors used in P2C-FCM? 3) Compare the velocity of CFCM, 2C-FCM and P2C-FCM algorithms

First, we use 2C-FCM to classify UNO dataset folowing by

2 context variables: Education Index- EI and GDP Per capita- GDPPC The number of clusters with respect to each context variable is 2 namely as “high EI”, “low EI” for EI context variable, “high GDPPC”, “low GDPPC” for GDPPC and “high”,

“low” for remain dataset The following figure shows the membership degree of all countries in the dataset to two first groups

For “high EI” countries, there is clear distinction between

“high GDPPC” and “low GDPPC” groups It is illustrated by the figure:

Even in “high EI” and “high GDPPC”, there still exists 2 groups: “high” and “low”

Figure 2 The classification of second context variable

Figure 1 2C-FCM for 2 first groups

Figure 3 The classification of remain dataset

Figure 5 The comparison of CFCM, 2C-FCM, P2C-FCM

Figure 4 Speed up and Efficiency on Cluster 1350

In CFCM, the return result is Figure 1 only while we

receive 23 - 1 figures in 2C-FCM method Obviously, more

information can be obtained in 2C-FCM in comparison with

CFCM

In what follows, we study the relation between the total

(CPU) time spent on the performance of a program, the speedup

and the efficiency of this performance The speedup of the

performance is defined as S = Ts/Tp, where Ts (Tp) is serial

execution time (parallel execution time), respectively The

efficiency of the performance is determined as E = S/P, where P

is the number of processors The result of an experiment with

P2C-FCM method on the dataset of UNO 2005 is reported in the

following figure

Normally, the running time of algorithm will be reduced

when the number of processors increases However, Figure 4

shows different results as stated in Efficiency line In this case,

the optimal number of processors is 4 and we will obtain best

results of Speed up and Efficiency in this number If more

processors are given, then more time we have to spend for

interaction and synchronization between processors As a result,

the running time of P2C-FCM will increase

Third, we compare the velocity of CFCM, 2C-FCM and P2C-FCM following by number of elements Notice that although the number of elements seem small, the other parameters such as the number of groups (C1, C2, C), the dimension of data X and

the accuracy or threshold  are quite large, for example, C1 = C2

= 15, C = 20, dimension r = 10 and  = 10-6 This makes our dataset big enough to test

Figure 5 shows that if we want to receive more information, the running time of 2C-FCM will be higher than the one in CFCM However, the parallel solution of 2C-FCM can help us overcome this situation It can both increase the knowledge in clustering and reduce the running time Futhermore, in most case, the P2C-FCM is even quicker than CFCM This shows the effectiveness of 2C-FCM and P2C-FCM in comparison with traditional CFCM

6 AN APPLICATION OF CONTEXT FUZZY CLUSTERING METHODS FOR GENERATING FUZZY RULES

6.1 Problem

There are many ways to generate fuzzy rules from a database, depending on specific tasks of building rules In this section, we illustrate the procedure using the statistics of the United Nation Organization in order to evaluate the comprehensive development level, especial the Human Development Index (HDI)

Every year, UNO collects more than 100 figures of 31 fields from 177 countries The highest aim of all the effects and activities of this organization is to upgrade the quality of human livings The most important general index to appraise that progress is the HDI This index is calculated through analytic functions of longevity index, education index and GDP (Gross Domestic Products) index and many others parameters In this study, we try to rule out some relations showing the affection of some attributes of 100 parameters that we most appreciate to HDI They are GDPPC, Life expectancy, Enrollment ratio, Number of telephones every 1000 people, and Public expenditure

on health In fact, these values are mutually dependent on each

others and on other factors of social economy and major forms of

services However, these 5 parameters say differently in different

groups of countries This is the point that we can exploit to apply

“context” concept As mentioned in the Introduction section,

“Groups of countries” are defined relatively, basing mainly on

annual GDPPC For example, if the ratio of illiteracy is

“medium”, level of internet popularization is “low”, etc , and

the expense for public education and public health are

“medium”, then, in the context of a developed country, HDI is

surely not “high”

Its reason is may be that the country does not attend to the

real quality of livings, or gets high incomes just due to extra

interest from possession of certain resources But if a developing

or underdeveloped country has the same set of statistics, its HDI

may be “fair” As usual, HDI is scaled to the unit interval [0,1]

In fact, strong developed countries generally have great

GDP, and hence have virtual conditions to upgrade the human

life Countries with high GDP and GDP per capita often have

high HDI However, the range of GDP per capita is rather wide;

the minimum is 561 USD, while the maximum reach 69961

USD This is a very important index and is the most dispersive

attribute among indices Thus, we can consider GDP per capita

(GDPPC) our first context variable

6.2 A Procedure To Find Fuzzy Rules

1 Choose some antecedent attributes for the rules Choose

the context variable and also the second important attribute for

the specific task

2 Use the context fuzzy clustering algorithms to the data set

which is constrained by the choosen attribues Receive the

corresponding partition matrix

3 Use FCM algorithm to other antecedent attributes

Receive the corresponding partition matrixes

4 Use FCM algorithm to the consequent attribute Receive

the consequent partition matrix

5 Use the received partition matrixes to calculate the rule

weight If the rule weight is no less than the given threshold, the

rule should be accepted

6.3 An initial experiment result

In this part, we illustrate the procedure using the statistics

of the United Nation Organization in order to evaluate the

comprehensive development level, especially the Human

Development Index (HDI) With initial experiment we limit the

calculation in a small set of following attributes:

1 Choose 4 antecedent attributes: GDPPC, Education Index

- EI, Longevity Index – LI, HIV Prevalence (% agee 15-49) –

HIV In these attributes, GDPPC is the context variable, EI is the

second important variable

2 Use the context clustering algorithms to classify the data

according to above two variables: (GDPPC, EI) into 6 clusters:

(high, high), (high, medium), (high, low), (low, high), (low,

medium), ( low, low), with the partition matrix is obtained

3 Use the FCM to classify the attributes LI, HIV and HDI

to 4 clusters: high, fair, medium, low We receive the

corresponding partition matrixes U

We calculate the rule weight from these partition matrixes

by using t-norms with memberships degrees of corresponding propositions Notice that a t-norm is a function T: [0,1] x [0,1]  [0,1] which satisfies the folowing properties:

 Commutativity: T(a,b) = T(b,a)

 Monotonicity: T(a,b) ≤ T(c,d) if a ≤ c and b ≤ d

 Associativity: T(a, T(b,c)) = T(T(a,b),c)

 The number 1 is the identity element: T(a,1) = a

An example

1 For Vietnam, from the partition matrix of (GDPPC, EI )

we have vector (0.000337, 0.000592, 0.0046999, 0.6138982,

0.35477337, 0.0256996) The membership degree 0.6138982 is

choosen to proposition (GDPPC, EI) is (low, high) The

membership degree 0.35477337 is choosen to proposition

(GDPPC, EI ) is (low, medium)

From the partition matrix of LI, we have vector (0.0032293,

0.0848, 0.913175, 0.0751175) The membership degree

0.913175 is choosen to proposition LI of Vietnam is medium

From the partition matrix of HIV Prevalance, we have vector (5.483 E-0.6, 1.6769E-0.5, 0.0004055, 0.9998) The

membership degree 0.9998 is choosen to proposition HIV

Prevalance of Vietnam is low

2 For Vietnam, from the partition matrix of consequent

attribute HDI we have vector: (0.0033686, 0.919263, 0.035644, 0.011406851) The membership degree 0.919263 is choosen to

proposition of consequent attribute: HDI of Vietnam is fair

To evaluate the comprehensive development level, especially the Human Development Index, we can generate the following fuzzy rules for Vietnam:

Rule 1: IF (GDPPC, EI ) is (low, high) and LI is medium and HIV is low THEN HDI is fair

If we use t-norm T(x,y) = min(x,y), we have: min (0.6138982, 0.913175, 0.9998, 0.919263) = 0.6138982 is the rule weight of Rule 1

Using t-norm T(x,y) = x.y, the weight of Rule 1 is 0.51523254

Rule 2: IF (GDPPC, EI) is (low, medium) and LI is medium and HIV is low THEN HDI is fair

If we use t-norm T(x,y) = min(x,y), we have: min (0.35477337, 0.913175, 0.9998, 0.919263) = 0.35477337 is the rule weight of Rule 2

Using t-norm T(x,y) = x.y the weight of Rule 2 is 0.29773548

3 Analogously, we can have the following fuzzy rules for South Africa From the partition matrix of (GDPPC , EI) we have vector (0.0071097, 0.0120878, 0.0764525, 0.465276,

0.41321257, 0.0258705) The membership degree 0.465276 is

choosen to proposition (GDPPC, EI ) is (low, high) The

membership degree 0.41321257 is choosen to proposition

(GDPPC, EI) is (low, medium)

From the partition matrix of LI, we have vector (0.004180,

0.009585, 0.151981, 0.834259) The membership degree

0.834259 is choosen to proposition LI of South Africa is low

From the partition matrix of HIV Prevalance, we have

vector (3.15609 E-0.5, 0.9992393, 2.729E-0.5, 1.72165E-0.5)

The membership degree 0.9992393 is choosen to proposition

HIV Prevalance of South Africa is fair

4 For South Africa, from the partition matrix of HDI, we

have vector (0.079156, 0.531274, 0.319693, 0.068976782) The

membership degree 0.531274 is choosen to proposition of

consequent attribute: HDI of South Africa is fair

To evaluate the comprehensive development level,

especially the Human Development Index (HDI), we can

generate the following fuzzy rule for South Africa:

Rule 1: IF (GDPPC, EI) is (low, high) and LI is low and

HIV is fair THEN HDI is fair

If we use t-norm T(x,y) = min(x,y), we have: min

(0.465276, 0.834259, 0.9992393, 0.531274) = 0.465276 is the

rule weight of rule 1 for South Africa

Using t-norm T(x,y) = x.y the weight is 0.20163398

7 CONCLUSION

This paper focuses on the classification problem and

knowledge-based fuzzy clustering methods Whilst traditional

Context FCM concentrates on a context variable, this paper has

presented the 2-context fuzzy clustering method (2C-FCM)

which broaden the number of context variables in order to extract

more meaningful information Besides, we also introduce a

parallel solution for 2C-FCM aiming to increase the velocity and

effectiveness of this algorithm These methods are carefully

tested through strictly evaluation and applied to generate fuzzy

rules on the classification problem using the data set of UNDP

On the other hands, 2C-FCM and P2C-FCM rely on original

CFCM and traditional FCM Therefore, all disadvantages of

FCM algorithm can be utilized to slow down these above

methods Some intensive researches on FCM ‘s weakness and

innovative methods such as Fast Generalized Fuzzy C-means

methods (FGFCM), FCM with spatial constraints (FCM_S),

can be found in [9] and other literatures

In the future, we will concentrate on the ameliorative solutions of FCM and CFCM algorithms in order to enhance the speed of our algorithms Moreover, we will try to adapt new local parallel method to 2C-FCM for better solutions Some findings

on extracting new fuzzy rules from 2 other context variables having specific meanings are also considered

8 ACKNOWLEDGEMENT

Some calculating results in Section 6.3 are obtained by Nguyen Quang Minh, Ha Noi University of Technology

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