Paper id 26201424

International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637

A New Kernelized Fuzzy C-Means Clustering Algorithm with Enhanced Performance Samarjit Das1, Hemanta K. Baruah2 1

Department of Computer Science &IT, 2Vice-Chancellor 1 Cotton College, Assam, India 2 Bodoland University, Assam, India 1 ssaimm@rediffmail.com, 2 hemanta_bh@yahoo.com

Abstract- Recently Kernelized Fuzzy C-Means clustering technique where a kernel-induced distance function is used as a similarity measure instead of a Euclidean distance which is used in the conventional Fuzzy C-Means clustering technique, has earned popularity among research community. Like the conventional Fuzzy C-Means clustering technique this technique also suffers from inconsistency in its performance due to the fact that here also the initial centroids are obtained based on the randomly initialized membership values of the objects. Our present work proposes a modified method to remove the effect of random initialization from Kernelized Fuzzy C-Means clustering technique and to improve the overall performance of it. In our proposed method we have used the algorithm of Yuan et al. to determine the initial centroids. These initial centroids are then used in the conventional Kernelized Fuzzy C-Means clustering technique to obtain the final clusters. We have also provided a comparison of our method with the Kernelized Fuzzy C-Means clustering technique of Hogo using two validity measures namely Partition Coefficient and Clustering Entropy.

Keywords: kernel-induced distance function, Random initialization, Partition Coefficient, Clustering Entropy.

1. INTRODUCTION Clustering is a technique which helps us to reveal the inherent grouping structure of data in an unsupervised manner. The conventional hard clustering techniques can not deal with the situations pertaining to non-probabilistic uncertainty. Hard clustering techniques are based on crisp set theory and therefore there is no possibility of partial belongingness of objects to multiple clusters. In other words the clusters revealed by a hard clustering technique are disjoint i.e. an object of a dataset, after the application of a hard clustering technique, either belongs totally to a particular cluster or does not belong to that cluster at all. The concept of partial belongingness was first introduced by Zadeh (1965) in his famous fuzzy set theory (FST). A complete presentation of all aspects of FST is available in the work of Zimmermann (1991). The applications of FST in dealing with ambiguous problems where nonprobabilistic uncertainty prevails have been reflected in the works of Dewit (1982) and Ostaszewski (1993).

Baruah (2011a, 2011b) has introduced a new approach to FST where he has justified that the membership value of a fuzzy number can be expressed as a difference between a membership function and a reference function and therefore the membership value and the membership function for the complement of a fuzzy set are not same. With the advent of FST, the conventional hard clustering techniques have unlocked a new way of clustering, known as fuzzy clustering, where due to the existence

of the concept of degree of belongingness, an object may belong exactly to one cluster or partially to more than one clusters depending on its membership value.

In the literature, out of the different available fuzzy clustering techniques the Fuzzy C-Means clustering technique (FCM) of Bezdek (1981) has been found to be widely studied and applied. Derrig and Ostaszewski (1995) have applied the FCM of Bezdek in their research work where they have explained a method of pattern recognition for risk and claim classification. Das (2013) has tried the fuzzy c-means algorithm of Bezdek with three different distances namely Euclidean distance, Canberra distance and Hamming distance which revealed that out of the three distances, the algorithm produces the result fastest as well as the most expected when Euclidean distance is considered and the slowest as well as the least expected when Canberra distance is considered.

Das and Baruah (2013) have shown the application of Bezdek’s (1981) FCM clustering technique on vehicular pollution, through which they have discussed the importance of application of a fuzzy clustering technique on a dataset describing vehicular pollution, instead of a hard clustering technique.

Although in most of the situations it is evident that the FCM clustering technique performs better than other fuzzy clustering techniques, due to the random initialization of the membership values the performance of FCM clustering technique varies significantly in its different executions. Yager and Filev (1992) proposed a simple and effective method,

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 called the Mountain Method, for estimating the number and initial location of cluster centers.

Although this technique, unlike the FCM clustering technique, did not depend on any randomly initialized membership value to estimate the initial locations of cluster centers, the problem with this method, mountain clustering, was that its computation grew exponentially with the dimension of the problem.

Chiu (1994) developed a new method called Subtractive Clustering (SC) with which he could solve this problem by using data points as the candidates for cluster centers instead of grid points as in the Mountain Clustering. Das and Baruah (2014b) have proposed a new method, named as SUBFCM, where they have first applied the Subtractive clustering technique of Chiu (1994) to find out the initial cluster centers and then used these initial cluster centers in FCM clustering technique of Bezdek (1981) for obtaining the final cluster centers along with the membership values of objects in different clusters. Das and Baruah (2014b) have justified that through the SUBFCM method not only the effect of randomness can be removed from the FCM of Bezdek (1981) but also the situations where the number of clusters is not predefined can properly be dealt with.

In addition to that the performance of SUBFCM is much higher than that of Subtractive clustering technique of Chiu (1994). Yuan et al. (2004) proposed a systematic method for finding the initial centroids where there is no scope of randomness and therefore the centroids obtained by this method are found to be consistent. Das and Baruah (2014c) proposed a method where they have used the algorithm of Yuan et al. (2004) as a preprocessor to FCM of Bezdek (1981) to remove the effect of random initialization from FCM and also to improve the overall performance of it. Das and Baruah (2014c) have justified that although the average performance level of their proposed method is higher than that of FCM of Bezdek (1981), it is advisable to optimize the performance level of their method with the best choice of the multiplication factor. In the recent past kernel methods (Carl (1999), Muller et al. (2001)) have earned popularity especially in the machine learning community and have widely been applied to pattern recognition and function approximation. A Kernel-based Fuzzy C-Means (KFCM) clustering technique is generally derived from the conventional FCM by using a kernel induced distance function instead of a Euclidean distance. Hogo (2010) proposed a KFCM algorithm where he used the Gaussian Radial Basic Function as a kernel induced distance function. His work presented the use of FCM and KFCM to find the learners’ categories and to predict their behaviour which could help the decision makers in e-learning system. Das and Baruah (2014a)

have made a comparison of FCM and KFCM clustering techniques through which they have shown that the performances of both FCM and KFCM vary significantly with randomly initialized membership values. They have also found that the performance of KFCM depends on the value of σ of Gaussian Radial Basic Function used in it as a distance function. Das and Baruah (2014a) have justified that the best choice of σ in KFCM provides better performance than FCM, however, with random value of σ the performance of KFCM may not always be better than that of FCM. Although KFCM is an attempt to improve the performance of FCM, the effect of randomness is still there in KFCM and as a consequence of which the performance of KFCM is found to be inconsistent. Looking into the inconsistent behaviour of KFCM due to the effect of randomness and the inability of KFCM to deal with the situations where the number of clusters is not predetermined, Das and Baruah (2014d) have proposed a method namely SUBKFCM where they have applied the Subtractive clustering technique of Chiu (1994) as a preprocessor to KFCM clustering technique with an intention to take care of both of these limitations of KFCM. This SUBKFCM algorithm of Das and Baruah (2014d) has the capability of handling these two limitations of KFCM in a much better way than that of SC algorithm, although in most of the cases it shows slightly poor performance than KFCM. Our present work proposes a new algorithm to remove the effect of randomness on KFCM and also to achieve improved performance of it. In our proposed work we apply the algorithm of Yuan et al. (2004), which uses data points to find the initial centroids, as a preprocessor to KFCM of Hogo (2010). We have also provided a comparison of our method with the KFCM clustering technique of Hogo (2010) using two validity measures namely Partition Coefficient (PC) and Clustering Entropy (CE) (Bezdek (1981) and Bensaid et al. (1996)).

In section-2 we define the problem of our present work. The mathematical calculations, algorithms used in our present work have been placed in section-3. We shall provide the results and analysis in section-4.

Finally we put the conclusions in section-5.

2. PROBLEM DEFINITION Two key limitations of KFCM clustering technique are – the inconsistency in its performance due to random initialization of objects and its inability to deal with the situations where the number of clusters is not predetermined. These two limitations of KFCM can be overcome by SC algorithm of Chiu (1994) which relies on the data points to find the initial cluster centers. But the problem with SC algorithm is that its performance is very low. Looking

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 into these limitations of KFCM and the poor performance of SC, Das and Baruah (2014d) have introduced an algorithm, namely SUBKFCM, which is capable of handling these two limitations of KFCM in a much better way than that of SC algorithm, although in most of the cases it exhibits slightly poor performance than KFCM. To remove the effect of randomness on KFCM and also to achieve improved performance of it, we propose a new algorithm namely YKFCM, in our present work, where we apply the algorithm of Yuan et al. (2004), which relies on data points in finding its initial cluster centers, as a preprocessor to KFCM algorithm of Hogo (2010).

Step 2: Calculate the fuzzy cluster centroids

{vi(l ) }i =1, 2,.....,c

given by the following formula n

vi

(l )

=

∑ (µ k =1 n

(l )

( xk )) m xk

si

(2)

∑ (µ k =1

(l ) si

for i = 1, 2 , ….. c;

( xk ))

m

k= 1, 2, , …..n.

Step 3: Calculate the new partition matrix (i.e. membership matrix)

U (l +1) = [ µ si

( l +1)

( xk )]1≤i ≤c ,1≤ k ≤ n ,

Where

3. OUR PRESENT WORK Before discussing our proposed algorithm we shall first discuss the algorithms which have been used, directly or indirectly, in our present work. In Section3.1 we shall describe the FCM of Bezdek (1981) followed by KFCM of Hogo (2010) and the algorithm of Yuan et al. (2004) in section-3.2 and section-3.3 respectively. Finally we illustrate our algorithm, YKFCM, in section-3.4.

1 2 (l ) || xk − vi || m −1 ( ) ∑ (l ) || j =1 || xk − v j

µs (l +1) ( xk ) =

(3)

c

i

for i=1,2,……..,c and k=1,2,……..,n.

If

xk = vi , formula (3) cannot be used. In this case (l )

the membership function is

µ s ( l +1) ( x k ) = {10ifkifk=≠ii ,i =1, 2,....,c i

Step 4: Calculate

3.1. Bezdek’s FCM Algorithm

( l +1)

Step 1: Choose the number of clusters, c, 2≤c<n, where n is the total number of objects or feature vectors in the dataset. Choose m, 1≤ m <α. Define the vector norm || || (generally defined by the Euclidean distance) i.e.

|| xk − vi || =

p

∑ (x j =1

kj

− vij ) 2

(1)

where xkj is the jth feature of the kth feature vector, for k=1,2,……,n; j=1,2,….,p and vij , j-dimensional centre of the ith cluster for i=1,2,……,c; j=1,2,….,p; n, p and c denote the total number of feature vector , no. of features in each feature vector and total number of clusters respectively.

Choose the initial fuzzy partition (by putting some random values)

U (0) = [ µ si ( xk )]1≤ i ≤ c ,1≤ k ≤ n (0)

Choose a parameter ∈>0 (this will tell us when to stop the iteration). Set the iteration counting parameter l equal to 0.

∆ = || U − U || (4) If ∆ >∈, repeat steps 2, 3 and 4. Otherwise, stop at (l )

*

some iteration count l .

3.2. The KFCM Algorithm Kernelized Fuzzy C-Means (KFCM) algorithm has been derived from traditional FCM of Bezdek (1981) by changing the vector norms, fuzzy cluster centers and the partition matrix as described in the following.

In step1 of KFCM the vector norm is defined by Gaussian Radial Basic Function (GRBF) (see equation (5)) instead of Euclidean distance (see equation (1)) of FCM.

k ( xr , vi ) = exp(

− || xr − vi || 2

σ2

)

(5)

In step2 of KFCM the fuzzy cluster centers are calculated using equation(6) instead of equation(2) of FCM. n

vi

(l )

=

∑ (µ r =1 n

(l ) si

∑ (µ r =1

( xr )) m k ( xr , vi ) xr (6)

(l ) si

m

( xr )) k ( xr , vi )

for i = 1, 2 , ….. c; r= 1, 2, , …..n.

In step 3 of KFCM the new partition matrix is calculated by using equation (7) instead of equation

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 (3) of FCM.

µs (l +1) ( xr ) = i

(1 − k ( xr , vi )) (l )

c

∑(1 − k ( x , v j =1

r

(l ) j

−1 ( m−1)

))

−1 ( m−1)

(7)

for i=1,2,……..,c and r=1,2,……..,n.

Remaining steps and calculations of KFCM are kept same as those of FCM. 3.3. Algorithm of Yuan et al.

Step 1: Set m =1; Step 2: Compute the distance between each data point and all other data points in the set X; Step 3: Find the closet pair of data points from the set X and form a data point set Am (1≤m≤c, c is the number of clusters) which contains these two data points, delete these two data points from the set X; Step 4: Find the data point in X that is the closet to the data point set Am, add it to Am and delete it from X; Step 5: Repeat step-4 until the number of data points in Am reaches 0.75*(n/c); (where .75 is a multiplication factor (MF)) Step 6: If m<c, then m = m+1, find another pair of data points from X between which the distance is shortest, form another data point set Am and delete them from X, go to step-4; Step 7: For each data point set Am (1≤m≤c) find the arithmetic mean of the vectors of data points in Am, these means will be the initial centroids.

3.4. Our proposed algorithm Step 1: Normalize the data points so that these are bounded by a hypercube.

Step 2: Find the initial centroids by the algorithm of Yuan et al. (see section-3.3)

Step 3: Take these centroids obtained in step-2 as input to KFCM. Step 4: Calculate the membership values of the objects by KFCM (see equation (7) of section-3.2) Step 5: Calculate the value of ∆. (see equation (4) of section-3.1) if (∆ > ∈) repeat steps-6, 7 and 5 else stop iteration.

Step 6: Update the centroids by KFCM. (see equation (6) of section-3.2) Step 7: Update the membership values by KFCM. (see equation (7) of section-3.2) The pictorial representation of the above algorithm has been provided in Figure-1. We have applied each of KFCM clustering algorithm and our proposed algorithm, YKFCM ten (10) times on the same dataset (see table1) and tried to make a comparison of the performances of these two clustering techniques. We have used two validity measures namely Partition Coefficient (PC) and Clustering Entropy (CE) (Bezdek (1981) and Bensaid et al. (1996)) and also the number of iterations to compare the performances of these two clustering techniques. The mathematical formulae of these two validity measures have been given in the following.

Partition Coefficient (PC): overlapping between clusters.

PC (c ) =

measures

the

1 c n ( µ ij ) 2 ∑∑ n i =1 j =1

Clustering Entropy (CE): measures the fuzziness of the cluster partition

CE (c ) =

1 c n ∑∑ µ ij log( µ ij ) n i =1 j =1

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637

Start

Read the normalized Dataset

Find the initial centroids by the algorithm of Yuan et al.

Take these centroids as the input to KFCM and calculate the membership values of the objects by KFCM.

Calculate ∆

Yes

if(∆ >∈)

No

Update the centroids by KFCM.

Update the membership values by KFCM.

Stop

Figure 1. Flowchart of our proposed model.

The dataset of our present work consists of fifty(50) Feature Vectors (FV) each of which is of dimension three(03) namely Intelligent Quotient (IQ) , Achievement Motivation(AM) and Social

Adjustment (SA). The numerical values of our dataset have been given in the following table.

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 Table 1. Data set of individual differences of fifty (50) feature vectors with dimension (feature) three (03).

FV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

IQ 91 85 120 90 92 82 95 89 96 90 97 125 100 90 100 95 130 130 90 91 140 92 101 85 97

AM 18 16 19 18 17 17 19 18 19 17 16 21 19 17 18 19 23 19 17 17 22 18 18 16 19

SA 55 40 74 75 74 55 75 74 75 55 54 74 75 54 84 75 85 75 55 56 82 75 55 54 54

4. RESULTS AND ANALYSIS We provide the results and analysis of our present work in this section. First we have applied the KFCM clustering algorithm of Hogo (2010) on a dataset (see table-1) which had been normalized so that it is bounded by a hypercube, by predetermining five (05) numbers of clusters. We applied this algorithm for ten (10) different random initialization of the membership values of the objects in the dataset. For each random initialization we have used three (03) different values of the adjustable parameter,σ, of the Gaussian Radial Basic Function to find the optimized result out of the three. In table-2 we provide the optimized values of different validity measures for the best choice among three (03) different values of σ of KFCM algorithm obtained in ten (10) different executions. Next we have applied our algorithm i.e. YKFCM on the same normalized dataset. We obtained the optimized results for the best choice of σ and Multiplication Factor (MF) and performed the same experiment for ten (10)

FV 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

IQ 110 100 100 70 105 79 80 125 100 125 80 85 145 80 92 120 145 95 80 90 115 100 80 105 120

AM 18 16 18 14 17 14 15 20 19 19 18 18 25 18 17 18 30 18 16 17 23 18 14 19 21

SA 55 40 75 30 55 35 34 75 75 85 60 70 90 74 55 70 80 50 36 55 84 80 35 75 74

times. The optimized values of different validity measures for the best choice of σ and MF of YKFCM algorithm obtained in ten (10) different executions have been given in table-3. Based on the values of the different validity measures (see tables – 2 and 3) obtained by these three different algorithms we have tried to make a comparison of these two algorithms.

In Figure-2 we see that there is significant variation of the validity measure PC obtained by KFCM algorithm in ten (10) different executions.

This is obviously due to the fact that the membership values of the objects had been initialized randomly to find the initial cluster centers. On the other hand there is no variation of the validity measure PC obtained by YKFCM algorithm in ten (10) different executions.

This consistent performance of YKFCM algorithm has become possible only because of the absence of randomness in it while finding the initial centroids.

Moreover in Figure-2 the line of performance of YKFCM algorithm is seen to be above (higher values of PC) that of KFCM. Figure-3 reveals that the

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 validity measure CE of KFCM varies significantly in ten (10) different executions, however no such variation is there in case of YKFCM. We also see in Figure-3 that YKFCM exhibits better performance (lower values of CE) than KFCM. In Figure-4 we see that KFCM needs different number of iterations to reach the final clusters in ten (10) different

executions. However it is seen here that YKFCM needs eight (08) iterations to accomplish the same in all the ten (10) executions proving it to be a consistent as well as fast executing algorithm when compared with KFCM.

Table 2: Optimized values of different validity measures for the best choice among three (03) different values of σ of KFCM algorithm obtained in ten (10) different executions.

KFCM ALGORITHM Itn PC RUN1 11 0.726 RUN2 8 0.683 RUN3 12 0.724 RUN4 9 0.682 RUN5 11 0.725 RUN6 10 0.682 RUN7 9 0.723 RUN8 9 0.683 RUN9 11 0.724 RUN10 12 0.725

CE 0.578 0.657 0.585 0.658 0.58 0.659 0.59 0.657 0.584 0.578

Table 3: Optimized values of different validity measures for the best choice among three (03) different values of σ & MF of YKFCM algorithm obtained in ten (10) different executions.

RUN1 RUN2 RUN3 RUN4 RUN5 RUN6 RUN7 RUN8 RUN9 RUN10

YKFCM ALGORITHM Itn PC 8 .727 8 .727 8 .727 8 .727 8 .727 8 .727 8 .727 8 .727 8 .727 8 .727

CE 0.576 0.576 0.576 0.576 0.576 0.576 0.576 0.576 0.576 0.576

MF = Multiplication Factor

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 0.74 0.72 0.7 0.68

PC KFCM

0.66

PC YKFCM

0.64

Figure 2: Comparison of KFCM and YKFCM based on the validity measure PC in ten (10) different executions.

0.68 0.66 0.64 0.62 0.6 0.58 0.56 0.54 0.52

CE KFCM CE YKFCM

Figure 3: Comparison of KFCM and YKFCM based on the validity measure CE in ten (10) different executions.

15 10 Iteration KFCM

5

Iteration YKFCM

0

Figure 4: Comparison of KFCM and YKFCM based on the no. of iterations in ten (10) different executions.

5. CONCLUSIONS KFCM has two major limitations- existence of inconsistency due to randomly initialized membership values of the objects in the dataset and the inability to deal with the situations where the number of clusters is not predetermined. Our previous algorithm, namely SUBKFCM, is capable of handling the limitations of KFCM in a much better way than that of SC algorithm of Chiu, but it has slightly poor performance than KFCM. Our present algorithm, YKFCM, removes the inconsistency due to

randomness from KFCM and exhibits better performance than KFCM although it is unable address the second limitation of KFCM i.e. the number of clusters needs to be predetermined in YKFCM.

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 [2] Baruah, H.K.(2011b): The theory of fuzzy sets: beliefs and realities. International Journal of Energy, Information and Communications, 2( 2),pp. 1-22.

[3] Bensaid, A.M.; Hall, L.O.; Bezdek, J.C.(1996): Validity- guided (re) clustering with applications to image segmentation. IEEE Trans. on Fuzzy Object, 2(2), pp.112-123.

[4] Bezdek, J.C.(1981). Pattern recognition with fuzzy objective function algorithms, Plenum Press, New York.

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Journal of Intelligent and Fuzzy Systems, 2, pp. 267-278.

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[12] Das, S.; Baruah, H. K.(2014d): Towards Finding a New Kernelized Fuzzy C-Means Clustering Algorithm. Journal of Process Management – New Technologies, 2(2), pp. 54-65.

[13] Derrig, R. A.; Ostaszewski, K. M.(1995): Fuzzy techniques of pattern recognition in risk and claim classification. Journal of Risk and Insurance, 62(3), pp.447-482.

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[15] Hogo, M. A. (2010): Evaluation of E-Learners Behaviour using Different Fuzzy Clustering Models: A Comparative Study. International Journal of Computer Science and Information Security, 7(2), pp. 131-140.

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based learning algorithms. IEEE Trans Neural Networks, 12(2), pp. 181—202.

[17] Ostaszewski, K. (1993). An Investigation into Possible Applications of Fuzzy Sets Methods in Actuarial Science, Society of Actuaries, Schaumburg, Illinois.

[18] Yager, R.R.; Filev, D.P. (1992): Approximate Clustering Via the Mountain Method, #MII-1305 Tech. Rep., Machine Intelligence Institute, Iona College, New Rochelle, NY. [19] Yuan, F. ; Meng, Z.H.; Zhang, H.X.; Dong, C.R.(2004): A new algorithm to get the initial centroids, Proc. of the 3rd International Conference on Machine Learning and Cybernetics, pp. 26-29.

[20] Zadeh, L. A.( 1965): Fuzzy sets. Information and Control, 8(3), pp. 338-353.

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