Exploring differential evolution and particle swarm optimization to...

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Exploring differential evolution and particle swarm optimizationto develop some symmetry-based automatic clustering techniques:application to gene clustering

Sriparna Saha1 • Ranjita Das2

Received: 17 April 2016 / Accepted: 8 November 2016 / Published online: 1 February 2017

� The Natural Computing Applications Forum 2017

Abstract In the current paper, we have developed two bio-

inspired fuzzy clustering algorithms by incorporating the

optimization techniques, namely differential evolution and

particle swarm optimization. Both these clustering tech-

niques can detect symmetrical-shaped clusters utilizing the

established point symmetry-based distance measure. Both

the proposed approaches are automatic in nature and can

detect the number of clusters automatically from a given

dataset. A symmetry-based cluster validity measure,

F-Sym-index, is used as the objective function to be opti-

mized in order to automatically determine the correct

partitioning by both the approaches. The effectiveness of

the proposed approaches is shown for automatically clus-

tering some artificial and real-life datasets as well as for

clustering some real-life gene expression datasets. The

current paper presents a comparative analysis of some

meta-heuristic-based clustering approaches, namely newly

proposed two techniques and the already existing auto-

matic genetic clustering techniques, VGAPS, GCUK,

HNGA. The obtained results are compared with respect to

some external cluster validity indices. Moreover, some

statistical significance tests, as well as biological signifi-

cance tests, are also conducted. Finally, results on gene

expression datasets have been visualized by using some

visualization tools, namely Eisen plot and cluster profile

plot.

Keywords Unsupervised classification � Particle swarm

optimization (PSO) � Differential evolution (DE) �Symmetry � Point symmetry-based distance � Geneexpression data

1 Introduction

In the field of data mining, clustering [22] has innumerable

applications for solving different real-life problems

[15, 23]. In the literature, many invariant clustering tech-

niques have been proposed [4] to cluster the dataset. To

identify clusters from a dataset, some proximity or simi-

larity measurements need to be defined among data points

to establish rules which can be used to assign points to the

domain of a particular cluster centroid. For recognition and

identification of most of the objects, ‘‘Symmetry’’ is useful

as it is an important characteristic of real-life objects. As

symmetry is a natural phenomenon, we can assume that

some kind of symmetricity exists in the cluster structure

also.

Symmetry measurements can be of two types, point

symmetry (PS) and line symmetry (LS). Point symmetry-

based measurements are more applicable for clusters which

are symmetric about their central point. In Fig. 1, some

objects having point symmetry and line symmetry prop-

erties are shown. Inspired by these observations, some

point symmetry-based measurements are developed in

[7, 39]. These distance functions are then utilized in [7] to

develop some clustering techniques which can determine

any kind of point symmetric clusters from different data-

sets. The symmetry in clustering is discussed in many

& Sriparna Saha

[email protected]

Ranjita Das

[email protected]; [email protected]

1 Department of Computer Science and Engineering, Indian

Institute of Technology Patna, Patna, India

2 Department of Computer Science and Engineering, National

Institute of Technology Mizoram, Aizawl, India

123

Neural Comput & Applic (2018) 30:735–757

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existing works on clustering, for example, in the analysis of

invariant clustering [4]. In [7], some genetic algorithm-

based techniques are developed for solving the clustering

problem using the properties of symmetry. The clustering

problem is modeled as an optimization problem and

genetic algorithm [19] was used to optimize the total

symmetrical compactness of the obtained clustering to get

the optimal partitioning. This algorithm overcomes some

drawbacks associated with SBKM and Mod-SBKM clus-

tering techniques [43].

1.1 Some automatic clustering techniques

In the literature, many genetic algorithm-based clustering

techniques are available which are capable of detecting the

number of clusters and the appropriate partitioning auto-

matically from any given dataset. Some examples are

variable string length genetic K-means algorithm

(GCUK)[6], hybrid niching genetic algorithm (HNGA)

[41] where Euclidean distance has been used for assigning

data points into different clusters. A variable string length

genetic clustering technique (VGAPS) [38] is also pro-

posed where point symmetry-based distance has been used.

In [6], a genetic algorithm-based K-means clustering

technique has been developed which is able to detect

clusters having equi-sized hyper-spherical shapes. GCUK

uses genetic algorithm-based K-means clustering technique

for automatic identification of clusters. In HNGA [41] to

prevent premature convergence, a niching method is

developed along with a weighted sum validity function for

optimization. Liu et al. [28] developed an automatic clus-

tering technique based on genetic algorithm and presented

a noising selection and division absorption-based mutation

technique to maintain the diversity of population and

selection pressure. Horta et al. [21] developed an evolu-

tionary technique based on fuzzy clustering for automati-

cally identifying the clusters present in the relational data.

In [1], authors have introduced a grouping-based evolu-

tionary approach which has used the idea of grouping

encoding and an adaptive exploration and exploitation

operator. Moreover, an elitist scheme is also applied to

ensure that the best solution is preserved by the algorithm.

He et al. [20] adopted for initialization of individual, a

variable length coding representation, and used the two-

stage selection and mutation operator. But when the

dimension of the dataset increases, the search ability gets

reduced. Kao et al. [24] presented a hybrid particle swarm

optimization algorithm for automatically evolving the

cluster centers and applied it to the problem of generalized

machine cell formation.

In recent years, some new optimization techniques like

cuckoo search technique [47], differential evolution (DE)

[34, 46], particle swarm optimization (PSO) [33] and ant

colony optimization [13] have been proposed in the liter-

ature. Recent studies have also revealed that these opti-

mization techniques converge much faster than the genetic

algorithms [34, 46]. Based on these observations, some

differential evolution-based and particle swarm optimiza-

tion-based clustering techniques are also developed in the

literature [29, 31, 35, 40]. In [31], a modified differential

evolution-based clustering technique is developed for

satellite image segmentation. In [40], a modified fitness-

based adaptive differential evolution algorithm is devel-

oped for clustering of image pixels. Here the control

parameters of the traditional DE-based approach are cal-

culated adaptively using the fitness-based statistics. In [36],

two variants of DE-based clustering techniques are pro-

posed. These are then applied for solving clustering prob-

lem from some real-life datasets. Zhang et al. [48] have

used DE to optimize the coordinates of the samples dis-

tributed randomly on a plane. Kernel-based approaches are

utilized here to map the data of the original space into a

high-dimensional feature space in which a fuzzy dissimi-

larity matrix is constructed. Cai et al. [11] combined tra-

ditional DE and one step K-means clustering for the

problem of unconstrained global optimization. Tvrdk

et al. [44] developed a hybrid method by combining DE

and K-means algorithm and applied it to non-hierarchical

clustering. In [26] authors have incorporated a local

improvement phase to the classical DE to get the faster

convergence and better performance and further applied in

the wireless sensor network to increase the lifetime of the

network. Liu et al. [27] combined two multi-parent cross-

over operators with differential evolution and it is pre-

sented to solve the problem of global optimization. A good

survey covering the existing particle swarm optimization-

based clustering techniques can be found in [2].

1.2 Motivation

All the existing DE- and PSO-based clustering techniques

are found to perform better than the corresponding genetic

algorithm-based versions. But in the earlier attempts, these

algorithms were used along with popular Euclidean dis-

tance for assignment of points to different clusters. As

mentioned earlier, symmetry-based measurements [7] are

Fig. 1 Point symmetric and line symmetric objects

736 Neural Comput & Applic (2018) 30:735–757

123

found to perform better than the popular Euclidean dis-

tance-based versions in detecting clusters having different

shapes and sizes. Thus, the incorporation of these sym-

metry-based measurements in the frameworks of differen-

tial evolution and particle swarm optimization-based

clustering techniques can help to increase the quality of the

partitions further.

In the current paper, we have made an attempt in this

direction. Two algorithms based on the search capabilities

of differential evolution and particle swarm optimization

are developed. Both the algorithms are able to detect the

number of clusters and the appropriate partitioning auto-

matically without having prior information about these.

Moreover, both the algorithms utilize the variable center-

based encoding to represent the partitions. Symmetry-

based similarity measurement [7] is utilized for the

assignment of points to different clusters. A symmetry-

based cluster validity index, namely F-Sym-index, a fuzzy

symmetry-based cluster validity index [38], is used as an

objective function in both the proposed clustering tech-

niques. In a part of the paper, another cluster validity

index, XB-index [45], is also used as the objective func-

tion for the purpose of comparison. Incorporation of point

symmetry distance in the evaluation of F-Sym measure

makes it capable of detecting all categories of clusters

irrespective of the shapes and sizes as long as those

contain some symmetrical properties. F-Sym values of the

obtained partitionings are optimized using the search

capabilities of DE and PSO.

1.3 Experimental results

The effectiveness of both the clustering techniques is

illustrated on several artificial and real-life datasets.

The performances are compared with respect to a

variable length genetic algorithm with point symmetry-

based clustering technique, VGAPS [38] and two other

genetic algorithm with Euclidean distance-based clus-

tering techniques, GCUK [6] and HNGA [41] in terms

of an external cluster validity index, Minkowski Score

[42]. We have also made a comparative study of the

number of clusters obtained by all these algorithms. In a

part of the paper, we have also conducted some statis-

tical significance tests. In order to show some real-life

applications of the proposed clustering algorithms, we

have shown results for gene expression data clustering.

We have used some gene expression datasets to show

results and evaluated the goodness of the obtained

partitions using an external cluster validity index, Sil-

houette index [37]. Finally, biological and statistical

significance tests have been conducted on the gene

expression datasets.

1.4 Major contributions

The followings are the key contributions of the current

paper:

• This is the first attempt where some differential

evolution or particle swarm optimization-based fuzzy

clustering techniques are developed using the proper-

ties of symmetry.

• First fuzzy clustering technique is based on the search

capabilities of differential evolution, and the second

one is based on the search capabilities of particle swarm

optimization.

• Both the proposed clustering techniques use point

symmetry-based distance for allocating points to

different clusters.

• Both DE- and PSO-based clustering techniques are able

to detect the number of clusters and the appropriate

partitioning automatically.

• Goodness of the partitioning measured in terms of point

symmetry-based cluster validity index, FSym-index, is

used as the optimization objective.

• Results on several artificial and real-life datasets show

that the performance of DE-based clustering technique

is the best compared to other symmetry-based

algorithms.

• Results on gene expression datasets show the superior

performance of DE in terms of cluster accuracy.

• Finally, some biological and statistical significance

tests have been performed to evaluate the biological

and statistical significance of the obtained results.

2 Existing point symmetry-based distancemeasure

In this section, at first the point symmetry (PS)-based dis-

tance developed in [7] is described.

2.1 Point symmetry-based distance

The PS distance or point symmetry-based distance [7]

dpsðx; cÞ associated with point x with respect to a cluster

center c of cluster cj, j ¼ 1; 2; . . ., C is described in this

section. Let the dataset contain all distinct points, and let x

be a point. The reflected or symmetrical point of x with

respect to a particular cluster center c is 2� c� x, and this

is denoted by x�. If knear number of unique nearest

neighbors of x� (calculated using Euclidean distances) are

at distances of dk, k ¼ 1; 2; . . ., knear. Then,

dpsðx; cÞ ¼ dsymðx; cÞ � deðx; cÞ ð1Þ

Neural Comput & Applic (2018) 30:735–757 737

123

where,

dsymðx; cÞ ¼Pknear

k¼1 dk

knearð2Þ

In Eq. 2, knear should not be chosen as equal to 1, because

if x� exists in the dataset, then the value of dpsðx; cÞ = 0, and

there will be no impact of the Euclidean distance. Again if

the value of knear is large, then also it will not be suit-

able because with respect to a particular cluster center it

may overestimate the amount of symmetry of a point. So

here we have kept knear = 2. Note that dpsðx; cÞ is a non-

metric distance measure which mainly calculates the point

symmetry distance between data point and a cluster center

unlike the popular Minkowski distances. Computation

complexity of dpsðx; cÞ is O(n). Hence, for n points and C

clusters, the complexity of assigning the points to different

clusters is O(n2C). In order to decrease the computational

complexity, Kd-tree-based approximate nearest neighbor

search is also proposed in [7].

2.2 Symbols used

Here the following symbols are used in describing the

proposed clustering techniques:

• C: number of clusters present in a particular string.

• Cmax: maximum value of number of clusters.

• D: dimension of the dataset.

• NP: population size.

• Pbest: best particle position in case of PSO-based

approach.

• Gbest: best global particle position in case of PSO-

based approach.

• par: current particle.

• G: current generation.

• Kgbest: Best vector till the current generation in case of

DE-based approach.

• Klbest: best vector of the current population.

• CR: crossover probability in case of DE-based

approach.

3 Proposed fuzzy symmetry-based automaticclustering technique using the search capabilityof DE

In this section, the description about variable vector length

differential evolution algorithm using a newly developed

point symmetry-based distance is given for automatic

determination of optimal clustering solution (Fuzzy-

VMODEPS scheme). A flowchart showing different steps

of the proposed approach is shown in Fig. 2.

Differential evolution (DE) is a meta-heuristic technique

developed by Storn and Price [34] to optimize real-life func-

tions. The idea behind the DE-based clustering technique is as

follows: Initial cluster centers are some randomly selected data

points from the dataset and those are encoded as cluster centers

in the vector. Similarly, all the vectors in the population have

been initialized. After initialization phase, centers have been

extracted to compute the fitness of a particular vector. Once

fitness has been calculated, all the vectors in the population are

gone through the mutation and crossover phase to generate the

mutant and crossover vectors.

3.1 Vector initialization and representation

In the proposed Fuzzy-VMODEPS scheme, population is

consisting of a collection of vectors. Each vector Vl

contains a collection of real numbers distinctly chosen

from given dataset where l ¼ 1; 2; . . .;NP, NP is the

maximum size of population. Here each vector Vl

encodes Cj number of clusters where minimum size of

Cj is 2 and maximum size is Cmax. Now Cj can be

Fig. 2 Flowchart of Fuzzy-

VMODEPS approach


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calculated by using the following equation Cj ¼ðrandðÞmodðCmax � 1ÞÞ þ 2 where rand() is a random

function returning an integer and Cmax denotes the

maximum value of number of clusters. Therefore, the

number of clusters present in the vector should be con-

fined between 2 to Cmax. Cj number of distinct points are

randomly selected from the given dataset. Let us con-

sider that Vl be the vector and it contains Cj number of

cluster centers. If the dimension of each data point in the

dataset is D, then the length of the vector will be

D� Cj. This is explained by an example:

Let a vector be represented by \1:2; 21:2; 13:2;

14:2; 5:3; 6:2; 4:2; 5:3; 6:3; 2:5; 2:3; 1:6[ : If the vector

contains Cj ¼ 3 number of clusters and each center is

having D ¼ 4 dimensions, then the centers of the clus-

ters will be: \1:2;21:2;13:2;14:2[; \5:3;6:2;4:2;5:3[and \6:3;2:5;2:3;1:6[:

After that, five iterations of fuzzy C-means algorithm

[10] will be executed on the whole dataset with the set of

cluster centers which has been encoded in each vector. This

generally replaces the centers in the corresponding vector

by the resultant centers so that centers get separated

initially.

3.1.1 Fitness computation

Fitness computation is a two step process, in the first step

using the point symmetry-based distance measure [7]

membership values of xi data points where i ¼ 1; . . .; n;

(n is the total number of data points) with respect to

C different cluster centers have been computed where C is

the number of centers encoded in a particular vector. Once

membership values have been calculated subsequently in

the second step using the membership matrix, fitness

measure is evaluated.

3.1.2 Computation of membership values

Let a particular vector contain C number of cluster centers

encoded in it. The centers are denoted by

cj; for j ¼ 1; . . .;C. The cluster center cmin among all the

cluster centers, cj; for j ¼ 1; . . .;C nearest to data point xihas been determined in terms of symmetry to compute the

membership values. The expression for determining cmin is

given below:

cmin ¼ argminj¼1;...;Cdpsðxi; cjÞ

dpsðxi; cjÞ, i.e., point symmetry distance between data point

xi and cluster center cj, has been calculated by using Eq. 1.

Here cj denotes the center of the jth cluster. In this context,

if dsym � h, i.e., dsymðxi; cminÞ is smaller than h, then the

membership values are calculated as follows:

uij ¼ 1; if j ¼ cmin

uij ¼ 0; if j 6¼ cmin

Otherwise membership values of uij will be updated

using the procedure as done in fuzzy C-means [10]

algorithm. Here m 2 ð1;1Þ is a weighting exponent

called the fuzzifier whose value has been considered,

m ¼ 2 and h value has been considered as the maximum

nearest neighbor distance among all the data points.

More details about h value calculation can be obtained

from [7].

3.1.3 Objective function used

In order to determine average symmetry present in a par-

titioning, an internal cluster validity measure FSym-index

[38] has been utilized as a fitness function. For each vector,

first the membership values are calculated using the above

discussed procedure. Finally, the FSym-index value is

calculated using this membership matrix. FSym-index has

been computed using following equation:

FSymðCÞ ¼ 1

C� 1

EC

� DC ð3Þ

Here C is the number of clusters encoded in the vector.

Ec ¼XC

j¼1

Ej ð4Þ

Such that

Ej ¼Xn

i¼1

uij � dpsðxi; cjÞ ð5Þ

and

DC ¼ maxCi;j¼1kci � cjkÞ ð6Þ

In this context in order to obtain the actual number of

clusters and to achieve proper clustering, FSym-index,

value needs to be maximized. Thus, the objective function

for a particular vector is FSym. This is maximized using the

search capability of DE.

3.1.4 Updation of centers

After computing the membership values, cluster centers are

updated. In order to update the cluster centers, following

equation has been used which is similar to the equation

used in fuzzy C-means [10]

cj ¼Pn

i¼1 umij xiPn

i¼1 umi;j

ð7Þ

for j ¼ 1. . .C.


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3.1.5 Mutation

The population of DE is composed of NP number of D

dimensional individuals, plðGÞ, l ¼ 1; 2; . . .;NP to attain an

optimal solution where G denotes the Gth generation. Here

plðGÞ, l ¼ 1; 2; . . .;NP are target vectors. Now each indi-

vidual vector plðGÞ in the population of target vectors at

Gth generation is gone through mutation phase. This leads

to generation of trial offsprings or mutant vectors. Mutant

vector is produced by the following equation.

MlðGþ 1Þ ¼ pkðGÞ þ FðpmðGÞ � pnðGÞÞ ð8Þ

For each individual in the population, pkðGÞ, pmðGÞ and

pnðGÞ are three vectors chosen randomly from the pop-

ulation of target vectors at the (G)th generation. Here

l 6¼ k 6¼ m 6¼ n and m; k; l; n 2 1; 2; . . .NP are mutually

distinct integers taken randomly. The mutant vector is

obtained by finding the difference of the two target

vectors multiplied by the scalar factor F where

F 2 ½0; 1�. Finally, this term is added with the values of

third individual target vector. Here in the equation, third

individual vector pkðGÞ is added with the weighting

difference of target vectors, pmðGÞ and pnðGÞ, which

leads to a generation of mutant vector, MlðGþ 1Þ for

ðGþ 1Þth generation. The above classical mutation

scheme is modified in the paper [31] and that modified

mutation operator is used in the proposed approach

(Fuzzy-VMODEPS). The detailed description about the

modified mutation scheme, the same which has been

adopted in the current paper, is given below. The mod-

ified mutation scheme is described as follows:

MlðGþ 1Þ ¼ KgbestðGÞ þ a KlbestðGÞ � KrðGÞð Þ ð9Þ

in the above equation, MlðGþ 1Þ represents (l)th mutant

vector generated at ðGþ 1Þth generation.MlðGþ 1Þ vectoris generated by adding the weighted difference vector of

KlbestðGÞ and KrðGÞ with the third vector KgbestðGÞ. HereKlbestðGÞ denotes the best vector of the current population

at Gth generation. KgbestðGÞ denotes the best vector gen-

erated till the Gth generation. KrðGÞ represents the (r)th

vector generated randomly from the current population at

Gth generation. Moreover, in the equation difference of

two target vectors, Klbest, Kr at the Gth generation is mul-

tiplied by the scalar factor, a. Calculation of a is given

below.

a ¼ 1

1þ exp �ð1GÞ

� � ð10Þ

Subsequently based on the calculated a value, classical or

modified mutation scheme is adopted for each of the

generations.

MlðGþ 1Þ ¼

KgbestðGÞ þ aðKlbestðGÞ � KrðGÞÞif randð0; 1Þ� a

pkðGÞ þ FðpmðGÞ � pnðGÞÞotherwise

8>>><

>>>:

Modified mutation scheme has been used in paper [31]

to accelerate the convergence of the proposed approach,

so that the trial vector can reach global optimum in

minimum number of generations. This is not same in

case of classical mutation scheme. As the generation

increases, a value gets decreased. Whenever a value is

high, the probability of adopting modi-mutation

scheme is high too. So, when modi-mutation function is

used, then the lbest vector, i.e., best vector in the current

population, has a greater influence for evolving the

mutant vector.

3.1.6 Crossover

Crossover function has a greater influence to increase the

diversity in the offspring vectors. Crossover operation is

performed on the individual vector or target vector and its

corresponding mutant vector. After crossover operation,

trail vector is generated. The trail vector is generated in the

following way:

CjlðGþ 1Þ ¼MjlðGþ 1Þ if randð0; 1Þ�CR or j ¼ randðlÞpjlðGÞ otherwise

�

Here j ¼ 1; 2. . . d and rand(l) is the randomly selected

index from 1; 2; . . .; d, where d ¼ D� C, C: number of

clusters encoded in the lth chromosome at Gth generation.

CR is the crossover rate and ClðGþ 1Þ is the trail vector

for ðGþ 1Þth generation. After that, fitness value is com-

puted for each of the trail vectors.

3.1.7 Selection

In this phase, the trail vector is compared with the target

vector, the vector which has maximum fitness value will be

survived for the next generation. The procedure is as

follows:

pl ¼Cl FSymðplÞ�FSymðClÞpl otherwise

�

If the fitness value (in this case the value of FSym-index

corresponding to the partitioning encoded in Cl vector) of

Cl is better than the fitness value of pl (FSym-index value

corresponding to the partitioning encoded in pl vector),

then update pl by Cl. Otherwise pl ¼ pl, previous value of

pl is preserved.


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3.1.8 Termination criteria

In this approach, the process of mutant vector generation in

the mutation phase, trail vector generation in the crossover

phase and the selection operation is performed for a con-

stant number of generations. At the final generation, a

population containing multiple solutions is generated. The

FSym-index values are calculated for individual vectors.

The best vector having the highest value of FSym is con-

sidered as the final solution. The corresponding set of

cluster centers is used to partition the given data, and the

obtained results are reported.

The basic steps of the proposed algorithm are enumer-

ated below:

• Generate the initial population NP randomly.

• Execute the steps of FCM algorithm five times.

• Evaluate the fitness of each individual or vector in NP

using Eq. 3.

• Set generation = 1, Maxgen: maximum number of

generations.

• Initialize gbest and lbest

• While the halting criteria is not satisfied

ðgenerationþþ�MaxgenÞ• do

• if randð0; 1Þ\a then

• (here a ¼ 1

1þexpð�ð1=generationÞÞ)

• Apply mutation operation using Equation

MlðGþ 1Þ ¼ KgbestðGÞ þ aðKlbestðGÞ � KrðGÞÞ.• Apply Crossover operation

• else

• Apply Mutation operation using Equation

MlðGþ 1Þ ¼ pkðGÞ þ FðpmðGÞ � pnðGÞÞ.• Apply crossover operation

• Evaluate the fitness of trial vector or offspring using

Eq. 3.

• Update Lbest by the best vector in the current

population

• if ðlbest[ gbestÞ

• Replace gbest with lbest

• If Cl (vector generated after application of genetic

operators) is better than pl (original vector)

• pl ¼ Cl

• otherwise previous value of pl will be retained

• End while

• Report the best vector

3.2 Time complexity

The time complexity of the proposed algorithm is analyzed

below:

• Initialization of Fuzzy-VMODEPS requires

OðPopsize� vectorlengthÞ time where Popsize and

vectorlength indicate the population size and the length

of each vector in Fuzzy-VMODEPS, respectively. Note

that vectorlength is OðCmax � DÞ where D is the

dimension of the dataset and Cmax is the maximum

possible number of clusters encoded in a string.

• Fitness computation is composed of three steps.

• In order to find membership values of each point

with respect to different cluster centers, minimum

symmetrical distance of that point with respect to

all clusters has to be calculated. For this purpose,

the Kd-tree [9]-based nearest neighbor search is

used. If the points are roughly uniformly dis-

tributed, then the expected case complexity is

Oðm� Dþ log nÞ, where m is a constant depending

on dimensions and the point distribution. This is

O(logn) if the dimension D is a constant [9].

Friedman et al. also reported O(logn) expected time

for finding the nearest neighbor [16]. So in order to

find the minimal symmetrical distance of a partic-

ular point, OðCmax � log nÞ time is needed. Thus,

total complexity of computing membership values

of n points to Cmax clusters is OðCmax � n� log nÞ.• For updating the centers, total complexity is

OðCmaxÞ.• Total complexity for computing the fitness values is

Oðn� CmaxÞ.So the fitness evaluation has total complexity =

OðPopsize� Cmax � n� log nÞ.

• Mutation and crossover require

OðPopsize� vectorlengthÞ time each.

• Selection step of the Fuzzy-VMODEPS requires

OðPopsize� vectorlengthÞ time.

Thus, summing up the above complexities, total time

complexity becomes OðCmax � n� logðnÞ � PopsizeÞ pergeneration. For maximum Maxgen number of genera-

tions, total complexity becomes OðCmax � n� logðnÞ �Popsize� MaxgenÞ.


123

4 Particle swarm optimization-based variablelength clustering technique using pointsymmetry-based distance (Fuzzy-VPSOPS)

Particle swarm optimization (PSO) is a population-based

stochastic search algorithm developed by Kennedy and

Eberhart [33]. This algorithm is developed after getting

inspiration by the swarm behavior of birds, bees and fish as

they search for food or communicate with each other. It

was mainly designed to solve optimization problems. The

PSO approach is highly decentralized and is based upon

interaction among the agents called particles [25]. Particles

are the agents which represent individual solutions and the

collection of particles is the swarm which represents the

solution space. Initially, the swarm is initialized by some

random solutions and the particle starts flying through the

solution space by maintaining a velocity value and keeping

track of its best previous position obtained so far which is

known as personal best position. Global best is another best

solution which corresponds to the best fitness value

obtained by any of the particles. In the current work, we

have proposed a fuzzy symmetry-based variable length

clustering technique using the search capabilities of parti-

cle swarm optimization (PSO). The algorithm is named as

Fuzzy-VPSOPS (fuzzy point symmetry-based variable

length clustering technique using particle swarm opti-

mization). A flowchart showing different steps of the pro-

posed approach is shown in Fig. 3.

The algorithmic flow of Fuzzy-VPSOPS is given below.

The parameters of the search space are encoded in the form

of particles and a collection of such particles is called

swarm. Initially, the process starts with a population of

particles whose positions represent the potential solutions

for the studied problem, velocities are randomly initialized

in the search space, and the population or swarm represents

different points in the search space. An objective function

is associated with each particle, and this will be the

particle’s position. In each iteration, the search for optimal

position is performed by updating the velocities and posi-

tions of particles. The velocity of each particle is updated

using Pbest and Gbest positions. The personal best posi-

tion, Pbest, is the best position the particle has visited and

Gbest is the best position the swarm has visited since the

first time step. The process of fitness calculation: Pbest,

Gbest calculations and velocity, position updation, con-

tinues for a fixed number of generations or till a termina-

tion condition is satisfied.

For the purpose of clustering, each particle encodes a

possible variable number of cluster centers. The goodness

of each partition is measured using a point symmetry-based

cluster validity index. Here we have used point symmetry-

based distance for cluster assignment and FSym-index as

the objective function. The details of this scheme are

described below:

4.1 Particle representation and population

initialization

In the proposed Fuzzy-VPSOPS scheme, population is a

collection of potential solutions of clustering the data

which are termed as particles. Each particle Parl contains a

collection of real numbers distinctly chosen from given

dataset where l ¼ 1; 2; . . .;NP, NP is the number of solu-

tions present in a population. Here each particle Parlencodes Cj number of clusters where the possible range of

Cj is ½2;Cmax�. Now Cj is determined by using the equation

Cj ¼ ðrandðÞmodðCmax � 1ÞÞ þ 2 where rand() is a ran-

dom function returning an integer and Cmax denotes the

maximum value of clusters. For initialization, Cj number of

centers for a particular particle are randomly selected dis-

tinct points from the given gene dataset. Let us consider

parl be the particle and it contains Cj number of clusters.

Let the dimension of each data point in the dataset be D,

then the length of the particle will be D � Cj. This is

Fig. 3 Flowchart of Fuzzy-

VPSOPS approach


123

explained by an example: let a particle be

\11:2; 22:2; 23:2; 14:2; 5:3; 7:2;

4:5; 4:3; 5:5; 7:5; 4:3; 7:6[ . If the particle contains Cj ¼ 3

number of clusters and each center is having D ¼ 4

dimensions, then the cluster centers will be:

\11:2; 22:2; 23:2; 14:2[ ; \5:3; 7:2; 4:5; 4:3[ and

\5:5; 7:5; 4:3; 7:6[ :

4.2 Fitness computation

Here again first the cluster centers encoded in a particle are

extracted. Thereafter, the steps mentioned in Sect. 3 are

executed.

4.3 Calculation of Pbest and Gbest vectors

At the beginning of the execution, Pbest and Gbest are

initialized by some small values. In order to calculate

Pbest, we need to compare particle’s fitness value with that

obtained by Pbest solution. If the current particle Parl’s

fitness value is better than that of Pbest, Pbest is replaced

by the current particle. In the similar manner, if the fitness

value of current particle is better than Gbest, then Gbest is

updated by the current particle’s fitness value and position.

4.4 Updation of velocity and position of particles

In order to search for optimal position in each generation,

velocities and positions of particles have been updated. A

velocity vector is assigned to each particle to regulate the

next transit of the particle. Each particle basically updates

it velocity on the basis of current velocity, personal best

position it has obtained so far and the global best position

which has explored by the swarm.

The velocity and position of the particle are updated as:

Vellðt þ 1Þ ¼w� VellðtÞ þ co1 � r1ðparlPbestðtÞ � poslðtÞÞþ co2 � r2ðParGbest � poslðtÞÞ

poslðt þ 1Þ ¼ poslðtÞ þ Vellðt þ 1Þ ð11Þ

Here, w is the inertia weight, VellðtÞ is the previous

velocity in iteration t of lth particle, co1 and co2 are

coefficients and r1 and r2 are random values in the range of

0 and 1. ðparlPbestðtÞ � poslðtÞÞ is the difference between

the local best parlPbest of the lth particle and the previous

position poslðtÞ. Similarly, ðparGbest � poslðtÞÞ is the dif-

ference between the global best parGbest and the previous

position poslðtÞ.In order to search for optimal position in each genera-

tion, velocity and position of the particles have been

updated. Each particle basically updates its velocity on the

basis of current velocity, personal best position it has

obtained so far and the global best position which has

explored by the swarm.

4.5 Termination criteria

In Fuzzy-VPSOPS method, the process of fitness computa-

tion, Pbest and Gbest calculations, update of velocity and

position of the particles is executed for constant number of

iterations. The best particle generated by the clustering

algorithm up to the last iteration will give the solution of the

problem of clustering. The steps of Fuzzy-VPSOPS method

are executed for constant number of iterations. The best

particle generated by the clustering algorithm up to the last

iteration will give the solution of the problem of clustering.

The basic steps of the proposed algorithm are enumer-

ated below:

1. Initialize the parameters including population size

NP, co1, co2, w, and the maximum iteration count.

2. Initialization of a swarm with NP particles, i.e., for

each particle arbitrarily select Cj number of clusters

from the n number of data points as the centroids.

3. Initialize position and velocity matrix, Pbest for each

particle and Gbest for the swarm.

4. Run the FCM algorithm for five iterations.

5. Calculate the fitness value of each particle using

point symmetry-based distance measure as men-

tioned in Sect. 3.

6. Calculate Pbest for each particle.

7. Calculate Gbest for the swarm.

8. Update the velocity matrix for each particle.

9. Update the position matrix for each particle.

10. Go to step 5 until the termination criteria are not

satisfied.

4.6 Time complexity

The time complexity of the Fuzzy-VPSOPS clustering

technique is analyzed below:

• Initialization of Fuzzy-VPSOPS needs Oðswarmsize�particlelengthÞ time where swarmsize and particlelength

indicate the population size and the length of each particle

in Fuzzy-VPSOPS, respectively. Note that particlelength

is OðCmax � DÞ where D is the dimension of the dataset

and Cmax is the maximum number of clusters.

• Fitness computation is composed of three steps.

• In order to find membership values of each point to

all cluster centers, minimum symmetrical distance

of that point with respect to all clusters has to be

calculated. For this purpose, the Kd-tree-based

nearest neighbor search is used. If the points are


123

roughly uniformly distributed, then the expected

case complexity is OðmD þ logðnÞÞ, where m is a

constant depending on dimension and the point

distribution. This is O(logn) if the dimension D is a

constant. Friedman et al. [16] also reported Oðlog nÞexpected time for finding the nearest neighbor. So

in order to find the minimal symmetrical distance of

a particular point, OðCmax � logðnÞÞ time is needed.

Thus, total complexity of computing membership

values of n points to Cmax clusters is

OðCmax � n� logðnÞÞ.• For updating the centers, total complexity is

OðCmaxÞ.• Total complexity for computing the fitness values is

Oðn� CmaxÞ.So the fitness evaluation has total complexity=

Oðswarmsize� Cmax � n� logðnÞÞ.

• Complexity of calculating Pbest for each particle is

Oðswarmsize� Cmax � nÞ.• Complexity of calculating Gbest for each particle is

OðswarmsizeÞ.• Complexity for updating the velocity matrix for each

particle is OðCmax � n� constantÞ, i.e., OðCmax � nÞ.• Complexity for updating the position matrix for each

particle is Oðn� constantÞ, i.e., O(n).Thus, summing up the above complexities, total time

complexity becomes OðCmax � n� log n� swarmsizeÞ periteration. For maximum tmax number of iterations, total

complexity becomes OðCmax � n� logðnÞ � swarmsize

�tmaxÞ:

5 Datasets chosen

For our experiments, we have chosen three artificial data-

sets: Sym_3_2, Sph_4_3 and Mixed_3_2, and six real-life

datasets obtained from UCI machine learning repository

[5]: Cancer, Glass, Iris, NewThyroid, Wine and

LiverDisorder. The total number of data points to be

clustered are 350, 400, 600, 150, 214, 683, 215, 178 and

345 for Sym_3_2, Sph_4_3, Mixed_3_2, Iris, Glass, Can-

cer, NewThyroid, Wine and LiverDisorder, respectively.

The dimensions of data points for nine datasets are 2, 3, 2,

4, 9, 9, 5, 13 and 32, respectively.

Sym_3_2: The actual distribution of clusters is shown in

Fig. 4. Here there are total 350 points distributed over three

different shaped clusters, ring-shaped, compact and linear

clusters.

Sph_4_3: This dataset consists of 400 data points in

3-dimensional space distributed over four hyperspherical

disjoint clusters where each cluster contains 100 data

points. This dataset is shown in Fig. 6a.

Mixed_3_2: The distribution of clusters is shown in

Fig. 7. Here there are 600 points spread over three equal

sized clusters.

Cancer: Here we use the Wisconsin breast cancer dataset

obtained from [5]. Each pattern has nine features corre-

sponding to clump thickness, cell size uniformity, cell shape

uniformity, marginal adhesion, single epithelial cell size,

bare nuclei, bland chromatin, normal nucleoli and mitoses.

There are two categories in the data: malignant and benign.

The two classes are known to be linearly separable.

Iris: This dataset, obtained from [5], represents different

categories of irises characterized by four feature values [5].

It has three classes: Setosa, Versicolor and Virginica. It is

known that the two classes (Versicolor and Virginica) have

a large amount of overlap, while the class Setosa is linearly

separable from the other two.

Glass: This is the glass identification data [5] consisting

of 214 instances having nine features (an Id feature has

been removed). The study of the classification of the types

of glass was motivated by criminological investigation. At

the scene of the crime, the glass left can be used as evi-

dence, if it is correctly identified. There are six categories

present in this dataset.

Newthyroid: The original database from where it has

been collected is titled as Thyroid gland data [5]. Five

laboratory tests are used to predict whether a patient’s

thyroid belongs to the class euthyroidism, hypothyroidism

or hyperthyroidism. There are a total of 215 instances and

the number of attributes is five.

Wine: This is the wine recognition data consisting of

178 instances having 13 features resulting from a chemical

analysis of wines grown in the same region in Italy but

derived from three different cultivars. The analysis deter-

mined the quantities of 13 constituents found in each of the

three types of wines.

−1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

1.5

2

Fig. 4 Sym_3_2


123

LiverDisorder: This is the LiverDisorder data consisting

of 345 instances having six features each. The dataset has

two categories.

6 Results and discussion

The obtained experimental results are provided in Table 3.

We have executed the following clustering algorithms:

Fuzzy-VMODEPS:Fuzzy variable length modified differ-

ential evolution with point symmetry-based clustering

technique; VGAPS: variable length genetic algorithm with

point symmetry distance-based clustering technique;

Fuzzy-VPSOPS: Fuzzy variable length modified particle

swarm optimization with point symmetry-based clustering

technique; GCUK: genetic algorithm-based K-means

clustering technique; HNGA: hybrid niching genetic

algorithm-based clustering technique for all three artificial

and six real-life datasets. For Fuzzy-VMODEPS, the fol-

lowing parameter combinations are used: population

size = 100, number of generations = 30, F = 0.8,

CR = 0.5. For Fuzzy-VPSOPS, the following parameter

combinations are kept: maximum number of itera-

tions = 30, swarm size = 100, co1, co2 ¼ 2; w: 0.9 to 0.4.

For the other clustering algorithms, parameters mentioned

in the corresponding papers are used.

6.1 Parameter study

In order to select the optimal values of parameters for these

two proposed clustering algorithms, we have performed a

thorough sensitivity studies of the parameters. We have

varied the parameters of these two clustering techniques

over a range. The results obtained with different values of

parameters for Fuzzy-VMODEPS are shown in Table 1.

Different values of F, CR, maximum number of generations

and population sizes are used. Here we have first performed

the sensitivity studies of parameters on Iris and Cancer

datasets. The optimal values are then used in Fuzzy-

VMODEPS while applying on different datasets. Table 1

clearly illustrates that the optimal parameter values for

Fuzzy-VMODEPS are population size = 100, number of

generations = 30, F = 0.8, CR = 0.5. Because with this

parameter setting, best results by Fuzzy-VMODEPS are

obtained for both the datasets. Similarly, the sensitivity

studies of parameters are also done for Fuzzy-VPSOPS

algorithm. We have varied the values of co1, co2, number of

iterations and swarm size over a range. The results obtained

by Fuzzy-VPSOPS algorithm with different parameter set-

tings for Iris and Cancer datasets are shown in Table 2. This

table illustrates that the best parameter values for Fuzzy-

VPSOPS are the following: maximum number of itera-

tions = 30, swarm size = 100, co1, co2 ¼ 2:

6.2 Discussion of results

For artificial datasets, we have shown the clustering results

visually. The artificial datasets are having either 2 or 3

dimensions. Thus, such kind of visualization is possible. The

final results obtained after application of GCUK, VGAPS/

Fuzzy-VMODEPS/Fuzzy-VPSOPS, HNGA for Sym_3_2

dataset are shown in Fig. 5a–c, respectively, where genetic

algorithm-based K-means clustering technique (GCUK) and

HNGA are found to fail in providing the proper clusters. The

Table 1 Minkowski Score (MS)

values corresponding to

different parameter values

obtained by Fuzzy-

VMODEPS:Fuzzy variable

length modified differential

evolution with point symmetry-

based clustering technique

Dataset F CR Max_gen Population size # Obtained cluster MS

Iris 0.5 0.4 40 40 3 0.633745

Iris 0.8 0.5 20 50 3 0.800680

Iris 0.8 0.5 30 100 3 0.61

Cancer 0.8 0.5 20 50 3 0.362014

Cancer 0.5 0.4 40 80 3 0.38640

Cancer 0.8 0.5 30 100 2 0.346018

Bold values indicate best performances

Table 2 Minkowski Score (MS)

values corresponding to

different parameter values

obtained by Fuzzy-

VPSOPS:Fuzzy variable length

modified particle swarm

optimization with point

symmetry-based clustering

technique

Dataset C1 C2 Max_iteration Swarm size # Obtained cluster MS

Iris 1.5 1.5 30 100 2 0.824786

Iris 2 2 40 80 3 0.657143

Iris 2 2 30 100 3 0.61

Cancer 2 2 20 50 4 0.639396

Cancer 2 2 40 80 2 0.370785

Cancer 2 2 30 100 2 0.3670



123

proposed Fuzzy-VMODEPS/Fuzzy-VPSOPS clustering

techniques are capable of detecting the correct partitioning

from this dataset. These algorithms behave similar to

VGAPS. As the clusters are having symmetrical structures,

the proposed point symmetry-based clustering techniques are

capable of detecting the partitionings correctly. Note that a

close investigation reveals that some points in the ellipsoidal

cluster are erroneously allocated to the ring cluster in the

partitionings identified by VGAPS/Fuzzy-VMODEPS/

Fuzzy-VPSOPS approaches. This is because of the overlap-

ping nature of these clusters. These fewpoints are very near to

the ring-shaped cluster. Thus, it is difficult to identify those

points properly as belonging to the ellipsoidal-shaped cluster.

Figure 6 shows the clusters obtained by GCUK, VGAPS/

Fuzzy-VMODEPS/Fuzzy-VPSOPS, HNGA clustering

techniques for Sph_4_3 dataset. As is evident, genetic

algorithm-based K-means (GCUK), HNGA and all the point

symmetry-based clustering techniques, VGAPS, Fuzzy-

VMODEPS, Fuzzy-VPSOPS, are successful in providing the

proper clusters. This is because this dataset possesses some

hyperspherical-shaped clusters. As in this dataset all the

clusters are well-separated, all the approaches used in the

current paper are able to identify those properly (Fig. 7).

Figure 8a–c shows the clusters obtained by GCUK,

VGAPS/Fuzzy-VMODEPS/Fuzzy-VPSOPS, and HNGA

clustering techniques, respectively, for Mixed_3_2 dataset.

Here again all the point symmetry-based clustering tech-

niques are capable of detecting the proper partitioning. But

GCUK and HNGA clustering techniques fail to detect the

same. This is because of the presence of some overlapping

clusters having symmetrical shapes and structures. The use

of this dataset shows the utility of point symmetry-based

distance for properly detecting symmetrical structures.

Again a close observation reveals that some points of the

big hyper-spherical cluster are erroneously allocated to the

ellipsoidal-shaped cluster as there is a large overlap

between these two clusters. This is because those points are

having low symmetrical distances with respect to the

hyperspherical cluster than the ellipsoidal cluster.

For real-life datasets, visualization is not possible as

these are high-dimensional datasets. For these datasets in

order to quantify the partitioning results obtained by

−1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

1.5

2

−1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

1.5

2

(b)(a)

−1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

1.5

2

(c)

Fig. 5 Clustering of Sym_3_2 for C ¼ 3 after application of (a) GCUK (b) VGAPS/Fuzzy-VPSOPS/Fuzzy-VMODEPS (c) HNGA clustering

technique


123

different clustering techniques, we have used an external

cluster validity index, Minkowski Score [8]. This is a

cluster validity measure responsible for checking the sim-

ilarity between the obtained partitioning and the available

true cluster information. Here for the real-life datasets, we

have the actual class information known to us. This is used

to measure the goodness of different obtained partitionings.

The Minkowski Score values obtained by these clustering

techniques for all the real-life datasets are shown in

Table 3. Results show that for Iris dataset, Fuzzy-VMO-

DEPS clustering technique performs the best. It attains the

minimum Minkowski Score. Fuzzy-VPSOPS clustering

technique also performs better than VGAPS clustering

technique. GCUK and HNGA clustering techniques per-

form slightly poorly for this dataset. This is because for this

dataset even though there exists total three clusters but

there is a big overlap between two clusters. Thus, most of

the algorithms detect only two clusters from this dataset.

But our proposed techniques are capable of determining the

appropriate partitioning with three clusters from this data-

set. For Cancer dataset, again Fuzzy-VMODEPS performs

well where as VGAPS and Fuzzy-VPSOPS perform simi-

larly but poorly as compared to Fuzzy-VMODEPS. So for

Cancer dataset all the point symmetry distance-based

clustering techniques perform well. Fuzzy-VPSOPS and

VGAPS attain the same Minkowski Score. But the perfor-

mance of GCUK and HNGA clustering techniques is

comparatively poor. For Newthyroid dataset, again Fuzzy-

VMODEPS and Fuzzy-VPSOPS perform well compared to

VGAPS, GCUK and HNGA. Fuzzy-VMODEPS attains the

lowest Minkowski Score among all the clustering algo-

rithms. But GCUK and HNGA clustering techniques again

perform poorly for this dataset. For Wine dataset, the per-

formance of Fuzzy-VMODEPS and Fuzzy-VPSOPS is

better than VGAPS. The Minkowski Score values attained

by Fuzzy-VMODEPS and Fuzzy-VPSOPS clustering

techniques are lower compared to VGAPS clustering

technique. But again GCUK and HNGA clustering tech-

niques perform poorly for this dataset. Those attain some

higher Minkowski Score values compared to other tech-

niques. For Glass dataset, again the newly developed point

symmetry-based clustering techniques perform better than

VGAPS. Both Fuzzy-VPSOPS and Fuzzy-VMODEPS

clustering techniques attain the same value of Minkowski

Score. But again GCUK and HNGA clustering techniques

fail to identify the proper partitioning. These approaches

attain some higher values of Minkowski Score. But for

LiverDisorder dataset, again Fuzzy-VMODEPS clustering

technique performs the best in terms of Minkowski Score.

Fuzzy-VPSOPS clustering technique performs the second

best in terms of the obtained Minkowski Score. VGAPS

performs poorly for this dataset. It attains the maximum

value of Minkowski Score.

−50

510

1520

−50

510

1520−5

0

5

10

15

20

−50

510

1520

−50

510

1520−5

0

5

10

15

20

(a)

(b)

Fig. 6 a Sph_4_3. b Clustering of Sph_4_3 obtained after application

of GUCK/VGAPS/Fuzzy-VPSOPS/Fuzzy-VMODEPS/HNGA clus-

tering technique

−10 −8 −6 −4 −2 0 2 4 6 8−2

0

2

4

6

8

10

Fig. 7 Mixed_3_2


123

We have also reported the number of clusters obtained

by different clustering techniques (refer to Table 4). It can

be seen from this table that the proposed Fuzzy-VPSOPS

and Fuzzy-VMODEPS clustering techniques are capable of

determining the appropriate number of clusters automati-

cally from different real-life datasets where most of the

existing techniques fail to do so.

In order to prove the efficacy of the used objective

function, FSym-index we have also performed experiments

with another cluster validity index as the objective func-

tion. Here in both the DE- and PSO-based frameworks, XB-

index [45], another well-known cluster validity index is

used as the objective function. Results obtained using XB-

index as the objective function are shown in Tables 3 and

4. The obtained results clearly show that FSym-index is

indeed a better cluster quality measure than the XB-index.

The use of FSym-index helps the proposed approaches to

automatically determine the appropriate partitioning from

all the datasets. The poor performance of XB-index is

because of the use of Euclidean distance in its computation.

In the proposed approaches, point symmetry-based dis-

tance is used for allocating points to different clusters.

Thus, there is a mismatch in the distance measure used for

allocation and for computing objective functions.

Thus, results on a wide variety of datasets reveal the

fact that the proposed DE- and PSO-based clustering

techniques using point symmetry-based distance are

much capable of automatically identifying partitionings

from some given datasets compared to another existing

genetic algorithm with point symmetry-based automatic

clustering technique, VGAPS clustering technique. The

proposed algorithms are more robust in detecting the

optimal partitionings. Those are capable of identifying

clusters having point symmetry properties without hav-

ing knowledge about the total number of clusters present

in the dataset. These DE- and PSO-based approaches

obtain better partitioning results than VGAPS in terms of

Minkowski Score. The comparison results with some

genetic automatic clustering techniques using Euclidean

distance for cluster assignment like GCUK and HNGA

−10 −8 −6 −4 −2 0 2 4 6 8−2

0

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8−2

0

2

4

6

8

10

(b)(a)

−10 −8 −6 −4 −2 0 2 4 6 8−2

0

2

4

6

8

10

(c)

Fig. 8 Clustering of Mixed_3_2 for C ¼ 3 after application of (a) GCUK (b) VGAPS/Fuzzy-VPSOPS/Fuzzy-VMODEPS (c) HNGA clustering

technique


123

prove that the point symmetry-based clustering tech-

niques are more capable of handling symmetrical-shaped

clusters. The proposed algorithms also perform better

than the traditional clustering techniques in partitioning

some real-life datasets. The real-life datasets used in the

current paper are some higher-dimensional datasets. But

the obtained results show that the proposed techniques

are capable of handling these real-life datasets as well.

But most of the existing traditional techniques fail to do

so. The developed symmetry-based clustering techniques

can partition any higher-dimensional datasets and are

able to detect clusters automatically.

6.3 Statistical test

In order to prove the effectiveness of the proposed point

symmetry-based clustering techniques statistically, we

have also conducted some statistical significance tests

guided by [12, 18]. Friedman statistical test [17] is per-

formed to establish whether the five clustering techniques,

Fuzzy-VMODEPS, Fuzzy-VPSOPS, VGAPS, GCUK and

HNGA used here for the experimental work are similar or

not. Each algorithm is assigned some rank after application

of this statistical test. There is a check to test whether the

difference between the calculated average ranks and the

Table 3 Minkowski Score values corresponding to different cluster-

ing techniques for different datasets; FVMODEPS: Fuzzy variable

length modified differential evolution with point symmetry-based

clustering technique; FVMODEPSXB: Fuzzy-VMODEPS clustering

technique using XB-index as the objective function; VGAPS: variable

length genetic algorithm with point symmetry distance-based

clustering technique; FVPSOPS: Fuzzy variable length modified

particle swarm optimization with point symmetry-based clustering

technique; FVPSOPSXB: Fuzzy-VPSOPS clustering technique using

XB-index as the objective function; GCUK: variable string length

genetic K-means algorithm; HNGA: hybrid niching genetic algorithm

Dataset FVMODEPS FVMODEPSXB VGAPS FVPSOPS FVPSOPSXB GCUK HNGA

Iris 0.61 0.840473 0.62 0.61 0.762738 0.847726 0.854081

Cancer 0.346018 0.4286 0.367056 0.367056 0.8534 0.386768 0.380332

Newthyroid 0.553785 0.8658 0.58 0.563478 0.894016 0.828616 0.838885

Wine 0.90095 1.6783 1.0854 0.943561 1.0027 1.2 0.97

Glass 0.8023 0.9262 1.106217 1.098560 0.9283 1.324295 1.117940

Liver disorder 0.968227 0.981873 0.987329 0.981923 0.982613 0.982611 0.981873

Table 4 Number of clusters obtained by different clustering techniques for different datasets: AC = actual number of clusters present in the

dataset

Dataset AC FVMODEPSXB FVMODEPS VGAPS FVPSOPSXB FVPSOPS GCUK HNGA

Iris 3 2 3 3 4 3 2 2

Cancer 2 2 2 2 3 2 2 2

Newthyroid 3 2 3 3 5 3 8 5

Wine 3 6 3 2 9 3 4 3

Glass 6 3 6 6 8 6 3 6

Liver disorder 2 2 3 3 2 2 2 2

Table 5 Ranking computations

for the five algorithms over six

datasets based on the Minkowski

Score values obtained

Dataset Fuzzy-VMODEPS VGAPS Fuzzy-VPSOPS GCUK HNGA

Iris 0.61 (1) 0.62 (2) 0.61 (1) 0.847726 (3) 0.854081 (4)

Cancer 0.346018 (1) 0.367056 (2) 0.367056 (2) 0.386768 (4) 0.380332 (3)

Newthyroid 0.553785 (1) 0.58 (3) 0.563478 (2) 0.828616 (4) 0.838885 (5)

Wine 0.90095 (1) 1.0854 (4) 0.943561 (2) 1.2 (5) 0.97 (3)

Glass 0.8023 (1) 1.106217 (3) 1.098560 (2) 1.324295 (5) 1.117940 (4)

Liver disorder 0.968227 (1) 0.987329 (5) 0.981923 (3) 0.982611 (4) 0.981873 (2)

average rank 1 3.17 2 4.17 3.5



123

mean rank is significant or not. Friedman test proves that

for the proposed algorithms, the measured average ranks

and the mean rank are different with a p value of 0.0106.

The ranks are reported in Table 5. At the end, Nemenyi’s

test [32] is also performed for the pair-wise comparison of

the clustering approaches. Here we have used a ¼ 0:05.

Results reveal that for all the datasets we can reject the null

hypotheses which state that pairing algorithms work in a

similar way (as the corresponding p values are smaller than

a). Results reported in Table 5 also reveal that Fuzzy-

VMODEPS is the rank 1 algorithm among all the algo-

rithms used here for the purpose of experiments. The sec-

ond best algorithm is Fuzzy-VPSOPS. Sometimes its

behavior is similar to Fuzzy-VMODEPS, but sometimes it

performs poorly compared to Fuzzy-VMODEPS.

7 Experimental results for gene expression dataclassification

This section discusses about results obtained by the pro-

posed clustering techniques for various publicly available

gene expression datasets which are used for the experi-

mental analysis. The different performance metrics are also

described in this section. Thereafter, obtained experimental

results are demonstrated quantitatively as well as by using

visualization tools. Here five gene expression datasets are

used. A small description of those datasets is provided

below.

Yeast Sporulation: This dataset [30] is having 6118 gene

expression levels and has been measured over seven time

points those are (0, .5, 2, 5, 7, 9 and 11.5 h). Out of total

6118 genes, some genes are ignored whose expression

levels are not changed during harvesting. Finally, 474

genes have been used for analysis. This dataset is available

from Web site .1

Yeast Cell cycle: Here in this dataset [30], approxi-

mately 6000 genes over 17 time points have been consid-

ered. The expression levels of genes where there are no

substantial changes have been rejected. Finally, from 6000

genes 384 have been chosen and others are ignored. This

dataset is down-loadable from Web site.2

Rat CNS: This dataset [30] is having 112 gene expres-

sion levels and has been measured over nine time points.

This dataset is available from Web site.3

Arabidopsis Thaliana: This dataset [30] is having 138

gene expression levels and has been measured over eight

time points. This dataset is available from Web site.4

Serum: This dataset consists of 8613 genes where each

gene is having total 13 dimensions corresponding to 13

time points. Out of total 8613 genes, 517 genes have been

considered for experimental analysis and other genes have

Fig. 9 Cluster profile plots for Serum dataset obtained by Fuzzy-VMODEPS approach

1 http://cmgm.stanford.edu/pbrown/sporulation.2 http://faculty.washington.edu/kayee/cluster.3 http://faculty.washington.edu/kayee/cluster.4 http://homes.esat.kuleuven.be/thijs/Work/Clustering.html.


123

http://cmgm.stanford.edu/pbrown/sporulation

http://faculty.washington.edu/kayee/cluster

http://faculty.washington.edu/kayee/cluster

http://homes.esat.kuleuven.be/thijs/Work/Clustering.html

been ignored because of no change in expression level over

12 time points. This dataset is available from Web site.5

7.1 Chosen validity measure

To measure the quality of obtained gene clusters, we have

chosen the following cluster quality measurement index. It

is described as follows.

Silhouette Index S(I) [37], an internal cluster validity

measure has been utilized to quantify the effectiveness of

the clustering solution obtained by the proposed

approach. Let p be one parameter of the S(I) index and

it has been calculated by the average distance of a point

from other points of the same cluster. Likewise q is also

another parameter of S(I) index and has been calculated

by the minimum average distance of a point from the

points of other clusters. Now the S(I) index value is

calculated based on the parameters p and q, which is

defined below:

SðIÞ ¼ q� p

maxðp; qÞ ð12Þ

Here Silhouette index S(I) is considered as the average

silhouette values over all the points. Silhouette index

measures the separability and compactness of clusters. The

value of silhouette index varies from -1 to þ1. So best

partition yields higher positive value of S(I) index.

7.2 Cluster profile plot

Cluster profile plot [3] (example Figs. 9, 10,11, 12) is used

to represent the expression values of genes over different

Fig. 10 Cluster profile plots for Yeast Sporulation dataset obtained by Fuzzy-VMODEPS approach

5 http://www.sciencemag.org/feature/data/984559.shl.


123

http://www.sciencemag.org/feature/data/984559.shl

time points. The expression value of a gene is denoted by

light blue color. Also, here red color is used to represent the

average expression value for each of the cluster of genes.

7.3 Eisen plot

In the Eisen plot [14] (example Figs. 13, 14, 15, 16) using

similar colors of spot on the microarray, a particular cell of

the gene data matrix is colored, and in the similar way, a

specific time point expression value of gene is specified. In

the figure, different expression levels of genes are specified

by shades of different colors. For example, the red color,

green color and black color shade represent higher and

lower expression levels of genes and also an absence of

expression values, respectively. In this paper before plot-

ting genes in the Eisen plot, all the genes have been ordered

in such a way that genes belonging to the same cluster are

put one after another. Here white color is used as a cluster

boundary separator.

7.4 Discussion of results

The proposed Fuzzy-VMODEPS and Fuzzy-VPSOPS clus-

tering techniques use the search capabilities of differential

evolution and particle swarm-based optimization technique,

respectively. The parameter combinations used for different

clustering techniques are provided below. For Fuzzy-

VMODEPS, the following parameter values are used: pop-

ulation size: 100, maximum number of generations: 30,

CR = 0.04 and F = 0.8. For Fuzzy-VPSOPS, the following

parameter combinations are used: maximum number of

iterations=30, swarm size=100, co1, co1 = 2, w: 0.9–0.4.

The experimental results of Fuzzy-VMODEPS and Fuzzy-

VPSOPS are shown over five real-life gene datasets like

Fig. 11 Cluster profile plots for Yeast Cell Cycle dataset obtained by Fuzzy-VMODEPS


123

Yeast cell cycle, Yeast sporulation, Serum, Thaliana and

RatCNS. Table 6 reports the Silhouette index values

obtained by the proposed approaches over five gene datasets

described in the current paper. Table 6 also reports the

Silhouette index values obtained by VGAPS [38], GCUK

[6], HNGA [41], some well-known automatic clustering

approaches. Table 6 reveals that the proposed Fuzzy-

VMODEPS clustering technique attains higher S(I) value for

all the datasets compared to other clustering techniques,

Fig. 12 Cluster profile plots for RatCNS dataset obtained by Fuzzy-VMODEPS

Fig. 13 Eisen plot for Serum dataset obtained by Fuzzy-VMODEPS

Fig. 14 Eisen plot for Yeast Sporulation dataset obtained by Fuzzy-

VMODEPS


123

namely Fuzzy-VPSOPS, VGAPS, GCUK and HNGA. In

Table 7, number of clusters obtained by each of those

algorithms has been reported. It can be seen from this

table that Fuzzy-VMODEPS approach identifies the correct

number of clusters from all the datasets used here for

experimental purpose. From Table 6, it is observed that

Fuzzy-VMODEPS approach performs the best among all the

algorithms for clustering the five gene expression datasets.

In order to visually inspect the obtained results by the

proposed Fuzzy-VMODEPS clustering technique, the Eisen

plots (see Figs. 13, 14, 15, 16 for example) and cluster profile

plots (see Figs. 9, 10,11, 12) have also been drawn. Cluster

profile plots show the distributions of expression values of

different genes which belong to a single cluster obtained by

the proposed approach over different time points. The com-

pactness or similarities of variations in the expression values

by different genes belonging to the same cluster proves that

genes are indeed similar in functionality. More is the com-

pactness, better is the cluster in terms of expression values

given. In case of Eisen plot, the presence of similar colors in

the same position represents the goodness of cluster. From the

correspondingEisen plot, we can see that genes having similar

expression profiles (denoted by similar colors) are grouped

together and placed in the same cluster by the proposedFuzzy-

VMODEPS clustering technique. The genes with different

expression values are placed in different clusters, which is

represented by cluster profile plot. The obtained plots clearly

show the effectiveness of the proposed technique, Fuzzy-

VMODEPS, for clustering five gene expression datasets, as

compared to other existing data clustering techniques, Fuzzy-

VPSOPS, VGAPS, GCUK and HNGA.

7.5 Biological significance test

In this paper, at 1% significance level biological signifi-

cance test has been conducted for Yeast Sporulation data-

set. Now in order to establish biological relevancy of

clusters, Gene Ontology annotation database (http://db.

yeastgenome.org/cgi-bin/GO/goTermFinder) is used. Here

all the six clusters obtained by Fuzzy-VMODEPS are

biologically significant, whereas for Fuzzy-VPSOPS,

VGAPS, GCUK and HNGA, number of biological signif-

icant clusters are 4, 4, 2 and 2, respectively. Most signifi-

cant GO terms and the corresponding p values for each of

the six clusters (obtained by Fuzzy-VMODEPS) of Yeast

Sporulation dataset have been reported in Table 8. From

Table 8, we can observe that all the clusters obtained by

the proposed Fuzzy-VMODEPS clustering technique are

Fig. 15 Eisen plots for Yeast cell cycle dataset obtained by Fuzzy-

VMODEPS

Fig. 16 Eisen plots for RatCNS dataset obtained by Fuzzy-

VMODEPS


123

http://db.yeastgenome.org/cgi-bin/GO/goTermFinder

http://db.yeastgenome.org/cgi-bin/GO/goTermFinder

biologically relevant because p values corresponding to

GO categories are less than 0.01.

7.6 Execution time

Both the proposed algorithms have been implemented

using standard C??, and the experiments are performed

on a Intel (Core i-5) processor having 2.4 GHz machine

with 4.0 GB RAM under Linux platform. Moreover,

MATLAB 7.5 version is used to draw the cluster profile

plot and Eisen plot and also for the computation of Sil-

houette index.

In Table 9, we have summarized the execution times

taken by different clustering algorithms used in the

current study for partitioning different datasets. This

table clearly shows that our proposed approaches are

much faster compared to other genetic algorithm-based

techniques.

Table 6 Silhouette Index

values obtained by five

clustering algorithms for five

gene expression datasets


Yeast Sporulation 0.7060 0.6391 0.6520 0.5781 0.6263

Yeast Cell Cycle 0.4531 0.3595 0.4329 0.1741 0.2379

Arabidopsis 0.3557 0.3453 0.3524 0.3194 0.2748

RatCNS 0.4386 0.411 0.4123 0.3125 0.2805

Human Fibroblasts Serum 0.3807 0.3506 0.3691 0.2681 0.2455

Table 7 Number of Clusters

obtained by five clustering

algorithms for different gene

expression datasets


Yeast Sporulation (C = 6) 6 4 4 2 2

Yeast Cell Cycle (C = 5) 5 2 5 10 9

Arabidopsis (C = 4) 4 3 2 5 8

RatCNS (C = 6) 6 3 6 9 12

Human Fibroblasts Serum (C = 6) 6 2 6 15 14

Table 8 Some of the most significant GO terms obtained by Fuzzy-VMODEPS clustering technique and the corresponding p values for each of

the six clusters of Yeast Sporulation dataset have been shown

Clusters Significant GO term p value

Cluster1 Anatomical structure formation involved in morphogenesisanatomical structure formation involved in morphogenesis

GO:0048646

2.60e-40

Sporulation GO:0043934 8.37e-40

Sporulation resulting in formation of a cellular spore GO:0030435 6.19e-39

Cluster2 Ribosome biogenesis GO:0042254 2.85e-15

Ribonucleoprotein complex biogenesis GO:0022613 2.14e-13

rRNA processing GO:0006364 2.32e-11

Cluster3 Meiotic nuclear division GO:0051327 1.52e-31

Meiosis GO:0007126 2.29e-29

Meiotic cell cycle GO:0051321 1.79e-26

Cluster4 Monocarboxylic acid metabolic process GO:0032787 8.80e-09

Oxoacid metabolic process GO:0043436 4.33e-07

Single-organism metabolic process GO:0044710 0.00222

Cluster5 Cytoplasmic translation GO:0002181 2.14e-56

Translation GO:0006412 1.86e-27

Peptide biosynthetic process GO:0043043 2.52e-27

Cluster6 Nicotinamide nucleotide metabolic process GO:0046496 2.89e-13

Pyridine nucleotide metabolic process GO:0019362 3.62e-13

Pyruvate metabolic process GO:0006090 5.35e-13


123

8 Discussion and conclusions

In this paper, we have proposed two Fuzzy bio-inspired

automatic clustering techniques which have utilized DE

and PSO as the underlying stochastic optimization tool,

respectively. Both the evolutionary techniques utilize the

established point symmetry-based distance for the alloca-

tion of points into different groups/clusters and an sym-

metry-based cluster validity index, F-Sym-index, as the

objective function. Moreover, proposed clustering tech-

niques can be able to detect total number of clusters present

in the dataset automatically. Recent studies show that PSO-

and DE-based approaches converge much faster than the

GA-based approach. Motivated by this fact, in the current

paper some automatic clustering techniques based on DE

and PSO have been proposed. Results on various synthetic

and real-life gene expression date sets indicate the superi-

ority of Fuzzy-VMODEPS-based technique over other

techniques like Fuzzy-VPSOPS, VGAPS, GCUK and

HNGA clustering techniques. In this context, five gene

expression datasets, namely Yeast Sporulation, Yeast Cell

Cycle, RatCNS and Serum, have also been clustered by the

proposed point symmetry-based DE and PSO clustering

techniques and the obtained results are compared with

other techniques, namely VGAPS, GCUK and HNGA. The

proposed algorithms often perform better than the existing

GA-based techniques in terms of cluster quality even for

gene expression datasets. The obtained results prove the

utility of using PSO- and DE-based algorithms as the

underlying optimization strategies. Results also prove that

DE-based approach is better than both PSO and GA-based

approaches. The results on gene expression datasets further

prove the applicabilities of the proposed clustering tech-

niques for solving some real-life problems.

As a scope of future work, some real-life applications of

the proposed clustering techniques can be done for classi-

fication of remote sensing images and MRI brain images,

etc. In the recent years, some multi-objective versions of

DE and PSO are also available. In general, multi-objective-

based algorithms perform better than their single objective-

based versions. In future, we would like to develop some

multi-objective-based clustering techniques using the

search capabilities of DE and PSO.

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