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International Journal of Advances in Engineering & Technology, May, 2014.
©IJAET
ISSN: 22311963

PERFORMANCE ANALYSIS OF FUZZY CLUSTERING
S.P. Priyadharshini1 and Ramachandra V. Pujeri2
1
Asst, Prof (IT), Ayyan Thiruvalluvar College of Arts & Science, Bharathiar
University, Coimbatore, India
2
Vice Principal, KGISL Institute of Technology, KGISL campus, Coimbatore, India

ABSTRACT
Abstract - Clustering algorithms is a process of break up the data objects into numerous groups which is
similar. A K-means clustering algorithm is one of the most popular to used data objects into K cluster analysis.
Fuzzy C-Means clustering algorithm is fashionable clustering method is more efficient, straightforward, easy to
implement and sensitive to initialization. A demerit of K-Means and Fuzzy C-Means both algorithms are easily
falling in local optima. Fuzzy Particle Swarm Optimization algorithm helps to solve local optima. Experimental
results show that Fuzzy PSO approach giving highly competitive results for fuzzy clustering and FPSO
outperform is better-quality the performance of other to be had algorithms.

KEYWORDS: K-means, Fuzzy C Means, Fuzzy Particle Swarm Optimization

I.

INTRODUCTION

Cluster is the procedure of assigning data stuff to a set of disjoint groups call Clustering [6].
Clustering methods are functional in many submission areas such as pattern recognition, machine
learning etc. In expansive, clustering is an unsupervised learning task as very tiny or no previous
knowledge is given excluding input data sets.
The fuzzy C means algorithm introduced by Jim Bezdek m [8]. It is earliest and most trendy fuzzy
clustering algorithms. Fuzzy C means simple says FCM. Fuzzy C Means clustering algorithm is
narrow hunt algorithm because it’s sensitive to initial value effectively.
Particle swarm optimization (PSO) is a population-based optimization tool, which could be
implemented and practical easily to explain various task optimization problems, or the problems that
can be transformed to task optimization problems (Kennedy & Eberhart) [11]. Pang, Wang, Zhou, and
Dong, proposed a version of particle swarm optimization for TSP called fuzzy particle swarm
optimization (FPSO). In this paper, a hybrid fuzzy clustering algorithm based on FCM and FPSO
called FCM–FPSO is proposed. The experimental results over six real-life data sets indicate the
FCM–FPSO algorithm is superior to the FCM algorithm and FPSO algorithm [14].
On the next step move, particle swarm optimization algorithm is denoted by PSO. A modified particle
swarm optimization is also known as fuzzy particle swarm optimization (FPSO) proposed by peng.
We discuss in one of the most popular clustering algorithm which objects k clustering. Then FCM
compare PSO to get better result which is best. We discuss about iris dataset. Many papers are
describes panel clustering algorithms using squared metric for lone key many facts. Iris information
have lot of inputs to create algorithm for K means, Fuzzy C means compare with other two techniques
Particle Swarm Optimization is best.

II.

K- MEANS

K-means clustering algorithm is the most popular paneling cluster algorithms. In earlier days, many
papers are describes paneling clustering algorithms using squared error metric for lone key many facts

488

Vol. 7, Issue 2, pp. 488-494

International Journal of Advances in Engineering & Technology, May, 2014.
©IJAET
ISSN: 22311963
hyper cubes. The square error for each cluster is defined as the sum of square distance between each
member in the class and the center, called intra-distance. Therefore the point function in the square
error clustering algorithm is to minimize the sum of all intra-distances. Traditional sequential
algorithm starts with initial set of K clusters, and move each pattern (or data) to a cluster if it
minimizes the square error.

2.1. K-Means Algorithm
K-Means is one of the clustering techniques, which panel for each cluster is represented by the mean
values of the object in the cluster. The algorithm has as an input a predefined number of clusters that
is the K from its name. The K-Means stands for an average, an average location of all the members of
a particular cluster. The K-means Algorithm is an iterative procedure whereby final results depend on
the values selected for initial centroid.
A Centroid is an artificial point in the space of records which represents an average location of the
particular cluster. The coordinates of this point are averages of attribute values of all examples that
belong to the cluster. Decision useful model in large datasets has attracted considerable importance
freshly and of the most widely studied problems in this area is the identification of clusters or densely
populated regions, in a multi-dimensional dataset. The major problem with the K-mean algorithms is
that of choice of initial centroids.
Finally, this algorithm aims at minimizing a point function, in this case a squared error function. The
k

2

n

J   xi  c j
j

point function

(2.1)

j 1 i 1

distance measure between a data point

x

( j)
i

where

x

( j)
i

2

 c j is a chosen

and the cluster centre c j , is an indicator of the distance of

the n data points from their respective cluster centre.
ALGORITHM 1: K MEANS
Step 1: Set as a cluster center and select randomly k points
Step 2: Allot each point to the collect centroid
Step 3: Recalculate the position of centroid for all the point have been assigning
Step 4: Except the centroid are not changes go to step 2

2.2. Fuzzy C Means
The Fuzzy C-Means algorithm (often abbreviated to FCM) is an iterative algorithm that find clusters
in data and which uses the theory of fuzzy membership instead of assigning a pixel to a single cluster,
each pixel will have different membership values on each cluster.
Fuzzy c-means panels set of n objects   o1 , o2 ,...on  in R d dimensional space into

c(1 c  n) fuzzy clusters with z   z1 , z 2 ,...z n  cluster centers or centroids. The fuzzy clustering of
objects is described by a fuzzy matrix  with n rows and c columns in which n is the number of data
objects and c is the number of clusters. ij , the element in the ith row and jth column in  , indicates the
degree of association or membership function of the ith object with the jth cluster. The characters of μ
are as follows:

i 0,1 i  1,2...n j  1,2...c

c

µ

ij

 1 i =1, 2…n

(2.2)

j1

c

o   µij  n  j =1, 2…c

(2.3)

j 1

The point function of FCM algorithm is to minimize the Eq. (4):
c

Jm  
j 1

489

n


i 1

ij

dij

(2.4)

Vol. 7, Issue 2, pp. 488-494

International Journal of Advances in Engineering & Technology, May, 2014.
©IJAET
ISSN: 22311963

dij  oi  z j

Where

(2.5)

In which, m (m>1) is a scalar termed the weighting exponent and controls the fuzziness of the
resulting clusters and d ij is the Euclidian distance from object z  {z1 , z2 ,...., zn } to the cluster
center z j . The z j , centroids of the jth cluster, is obtained using below equation
n

z


i 1
n

j

i

m
ij

(2.6)

 ijm
i 1

ij 

1
 dij 

k 1  ik 
c

 d

2
m 1

(2.7)

The Fuzzy C-Means algorithm (FCM) is an iterative algorithm that finds clusters in data and which
uses the concept of fuzzy membership, [9] instead of assigning a pixel to a single cluster, each pixel
will have different membership values on each cluster. It panels set of n objects   o1 , o2 ,...on  in

R d dimensional [15] space into c(1 c  12,,... n) fuzzy clusters with z   z1 , z 2 ,...z n  cluster
centers.
The fuzzy clustering of objects is described by a fuzzy matrix  with n rows and c columns in which
n is the number of data objects and c is the number of clusters. The element in the ith row and jth
column in  , point out the degree of association or membership function of the i th object with the jth
cluster. The characters of  are as follows:
ALGORITHM 2: Fuzzy C Means
Step 1: Initialize the membership function values to be choose m (m>1) ij i=1,2,…n, j=1,2…c
Step 2: The Eq. (2.6) to Compute the cluster centers z j , j = 1,2,..., c
Step 3: Euclidian distance to work out d ij , i = 1,2,..., n; j = 1,2,..., c by using Eq. (2.5).
Step 4: Modernize the membership function µij , i = 1, 2... n; j = 1,2,..., c by using

ij 

1
 dij 

 
k 1  d ik 
c

2
m 1

(2.7)

Step 5: If not converged, go to step 2.

III.

PARTICLE SWARM OPTIMIZATION

Kennedy and Eberhart, introduced Particle Swarm Optimization (PSO) [11] which is population based
stochastic optimization technique moved by parts of body originally designed and introduced.
The PSO starts with a population of particles whose positions represent the potential solutions for the
studied problem and some human categories are two types male and female in those velocities are
randomly initialized in the search space. It searches optimal personal best position and global best
position. The personal best position, pbest, is the best position the particle has visited and gbest is the
best position the swarm has visited since the first time step. A particle’s velocity and position are
updated by using (3.1)

490

Vol. 7, Issue 2, pp. 488-494

International Journal of Advances in Engineering & Technology, May, 2014.
©IJAET
ISSN: 22311963
v t  1  wv
. t  .c1r1  pbest t    t    c2r2  gbest t    t   k=1,2…P

(3.1)

  t  1    t   v  t  1
X and V are position and velocity of particle respectively, w is inertia weight, c1 and c2 are positive
constants, called acceleration coefficients which control the influence of pbest and gbest on the search
process, P is the number of particles in the swarm, r1 and r2 are random values range between 0 and
1.

3.1. Fuzzy Particle Swarm Optimization
A particle swarm optimization with fuzzy set theory is called Fuzzy Particle Swarm Optimization
(FPSO), which is proposed by Peng [10] et al.,. Using fuzzy relation between variables, FPSO
redefines the position and velocity of particles. And it also applied for clustering problem. In this
method X is the position of particle, the fuzzy relation for the set of data objects,   o1 , o2 ,...on  to
set of cluster centers, z   z1 , z 2 ,...z n  , Can be expressed as follows
ALGORITHM 3: FUZZY PARTICLE SWARM OPTIMIZATION (FPSO)
Input: Dataset
Output: Intention Values
Step 1: Initialize the parameters including hand or leg size P, c1, c2, w and then maximum iterative
count.
Step 2: Swarm matrices to create P particles (X, pbest, gbest and V are n × c matrices).
Step 3: Initialize X, V, pbest for each particle and gbest for the swarm.
n

Step 4: Calculate the cluster centers for each particle using by (6)

z


j

i 1
n


i 1

Step 5: Calculate the fitness value of each particle using by (14) f

i

m
ij

 x 

m
ij

K
Jm

Step 6: pbest compute for each particle.
Step 7: gbest compute for the swarm.
Step 8: Update the velocity matrix for each particle using by (11)
Step 9: Update the position matrix for each particle using by (12)
Step 10: If terminate state is not met, go to step 4

3.2 Fuzzy Particle Swarm Optimization for fuzzy clustering
Customized particle swarm optimizations are called Fuzzy Particle Swarm Optimization (FPSO)
proposed by Peng et al. In which method the position and velocity of particles redefined to represent
the fuzzy relation between variables. In this part we describe the system for fuzzy clustering problem.
In FPSO algorithm X, the position of particle, shows the fuzzy relation from set of data objects,
o  {o1 , o2 ,...., on } to set of cluster centers, z  {z1 , z2 ,...., zn } , X Can be expressed as follows:

 11



 n1

1c 


nc 

(3.2)

In which ij is the membership function of the ith object with the jth cluster with constraints stated Eq.
(2.1) and Eq. (2.2) therefore we can see that the position matrix of each particle is the same as fuzzy
matrix  in FCM algorithm. Also the velocity of each particle is stated using a matrix with the size n

491

Vol. 7, Issue 2, pp. 488-494

International Journal of Advances in Engineering & Technology, May, 2014.
©IJAET
ISSN: 22311963
rows and c columns the elements of which are in range between -1 and 1. We get the Eq. (3.3) and Eq.
(3.4) for update the positions and velocities of the particles base of the matrix.
v  t  1  w  v  t   c1r1  pbest  t    t   c2r2  gbest t    t  k=1, 2…P (3.3)





  t  1  x  t   v  t  1

(3.4)

After update the arrangement matrix, it may go against the constraint given Eq. (2.1) and Eq. (2.2). So
it is necessary to normalize the position matrix. First we make all the negative elements in matrix to
become zero. If all elements in a row of the matrix are zero, they need to be re-evaluated using series
of random numbers within the interval between 0 and 1, and then the matrix undergoes the following
transformation without violating the constraints:
1c
 11

c

ij


j 1

normal  

 n1 c

nj


j 1




j 1



nc

c

nj 

j 1

c



1j

(3.5)

In FPSO algorithm the same as other evolutionary algorithms, we need a function for evaluating the
generalized solutions called fitness function. In below equation is used for evaluating the solutions.

f  x 

K
Jm

(3.6)
K is a stable and J m is the point function of FCM algorithm (Eq. (3.6)). The minor is J m , the better is
the clustering result and the upper is the individual fitness f (X). The FPSO algorithm for fuzzy
clustering problem can be stated as follows:
ALGORITHM 4: FUZZY PSO FOR FUZZY CLUSTERING
Step 1: Initialize the parameter together with human part size P, c1, c2, w and the maximum iterative
count.
Step 2: Create a swarm with P particles (X, pbest, gbest and V are n× c matrices).
Step 3: Initialize X, V, pbest for each particle and gbest for the swarm.
Step 4: Calculate the cluster centers for each particle using by Eq. (2.6).
Step 5: Calculate the fitness value of each particle using by Eq. (3.6).
Step 6: Calculate pbest for each particle.
Step 7: Calculate gbest for the swarm.
Step 8: Update the velocity matrix for each particle using by Eq. (3.3).
Step 9: Update the position matrix for each particle using by Eq. (3.4).
Step10: If terminating condition is not met, go to step 4.
The termination condition in proposed method is the maximum number of iterations or no
improvement in gbest in a number of iterations

3.3 EXPERIMENTAL RESULTS AND DISCUSSION
TABLE I: Values different from K- Means, Fuzzy C Means and Fuzzy PSO
K means

Fuzzy C - Means

Fuzzy PSO

Accuracy

6.95

6.15

10.35

Sensitivity

7.36

11.1

9.55

Specificity

10.26

10.74

12.22

Table 1 show we compared K- means, Fuzzy C- Means and Fuzzy Particle Swarm Optimization
(FPSO) these three algorithms to calculate take fifteen iteration to find out these Accuracy, Sensitivity

492

Vol. 7, Issue 2, pp. 488-494

International Journal of Advances in Engineering & Technology, May, 2014.
©IJAET
ISSN: 22311963
and Specificity. Iteration takes 2 * 2 matrixes to get the final values. So finally we conclude Fuzzy
PSO is best algorithm.
TABLE II: Show the which methods is best

Methods

Worst

Average

Best

K - Means

0.46

0.49

0.68

Fuzzy C Means

0.41

0.74

0.71

Fuzzy PSO

0.69

0.63

0.82

Table II we compared to K- Means, Fuzzy C Means algorithms compare these two algorithm Fuzzy
PSO best. Above the Table I to find Accuracy, Sensitivity and Specificity take iteration values and
calculate the total number of iteration. Finally Fuzzy Particle Swarm Optimization is best values are
(0.69,0.63,0.82)
Fig 1. Performance analysis of clustering algorithms

Performance evalution of K means, Fuzzy C means and Fuzzy PSO with their parameter values got
from confussion matrix with implementation of MATLAB 6.0. In this chart includes our Accuracy,
Sensitivity and Specificity values.
IV.

CONCLUSION

The PSO is a global optimizer for continuous variable problems with efficient. Small number of
parameters to deal and the large number of processing elements is the advantages of the PSO. Enable
by FPSO to fly around the solution space effectively.

V.

FUTURE WORK

This research seeks at efficient clustering the traditional paneling clustering techniques K-Means,
Fuzzy-C Means and FPSO. FPSO is giving the better results as compared to all other algorithms on
iris datasets. The future work includes, Ant Colony Optimization (ACO), Bee Colony Optimization
(BCO). Etc.,

493

Vol. 7, Issue 2, pp. 488-494

International Journal of Advances in Engineering & Technology, May, 2014.
©IJAET
ISSN: 22311963

REFERENCES
[1]. Bezdek, J. (1974). Fuzzy mathematics in pattern Classification. Ph.D. thesis. Ithaca,
NY:Cornell University.
[2]. Chun-Wei Tsai, Kai-Cheng Hu, Ming-Chao Chiang Chu-Sing Yang, “Ant colony
optimization with dual pheromone tables for clustering” Fuzzy Systems (FUZZ), 2011 IEEE
International Conference’,978-1- 4244-7315-1.
[3]. Chen Yanyum, Qiu Jianlin, Gu Xiang, Chen Jianping, Ji Dan Chen Li,”Advances in Research
of Fuzzy C- Means Clustering Algorithm”, IEEE, 10.1109/NCIS.2011.10
[4]. Eduardo Silva Palmeira,”A Comparison between K- Means, FCM and CKmeans Algorithms,
2011 IEEE,978-0 -7695-4628-5
[5]. Erol Egrioglu, Ufuk Yolcu, Cagdas Hakan Aladag, and Cem Kocak, “An ARMA Type Fuzzy
Time Series Forecasting Method Based on Particle Swarm Optimization,” Mathematical
Problems in Engineering, vol. 2013, pp. 1–12, 2013.
[6]. Gan, G., Wu, J., & Yang, Z. (2009). A genetic fuzzy k- modes algorithm for clustering
categorical data. Expert Systems with Applications (36), 1615–1620.
[7]. Han and Kamber, Data Mining Concepts and Techniques, Morgan Kauffman Publishers
[8]. Hathway, R. J., & Bezdek, J. (1995). Optimization of clustering criteria by reformulation.
IEEE transactions on Fuzzy Systems, 241–245.
[9]. H. Izakian, A. Abraham / Expert Systems with Applications (2011) Fuzzy C-means and fuzzy
swarm for fuzzy clustering problem1835–1838
[10]. Hesam Izakain a, Ajith Abraham, “Fuzzy C-mean and fuzzy swarm for fuzzy clustering
problem,” 0957- 4171 2010 Elsevier Ltd., doi:10.1016/j.eswa.2010.07.112
[11]. Kennedy,J., & Eberhart R.”Swarm Intellignece. Morgan Kaufmann.
[12]. K.Sankar, Dr.K.Krishnamoorthy Cluster Mining with Ant Colony Optimizer using Fuzzy
inferences. International Conference on Computing, Communication and Networking
Technologies
[13]. V. Ramesh, K. Ramar, S. Babu (2013) Parallel K-Means Algorithm on Agricultural
Databases, IJCSI International Journal of Computer Science.
[14]. Stuti Karol, Veenu Mangat Evaluation of a Text Document Clustering Approach based on
Particle Swarm Optimization, IJCSNS International Journal of Computer Science and
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[15]. Semantically Enhanced Document Clustering Based on PSO Algorithm Sridevi.U. K.,
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(2011), pp.485-493
[16]. Sanya,Hainan China,”SVM Combined with FCM and PSO for Fuzzy Clustering, 7 th
International Conference on Computational Intelligence and Security” IEEE, 978-0-76954584 - 4.
[17]. K.Velusamy, R.Manavalan Performance Analysis of Unsupervised Classification Based on
Optimization International Journal of Computer Applications (0975 – 8887) Volume 42–
No.19, March 2012.

AUTHORS
S. P. Priyadharshini is a Research Scholar at the Bharathiar University
Coimbatore. She is now working as a Asst. Prof. in Computer Science, Ayyan
Thiruvalluvar College of Arts & Science, P.Puliampatti. Her Research interests are
in the field of Data Mining and Optimization Techniques.

Ramachandra V. Pujeri Working as a Vice Principal at KGISL Institute of
Technology, KGISL campus, Coimbatore. His Research interests are in the field of
Computer Network, Data Mining and
Optimization Techniques.

494

Vol. 7, Issue 2, pp. 488-494


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