Sci.Int.(Lahore),27(3),2055-2061,2015
ISSN 1013-5316; CODEN: SINTE 8
2055
PROPOSED DYNAMIC CLUSTERING BASED ON DISCRETE PSO-GA
Hamid Tabatabaee 1,* ,Maliheh Hassan Pour2,Ameneh fakhraee3, ,Mohammad Reza Baghaeipour4
1,* Young Researchers and Elite Club, Quchan Branch, Islamic Azad University, Quchan, Iran.
3,2 Department of Computer Engineering ,Mashhad Branch, Islamic Azad University, Mashhad, Iran.
4 Department of Computer Engineering, Ferdows Higher Education Institute, Mashhad, Iran.
ABSTRACT: Data clustering is the process of identifying clusters or natural groups based on similarity measures. Despite the
improved clustering analysis algorithms, most of the clustering algorithms likewise require number of clusters as an input
parameter. Hence, this paper describes a new method of clustering, called dynamic clustering based on discrete particle
swarm optimization and genetic algorithms (DPSO-GA). The proposed algorithm automatically determines the optimal
number of clusters and at the same time clusters the data set with minimal user intervention. In this method, the discrete
PSO algorithm as well as genetic are employed with a new display solutions which are of variable length to determine the
number of clusters and initial cluster centers for K-Means algorithm to improve its performance. The results obtained from five
UCI evaluation datasets show that the proposed algorithm has higher performance in comparison with other methods.
Keywords: Clustering, Particle swarm optimization, Genetic algorithms.
1. INTRODUCTION
Clustering is one of the important category of Unsupervised
learning methods for grouping objects into groups distinct
from the clusters [ 1]. Cluster grouping is usually defined as
the objects grouped into one cluster are similar according to
the given criteria while they are different with the objects
from other clusters according to the criteria given [ 1] .
In recent years, clustering is widely used in various fields
such as pattern recognition, machine learning and market
analysis [ 2 ]. Analysis of clustering is a significant technique
in data mining [ 3 ] . The primary objective of clustering is to
put similar data within a cluster and then collect the highsimilarity clusters [ 3 ]. Through clustering, valuable
information such as the distribution and characteristics of
data which come from a very large data is obtained and
hidden information is extracted by reducing the complexity of
the data. Thus, the results of clustering can be effectively
applied to solve problems or to help with decision making.
Four types of clustering algorithms have been identified: the
first group of clustering algorithms is based on the idea that
the neighborhood data should be placed in the same
cluster. As it is stated in [ 4 ] this type of algorithms is robust
to recognize the clusters of any shape, but it fails when there
is a small spatial separation between the clusters. The second
class of clustering algorithms has been made by means of the
dynamic changes in the cluster to make the ultimate
solution. These algorithms are the most popular clustering
algorithms contains k-means [ 5 ] and the other category is
based on model-based clustering [ 6 ]. The fourth class
of clustering algorithms characterized by different methods
which optimize the different attributes of data set [ 7 , 8 ].
So far, the speed and efficiency of data processing has been
the focus of clustering development and the most clustering
methods such as k-means need to get the number of clusters
as an input parameter. Appropriate number of clusters may
have an impact on clustering results. For example, the smaller
number of clusters can help the clarity of the
original data structure but the hidden information cannot be
detected. On the other hand, using large number of clusters
may result in high heterogeneity of the same cluster sets but it
ignores the basic structure of data [ 3] .
In recent years, due to their ability to solve various problems
with very little change, evolutionary computation algorithms
are widely used for clustering problems. On the other hand,
these algorithms can manage the limitations in a good
way. Hruschka et al. [ 9 ] have applied evolutionary
algorithms to clustering problems in which the genetic
algorithm is used employed. In this study, a simple coding
procedure manipulates the fixed length individuals. The
objective function maximizes the between-cluster similarity
and between-cluster dissimilarity. In addition, this procedure
finds the true number of clusters based on external
criteria. There are various kinds of evolutionary methods that
have studied clustering problems, such as evolutionary
programming [ 10 ], particle swarm optimization [ 11 , 12 ],
ant colony algorithm [ 13 ] or bee colony algorithm [ 14 ].
When
the
length
of
display
solutions
in evolutionary algorithms is equal to the number of data,
then clustering efficiency will be reduced when there is a
large number of data [ 15 ].So this paper suggests the
dynamic clustering method based
on discrete
particle
swarm optimization and genetic algorithm (DPSO-GA) . The
proposed algorithm automatically determines the optimal
number of clusters and simultaneously clusters the data set
with minimal user intervention.
The proposed method uses discrete PSO algorithm [ 16 ] and
genetic algorithms with a new display solutions which are of
variable length, we use the number of clusters and the
primary centers of the clusters to specify the algorithm of our
K-Means and thus raise the efficiency of clustering.
The rest of the paper is organized as follows: In Section 2,
particle swarm optimization algorithm and discrete particle
swarm optimization algorithm is described. In Section 3 and
Section 4, the genetic algorithms are briefly explained. In
Section 5, the details of the proposed algorithm are
given. Experimental results on five databases are reported in
space Section 6. Finally, concluding remarks are presented in
Section 7.
2.Particle swarm optimization Algorithms (PSO)
PSO Algorithms [17 ] acts on a population of particles.
In PSO the practical solution is called a particle and a fitness
value of each particle is given through the objective
function. Each particle has its own velocity (V id) and
location (X id). The velocity and position of particle is
modified according to its own flying experience and the
population of particles. Thus, each particle moves towards the
probability of best given ( pbest ) and the global best
( gbest ). Initially, the algorithm randomly produces the initial
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velocity (V id) and the initial position (X id Points) of each
particle. Each particle remembers its best fitness value, and
further, the individual values are compared to determine the
best of the best called global best ( gbest ). Finally, the
velocity and position of each particle is modified by the
optimal individualal solution and the global best
solution. The optimal solution is determined by repeating the
calculations. Therefore, the update equation of the particle
velocity and position are the relations (1 ) and (2).
𝑣𝑖𝑑 = 𝑤𝑣𝑖𝑑 + 𝑐1 × 𝑟𝑎𝑛𝑑1 × (𝑝𝑏𝑒𝑠𝑡𝑖𝑑 − 𝑋𝑖𝑑 ) + 𝑐2 ×
𝑟𝑎𝑛𝑑2 × (𝑔𝑏𝑒𝑠𝑡𝑑 − 𝑋𝑖𝑑 )
(1)
Xid = X id + vid
(2)
Where c 1 and c 2 are the individualized and global learning
factors respectively and rand 1 and rand 2 are random
variables between 0 and 1, and w 1is the inertia factor.
3.Discrete PSO Algorithm
Discrete PSO algorithm deals with discrete variables and the
length of solutions (particles) are according to the number of
data (p = number of data). Amount of each component of the
particle (vector) is between 1 and n (k = number of clusters)
that represent the clusters that have been assigned to
it. Particle X i records the best acquired place into a separate
particle called pbest i. The Particle populations, in addition,
save the best global position demonstrated by gbest [ 16 ].
The initial population of particles is produced by a set of
randomly generated integers between 1 and n. The particles
are fixed-length strings of integers are encoded as follows:
Xi = {xi1 , xi2 , … , xip }
in
which xij ∈ {1,2, . . , n}, i =
1,2, . . , m , j = 1,2, … , p.
For example, given the number of 3 clusters (k = 3) and 10
data (p = 10), the particle X1 = {1,2,1,1,2,2,1,3,3,2} indicates
a candidate solution that in this way, data solutions 1, 3, 4, 7,
have been assigned to Cluster 1and data 2, 5, 6 and 10 to
Cluster 2 and data 8 and 9 to Cluster 3. The particles
population thus is produced and fitting of each particle is
calculated.
The particle velocity is a set of transfers applied to
solutions. The rate of (transfer) each particle is calculated by
subtracting the two positions (of particles). The difference
between X i and pbesti suggest changes that will be necessary
for particle i to move from X to pbest. μ is the number of
elements against zero in subtraction. If the difference
between an element of X i and pbesti is zero, it means
that there is the possibility to change the location and that
element is subjected to change by the operation described
below.
First, the new vector P is generated in which the locations of
elements in subtracting vectors against zero are stored. We
generate a random number β. This number (β ) corresponds to
the number of changes that must be made on the basis of the
difference between X i and X i pbest i , so β lies in the range
(0, μ ). Then we generate the binary vector Ψ according to
measure μ where each component is associated with a
component of vector P. The β number of elements in the
vectorΨ is randomly initialized with 1 and the remainder is
equal to zero. If the binary number is equal to
1 means that the component must change, and if the binary
number is equal to zero, there is no need to change . A similar
process is run based on the updated particle position and the
Sci.Int.(Lahore),27(3),2055-2061,2015
global best position to calculate the particle in next step [16].
For instance, if
(𝑋𝑖 − 𝑝𝑏𝑒𝑠𝑡𝑖 ) = (1 − 1, 2 − 3, 1 − 2, 1 − 1, 2 − 2, 2 −
1, 1 − 2, 3 − 3, 3 − 2, 2 − 1) =
(0, −1, −1, 0, 0, 1, −1, 0, 1, 1)
(3)
The new P vector is equal to P = (2,3,6,7,9,10) and
suppose μ = 6 , β = 5 and ψ = (0,1,1,1,1,1).
This means that the element positions 3, 6, 7, 9 and 10 in the
vector X i are initialized by the same element positions from
pbest i. vector. The process is repeated with new particle and
gbest until the final particle is achieved.
4.Genetic Algorithms
GA algorithm is based on the idea of "survival of the fittest in
natural selection" suggested by Darwin. Algorithms GA
consists of selection operators 2 , reproduction operators 3 crossover 4 and mutation operators 5 . The fitted genes
passed on the offspring by the best parents of the generation
to reach the optimal solutions during the search relatively
quickly. GAs is employed in many applications.
5.The proposed algorithm
In the proposed method, a new display is used to show the
particles in order to reduce the size of particles. In the
following sections, first the new display and
crossover, mutation and elitism operators will be described
and then the proposed algorithm is given.
5.1
Display
In the proposed method for clustering procedure the coding
method by Falkenaure [ 18 , 19 ] in combination with the
aforementioned coding method [ 15 ] are used.. This means
that
every
individual
has
a
variable
length
for displaying solutions.
In the proposed display each individual is composed of two
parts c = [l, g] in which the first part refers to element section
and the second part shows the grouping section of the
individual.
In element section, every position is assigned to Element in
every individual, every place into a small group of data (sub
X). In this method, the stored value in this position shows the
cluster to which the data in the X group belong. For example,
the different parts of the solution (individual) for clustering
of N subgroups and k clusters is shown by following
relationship (4 ).
l1 , l2 , … , lN |g1 , g 2 , … , g k
(4)
li Indicates a cluster that the data in the subgroup i have been
assigned to it, while the group section represents the list of
tags associated with solution clusters.
lj = g i ⇔ ∀xk ∈ Xj , xk ∈ Gi
(5)
Note that the length of the element for the given problem is
fixed (equal to N), but it is not constant throughout the length
of the group and will vary from individual to
individual. Therefore, the proposed method does not require
the number of clusters as an input parameter but investigates
the fitted value of k based on the objective function.
For example, Figure 1 shows a model of proposed method
solution to the problem of clustering. In this problem, of data
clustering lies within five subgroups so the length of the
element equals to 5.In this solution, the subgroup data X 1
belongs to cluster G1 , the subgroup data X 2 and
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Figure 1 Particle display in the algorithm DPSO-GA.
X 5 belongs to G2 and the subgroup data X 3 and
X 4 belongs to cluster G3 .
With this display methodology, the near data must first be
grouped into Xi subgroups. The closest neighbor for each
data, k methodology can be used to use to get the groups X i.
5.2
Crossover Operator
Crossover operator used in genetic algorithm of this paper is
similar to [ 1 ] which has been accepted for the clustering
problem. This crossover operator creates a child from two
parents composed of the following steps:
Initially, two parents are chosen randomly and crossovering
is done on two points of their group.
Elements related to selected groups of individuals within the
child come first.
Elements from the selected group of second individual enter a
second child if the elements are not initialized previously by
the first individual.
Elements that have not been initialized before are assigned to
the current groups.
Empty clusters are removed.
Current group labels in children are modified from 1 to k.
Relationship (6 ) is another example of the crossover operator
implemented in this paper.
ind1 = [1 3 2 1 4 1 1 2 3 2 1 3 4 2 1|1↓ 2 3↓ 4]
a)
ind2 = [3 1 2 1 3 2 2 1 3 1 2 3 2 2 2|↑ 1 2↑ 3]
b)off = [− 3 2 − − − −2 3 2 − 3 2 3|2 3]
c)off = [− 3 2 1́ − 2́ 2́ 2 3 2 2́ 3 − 2 2́|2 3 1́ 2́]
d)off = [3 3 2 1́ 1́ 2́ 2́ 2 3 2 2́ 3 2 2 2́|2 3 1́ 2́]
e)off = [2 2 1 3 3 4 4 1 2 1 4 2 1 1 4|1 2 3 4]
(6)
5.3
Mutation Operator
Mutation operator makes small modifications in each
individual of population with a low probability of occurring
in A and D to discover the new areas of the search and when
the algorithm is close to the convergence escapes the local
optimum [1].
1-Mutation by dividing cluster: dividing the selected cluster
into two different clusters. Samples which belong to the
main cluster with equal probability are assigned to the new
cluster. Note that one of the new clusters produced keeps its
class tag in the group section while new label of k + 1 is
assigned to the other group.
2057
Selecting the first cluster to divide the cluster
size is associated with larger clusters are more
likely
to
split. For example, the application of this operator on the
individual used in the mutation operator of relation (6 ) is
shown in equation (7 ) in which cluster 1 is selected to divide.
22 1 3 3 4 4 5 2 1 4 2 5 1 4|1 2 3 4 5
(7)
Mutation operator and merging: Merging two existing
clusters selected randomly is done within a cluster. As for the
mutation in cluster division, the probability of cluster
selection depends on the size of clusters. To show the
function of this operator, this operator is implemented on the
individual whom the mutation operator is performed on
(relation (6 )). In this case, suppose two clusters 2 and 4 have
been selected for merging.
2 2 1 3 3 2 2 1 2 1 2 2 1 1 2| 1 2 3
(8)
Note that the two mutation operators are applied in a
sequence (one after another) with independent probabilities.
Comparative form of the mutation operator probability is
implemented. In this case, the mutation probability in the first
generations is lower and get higher in the last generations to
have the opportunity to escape from the local optimum.
j
Pm (j) = Pmi + (Pmf − Pmi )
(9)
TG
Pm (j) is the implemented mutation probability the
j generation, TG indicates the total number of generations
and Pmi and Pmf show the initial and final
values, respectively.
5.4
Replacement and elitism
In the proposed approach, the elitist approach has
been used for the construction of the next generation. In
elitism, the best individuals in j generation automatically are
placed to the j + 1 generation to ensure that the best
individuals found so far remains in the evolution by
the algorithm
5.5 Proposed algorithm
The algorithm proposed in this paper uses a discrete PSO
algorithm and genetic algorithm in which the mutation and
genetic crossover operators are added by discrete PSO
method. It is expected that the combined algorithm increase
the global search ability and escape from local optimal
solution. Discrete PSO algorithm which has memory
can establish the rapid correlations based on current optimal
solution with the velocity of the rapid convergence. But when
the optimal particle lens is a local optimal solution, the global
optimal
solution
cannot
be
found. Therefore,
this paper uses genetics to overcome this drawback. So, a
parent is produced by PSO and then the parent uses the
mutation and crossover operator in genetics to produce
another parent.
Finally, the next generation of parents selected by elitism is
determined. This selection process continues until the
terminating condition is met. Then k-means is used to modify
the cluster centers. Figure 3 shows the flowchart of the
proposed algorithm.
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Start
Subgroups formation: The subgroup to which
each data belongs is calculated by use of nearest
neighbors.
Generation of initial
population
Determine the best overall solution (gbest) and the best solution of
each particle (Pbesti)
Update the position of each particle
(Solution)
Update the population 1
Generation of
population 1
Update the population 2
Crossover between the best of
each particle (gbest) and the
general (pbest)
Mutation of the general best (gbest)
Generation of population 2
Elitism selection
Is the end
condition
satisfied?
K-means algorithm
End
Figure 2 Flowchart of the proposed algorithm.
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At first, the algorithm k is used to define the subgroup
nearest neighbors of each data to which they belong. Then,
the particle population is randomly generated based on the
display method mentioned in section 5.1. The length of each
particle is equal to the total number of subgroups where data
is located and the number of clusters that the subgroups were
assigned to, the length of the particles is changing. Then, a
discrete PSO and genetic algorithms are used to find the
initial cluster centers and the number of clusters after a
specified number of iterations. Then K-means is used for
setting more precise (better) results.
The steps of the proposed algorithm are as follows:
Step 1- Parameters values include population size (number of
particles), the maximum number of clusters (N c), and the
mutation rate are initialized.
Step 2- The subgroups of data using algorithm k, the nearest
neighbors are determined.
Step 3- The initial position (X i) of a single particle in a
population is generated according to equation (4 ).
Step 4: The values of the fitness of all particles are calculated.
Step 5- Pbest i (the probability of the best given particle i so
far) and gbest (the global best solution found so far) is
specified.
Step 6: The place of each particle is updated according to the
procedures set forth in Section 3.
Step 7- Perform steps 1 and 2 below on the parent updated in
Step 6:
Copy all the particles to generate the population 1.
Do Crossovering on two points calculated in step 5 on Pbest i
and gbest and the mutation on gbest. The population 2 is
produced by this process.
Step 8- Populations 1 and 2 are combined and the fitness
value of each particle is calculated.
Step 9-Elitism selection is applied on populations 1 and 2 to
generate the next generation
Step 10- Return to step 4 until a predefined number of
iterations will be obtained.
Step 11- According to gbest, the initial cluster centers and the
number of clusters are identified and then the identified
parameters are given to k-means algorithm as an input to
group the data into clusters.
6. EXPERIMENTS
Five data sets are used for clustering analysis and the results
of the proposed algorithm in comparison with DCPG
algorithms [ 3 ] is presented according to these 5 databases.
6.1 Databases
In this paper, different methods are tested on 5 databases
from UCI database. The database properties are provided
in the table .
Table 1
Database Properties
Number of
Number of
Database
data
Properties
Iris
150
4
Wine
178
13
Number of
Classes
3
3
Glass
214
9
6
Seed
210
7
3
Tissue
106
9
6
2059
6.2
Preprocessing
Data preprocessing is done on the basis that the maximum
and minimum values for each attribute in the data set is found
and then each attribute is subtracted from its minimum value
and the obtained value is divided by the difference between
maximum and minimum value of that attribute.
[3]. Equation (10) described methods to obtain a normalized
value of attribute i of data x:
x −x
xi = i min
(10)
xmax −xmin
6.3
Experiments Results and the Analysis
Setting parameters for the algorithm are shown in Table 2:
Table 2
Setting the values of algorithm parameters DPSO-GA
Value
Description of parameters
500
The total number of iterations
0.9- 0.1
Minimum and maximum values of the
mutation rate
20
Population size
1
crossover rates
In addition to the parameters listed in Table 1 , the maximum
number of clusters should be identified for the
implementation of the algorithm. Zhang et al. [ 20 ] stated
that the number of maximum clusters should not be more
than the root of the number of data in the dataset ; so the
maximum values of clusters for iris and Wine databases
equals to 13, for the Glass database is 15, Tissue database
equals to 11 and for the seed database is 15.
The number of experiments for each database is equal to 20
and the result average of these 20 times is reported as the
results of the algorithm.
6.4
Estimation Algorithm
Clustering Quality measurement method is based on the fact
that how well-connected the things are to each other. Former
clusterings stressed high similarity clustering of data set in
the same cluster and attempted to find the shortest distance
between data and the cluster center. In dynamic clustering,
the distance between and within clusters are considered as
measurement index. As a result, the modified index Turi
[ 21 ] is accepted in relation (11 ) .
intra
VI = (c × N(0,1) + 1) ×
(11)
inter
Where (c × N(0,1) + 1) is considered as penalty term to
prevent too many clusters; c is a constant equals to 30 and
N(0,1) shows a Gaussian function with the mean of 0 and the
standard deviation of 1 from the number of clusters.
N(μ, σ) =
1
√2πσ
e
2
(k−μ)2
]
2σ2
[−
(12)
Equation (12 ) shows the Gaussian distribution function with
the mean of µ and the standard deviation of σ.
Turi shows that the result of dynamic clustering is placed in
range 2 of the maximum number of clusters. Penalty term can
prevent the data to group in two clusters and enable the
clustering algorithms to find the appropriate number of
clusters [ 21 ] .
In this experiment, since Seed , Iris and Wine have a small
number of clusters, N (0.1) is accepted to calculate the
VI index and since Glass and Tissue have more clusters N
(2,1) is admitted instead of N (0, 1) to calculate the VI index
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Np
Where Np represents the total number of data. Smaller value
of intra indicates better performance of clustering for
algorithms. Finally, inter is the distance between two clusters
defined by the equation (14 ) .
inter = min{‖mk − mkk ‖2 }
(14)
∀k = 1,2, . . , K − 1 , kk = k + 1, . . , K
Inter focuses on the minimum distance between clusters and
is defined as the minimum distance between cluster centers.
Another criterion used to evaluate clustering is the Rand
index [ 22 ] calculated the similarity between the obtained
partition and the optimal solution, it means the percentage of
right decisions taken by the algorithm.
TP+TN
R(U) =
(15)
TP+FP+TN+FN
Where TP and TF are, respectively, the number of correct or
incorrect attributions. When the decision involves assigning
two elements to the same cluster. TN and FN show the
number of correct and incorrect attributions when the
decision involves assigning two elements to different clusters.
Note that R is in the range [ 1,0 ] and the R-value is closer to
1 indicates better quality of solutions .
6.5
EVALUATION OF RESULTS
Clustering performance indicators are the VI measured values
, if the VI is smaller (larger) the results of clustering
algorithm will be better (worse). VI value shows that the
training process obtains convergence algorithm.
Considering the number of clusters, the obtained number of
clusters by the algorithm nearer to the correct number of
clusters shows the better results of clustering algorithm.
In the experiment, some comparisons are also made on the
Rand-index values and the number of clusters. Since the
initial solution is generated randomly, evaluation is done
based on the mean values of 20 times experiments and
standard deviations.
The results based on the Rand index, the number of clusters
and the standard deviation from the implementation of
various algorithms on five databases are reported in
Tables (3)- (7). As a result, the proposed method achieved
better performance in terms of determining the number of
clusters compared to other methods.
Table 3
Comparison of results obtained by different
methods on Iris Database
Rand
0:01 ± 0.8 7
12:03 ± 0.86
Number of clusters
Methods
0.3 ± 3.1
DPSO-GA
0.4 ± 2.95
DGPSO
Table 4
Comparison of results obtained by different
methods on Glass Database
Rand
Number of clusters Methods
0:01 ± 0.66
0.6 ± 5.45
DPSO-GA
.02 ± 0.63
0.0 ± 5
DGPSO
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Table 5
Comparison of results obtained by different
methods on Seed Database
Rand
Number of clusters Methods
0.0 5 ± 0.84
0.48 ± 2.85
DPSO-GA
0.05 ±0.75
0.37 ±2.15
DGPSO
Table 6
Comparison of results obtained by different
methods on Tissue Database
Rand
Number of clusters Methods
0.02 ± 0.77
0.31 ± 5.1
DPSO-GA
0.08 ±0.70
0:44 ± 5.25
DGPSO
Table 7
Comparison of results obtained by
different methods on Wine Database
Rand
Number of clusters Methods
0.01 ± 0.93
0.22 ± 3.05
DPSO-GA
0.07 ±0.91
0.39 ± 2.95
DGPSO
. The value of Intra is shown in relation (13 ) that is the
average of intra-cluster distance.
1
intra = ∑Kk=1 ∑u∈Ck‖u − mk ‖2
(13)
7. CONCLUSION AND FUTURE WORKS
This paper proposed the DPSO-GA algorithm to solve the
problem of setting the number of clusters and also to find the
appropriate number of clusters according to the data
characteristics. Five UCI datasets are employed with different
numbers of clusters, different sizes and different types of data
to verify that the DPSO-GA algorithm can offer better
clustering results. Using a discrete PSO and GA algorithm
DPSO-GA achieved better results.
Clustering is a data mining method, which focuses on
identifying relationships between data. In future work, we
plan to suggest a multi-objective method based on DPSO-GA
to identify the relationships with multiple criteria.
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