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ISCMI15
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
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
https://doi.org/10.1007/s00521-016-2710-0
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
738 Neural Comput & Applic (2018) 30:735–757
123
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.
Neural Comput & Applic (2018) 30:735–757 739
123
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.
740 Neural Comput & Applic (2018) 30:735–757
123
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.
Neural Comput & Applic (2018) 30:735–757 741
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
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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
Neural Comput & Applic (2018) 30:735–757 743
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
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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
Bold values indicate best performances
Neural Comput & Applic (2018) 30:735–757 745
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
746 Neural Comput & Applic (2018) 30:735–757
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
Neural Comput & Applic (2018) 30:735–757 747
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
748 Neural Comput & Applic (2018) 30:735–757
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
Bold values indicate best performances
Neural Comput & Applic (2018) 30:735–757 749
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.
750 Neural Comput & Applic (2018) 30:735–757
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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.
Neural Comput & Applic (2018) 30:735–757 751
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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
752 Neural Comput & Applic (2018) 30:735–757
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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
Neural Comput & Applic (2018) 30:735–757 753
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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
754 Neural Comput & Applic (2018) 30:735–757
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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
Dataset Fuzzy-VMODEPS VGAPS Fuzzy-VPSOPS GCUK HNGA
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
Dataset Fuzzy-VMODEPS VGAPS Fuzzy-VPSOPS GCUK HNGA
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
Neural Comput & Applic (2018) 30:735–757 755
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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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