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Evolutionary Intelligence
https://doi.org/10.1007/s12065-021-00590-1
RESEARCH PAPER
Chaotic vortex search algorithm: metaheuristic algorithm forfeature
selection
FarhadSoleimanianGharehchopogh1 · IsaMaleki1· ZahraAsheghiDizaji1
Received: 2 July 2020 / Revised: 2 November 2020 / Accepted: 2 March 2021
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
Abstract
The Vortex Search Algorithm (VSA) is a meta-heuristic algorithm that has been inspired by the vortex phenomenon proposed
by Dogan and Olmez in 2015. Like other meta-heuristic algorithms, the VSA has a major problem: it can easily get stuck in
local optimum solutions and provide solutions with a slow convergence rate and low accuracy. Thus, chaos theory has been
added to the search process of VSA in order to speed up global convergence and gain better performance. In the proposed
method, various chaotic maps have been considered for improving the VSA operators and helping to control both exploitation
and exploration. The performance of this method was evaluated with 24 UCI standard datasets. In addition, it was evaluated
as a Feature Selection (FS) method. The results of simulation showed that chaotic maps (particularly the Tent map) are able
to enhance the performance of the VSA. Furthermore, it was clearly shown the fitness of the proposed method in attaining
the optimal feature subset with utmost accuracy and the least number of features. If the number of features is equal to 36, the
percentage of accuracy in VSA and the proposed model is 77.49 and 92.07. If the number of features is 80, the percentage of
accuracy in VSA and the proposed model is 36.37 and 71.76. If the number of features is 3343, the percentage of accuracy in
VSA and the proposed model is 95.48 and 99.70. Finally, the results on Real Application showed that the proposed method
has higher percentage of accuracy in comparison to other algorithms.
Keywords Vortex Search Algorithm· Feature Selection· Chaotic Maps· Exploration· Exploitation· Accuracy
1 Introduction
The metaheuristic optimization algorithms were proposed
over the past decades and implemented extensively to the
problem of the complicated [1, 2]. The essential target in the
optimization is the candidate the problem variables to mini-
mize or maximize the objective function based on the global
and local search [3, 4]. So as to triumph over the state-of-
the-art goals in any problem, most of such algorithms were
applied as an attempt to establish an approximate technique
for attaining the optimum solution [5, 6]. A number of well-
known new nature-inspired algorithms include the Invasive
Weed Optimization (IWO) [7], the butterfly optimization
algorithm (BOA) [8], the Artificial Bee Colony (ABC) [9],
the Fruit Fly Optimization Algorithm (FOA) [10], the Firefly
Algorithm (FA) [11], the Krill Herd (KH) algorithm [12],
the Differential Evolution (DE) algorithm [13], the Flower
Pollination algorithm (FPA) [14], etc. The distinction in
nature is an essential factor why the algorithms have an
alternate dimension of execution in delivering results [15,
16]. Besides, this factor might be the motivation behind why
a few algorithms can best item an answer for specific issues,
while others don’t. Thus, it is according to this limitation
that one algorithm is not good enough for solving every kind
of problem.
During the past decade, an arithmetic framework and
scientific branch, namely chaos, has been proposed, and
is connected deeply with different scientific fields. Chaos
involves three major dynamic properties: the quasi stochas-
tic property, being sensitive against initial conditions, and
ergodicity. The application of chaos theory in optimization
research areas has attracted a lot of attention over the recent
years. The Chaotic Optimization Algorithm (COA) [17]
is among the applications of chaos, and uses the nature of
chaos sequences. It has been indicated if random variables
are replaced with chaotic variables, the performance of COA
can be enhanced. Therefore, in the literature, there are a
* Farhad Soleimanian Gharehchopogh
bonab.farhad@gmail.com
1 Department ofComputer Engineering, Urmia Branch,
Islamic Azad University, Urmia, Iran
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number of studies on the hybridization of chaos with other
algorithms for the purpose of improving the performance of
COA. Some instances include the chaotic ACO [18], cha-
otic DE algorithm [19, 20], chaotic KH algorithm [21, 22],
chaotic FPA [23], chaotic genetic algorithm [24, 25], chaotic
PSO [26–28], chaotic gravitational search [29–31], chaotic
bat algorithm [32], and etc.
FS is the procedure of selecting a subset of features from
an original feature set; it may be considered the most impor-
tant pre-processing instrument to solve classification issues
[33]. Figuring out a superior subset of features is a quite
complicated challenge, and is decisive in the final results of
the rates of classification error. The finalized feature subset
will retain high rates of classification accuracy. The pur-
pose is choosing an applicable subset including d features
from a set of D features (d < D) in a given dataset [34]. D
is made out of all features that are present in a given data
set; it can encompass redundant, noisy, and misleading
features. Consequently, an exhaustive search is performed
within the whole solution environment, which usually takes
a lot of time and cannot often be implemented in practice.
To remedy this FS strategy, maintaining the best subset of
d relevant features was taken into consideration. Inappro-
priate features are not only useless, but also can certainly
worsen the classification performance. If irrelevant features
are deleted, the computational efficiency can be advanced
and classification accuracy expanded.
As indicated by search techniques of feature subsets, the
current FS strategies can be classified into two classes: the
filter-based approach and the wrapper-based approach. The
filter method depends fundamentally on general qualities of
datasets to assess and choose include subsets without tak-
ing into account an uncommon learning approach. Thus,
the productivity of this methodology depends predominantly
on the dataset itself instead of on the classifier [35, 36].
The wrapper method utilizes a classification calculation to
assess feature subsets and embraces a search system to look
for ideal subsets. It often leads to better since the wrapper
approach takes into consideration a classifier with the evalu-
ation or search process [37].
Each meta-heuristic algorithm has a unique search strat-
egy. Meta-heuristic algorithms can find optimal solutions
based on its own strategy such as balance between explo-
ration and exploitation. Furthermore, VSA has advantages
such as the smaller number of parameters and easy imple-
mentation. The VSA was embedded with chaotic maps to
obtain a better compromise between exploitation and explo-
ration. This paper uses hybrid methods based on CMs with
the VSA for FS. The major contribution of the current paper
is that a CMs model of VSA has been proposed to enhance
the performance of VSA. In proposed methods, the chaotic
seek method are followed to choose the ideal characteristic
subset that maximizes the category accuracy and minimizes
the feature subset duration. Ten one-dimensional CMs are
adopted and changed with random movement parameters
of the VSA. The performance of the proposed methods is
tested on 24 benchmark datasets. Similarly, the performance
of VSA is comparison with seven other metaheuristic algo-
rithms. Based on mean criterion, the proposed method can
obtain better solutions using the Tent Map in comparison
with other metaheuristic algorithms.
The main contributions of this paper are as follows:
• VSA and Chaotic Maps are defined to FS.
• The proposed method has a faster convergence perfor-
mance than the other algorithms. The proposed method
has better convergence results on different datasets.
• The proposed method has been evaluated with 24 UCI
standard datasets.
• The best VSA is State2 with VSAC101 that obtained by
using the Tent map.
• The proposed method has been tested on author identifi-
cation datasets
• The obtained results confirmed the validity and supe-
riority of the proposed method in comparison to other
algorithms.
The organization of this paper is as follows: Sect.2 gives
related works about chaotic and FS. Section3 provides an
introduction to VSA. The detailed description of the pro-
posed method has been provided in Sect.4, while the experi-
mental results and discussion of the proposed VSA have
been provided in Sect.5. In Sect.6, the proposed method
has been applied on a real application (i.e., author identifi-
cation). Finally, the conclusion and future work have been
discussed in Sect.7.
2 Related works
The Moth Swarm Algorithm (MSA) is among the most
recently-developed nature-inspired heuristics for the pur-
pose of the optimization problem. However, its shortcoming
is that it has slow convergence rate, and the Chaos theory
has been incorporated into it to eliminate this drawback.
In [38], ten CMs have been embedded within the MSA for
the purpose of finding the ideal number of prospectors to
increase exploiting the most promising solutions. The pro-
posed method was applied in solving the famous seven
benchmark test functions. The results of simulation showed
that CMs can enhance the performance of the original MSA
with regard to the convergence speed. In addition, the sinu-
soidal map was found to be the best map for enhancing the
performance of MSA.
The Cuckoo search algorithm (CSA) is a metaheuristic
algorithm that has been inspired by nature and imitates
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the obligate brood parasitic behavior of the cuckoo spe-
cies. The method has been proven to have promising over-
all performance in solving optimization problems. Chaotic
mechanisms were incorporated into CSA to make use of the
dynamic features of the chaos theory, to further improve
its search overall performance. However, in chaotic CSA
(CCSA) [39], the best CM was applied in a single search of
the new release, which restrained the exploitation capabil-
ity of the search. The researchers considered utilizing mul-
tiple CMs at the same time to perform the nearby search
inside the community of the global best solution that is
found by CSA. To attain this goal, three kinds of multiple
chaotic CSAs (MCCSA) were proposed via incorporating
several CMs into the chaotic local search (CLS) parallel in
a random or selective manner. The overall performance of
MCCSA was validated using 48 broadly-used benchmark
optimization features. The experimental results indicated
that MCCSAs are generally better than CCSAs, and the
MCCSA-P that makes use of the CMs has the best quality
among all sixteen editions of the CSAs.
In [40], a chaos-based Crow Search Algorithm (CCSA)
has been proposed to solve the fractional optimization prob-
lems (FOPs). The proposed CCSA integrated the chaos the-
ory (CT) into the CSA for the purpose of refining the global
convergence velocity and enhance the exploration/exploita-
tion inclinations. CT was utilized to track the standard CSA
parameters, which yielded four versions and the high-quality
chaotic variant was investigated. The incorporation of CT
was able to improve the overall performance of the proposed
CCSA and allow the search process to perform better speeds.
The overall performance of the CCSA method was proven
on twenty fractional benchmark problems. Furthermore, it
was further tested on a fractional monetary environmental
power dispatch problem via attempting to limit the ratio of
the overall emissions to general gasoline cost. Ultimately,
the proposed CCSA was compared with the PSO, standard
CSA, FA, Dragonfly Algorithm (DA), and GWO. In addi-
tion, the efficiency of the proposed CCSA was justified by
the non-parametric Wilcoxon signed-rank test. The experi-
mental results proved that the proposed CCSA performs
better than similar algorithms with regard to efficiency and
reliability.
In [41], a new hybrid algorithm for solving optimization
problems based on chaotic ABC and chaotic simulated
annealing has been proposed. The chaotic ABC reveals
new locations chaotically. Chaos may additionally improve
the exploration of the search space. Really, the proposed
hybrid method affords a hybrid of nearby search accu-
racy of simulated annealing and the capacities of global
seek of ABC. Moreover, they used an exclusive method
for producing the initial population. Sincerely preliminary
populace is of brilliant significance for populace-based
techniques, because it immediately influences the rate of
convergence and nice of the outcomes. It is established
the usage of 12 benchmark functions. The effects are as
compared with those of the artificial bees’ algorithm, the
hybrid algorithm of ABC and simulated annealing and
PSO. Simulation effects display the performance of the
proposed method.
In [42], an adaptive chaotic Bacterial Foraging Optimi-
zation (BFO) is presented. The improved BFO consisted
of two new features, the adaptive chemotaxis step setting,
and the chaotic perturbation operation in all chemotactic
events. The former feature results in fast convergence rate
and the acceptable convergence accuracy in the algorithm,
while the latter further allows the search to avoid the local
optima and attain better convergence accuracy. Firstly, an
idea of adaptive exponential decease chemo taxis step is pre-
sented, in which the natural exponential function variable is
a function about the iterations and nutritive ratio between the
current bacterium position and the best bacterium position
in each iteration. Secondly, when each bacterium reaches a
new position through swim behavior, chaotic perturbation
is applied to avoid entrapping into local optima. With five
benchmark functions, Chaotic BFO is proved to have a bet-
ter performance than the original BFO and BFO with linear
deceasing chemo taxis step (BFO-LDC).
Jia etal. [43] proposed an effective memetic DE algo-
rithm (DECLS), which makes use of a CLS with a ‘shrink-
ing’ strategy. The shrinking strategy for the CLS search
space was introduced in that paper. In addition, the local
search length was determined according to the feedback of
the fitness of the objective functions in a dynamic manner
in order to save the function evaluations. Furthermore, the
parameter settings of the DECLS were adapted in the pro-
cess of evolution so as to further enhance the optimization
efficiency. The hybrid form of the DE and a CLS as well as
a parameter adaptation mechanism seemed very reasonable.
The CLS is helpful in enhancing the local search capability
of DE, whereas the parameter adaptation can improve the
global optimization quality. The CLS is helpful in improving
the optimization performance of the canonical DE through
exploring a very large search space in the early phases so
as to avoid the occurrence of premature convergence, and
exploiting a tiny region in later phases to refine the final-
ized solutions. In addition, the settings of parameters in the
DECLS were controlled adaptively to further improve the
search capability. To assess the efficiency and effectiveness
of the proposed DECLS algorithm, it was compared with
four state-of-the-art DE variants and the IPOP-CMA-ES
algorithm on a set of 20 selected benchmark functions. The
findings showed that the DECLS is significantly superior, or
at least comparable, to other optimizers with regard to the
convergence performance and solution accuracy. Further-
more, the DECLS was shown to have certain advantages in
terms of solving problems with high dimensions.
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In [44], a modified DE algorithm based on the Opposi-
tion-based Learning (OBL) and a chaotic sequence named
the OBL Chaotic DE (OBL-CDE) was proposed. The pro-
posed OBL-CDE algorithm is different from the basic DE
in two ways. The first one is related to the generation of
the initial population that follows the OBL rules, while the
second one is the dynamic adaption of the scaling factor F
through using the chaotic sequence. The numerical results
obtained by the OBL-CDE compared to the results of DE
and the opposition-based DE algorithms on 18 benchmark
functions indicated that the OBL-CDE is capable of finding
more superior solutions and maintaining reasonable conver-
gence rates at the same time.
The standard Glowworm Swarm Optimization (GSO)
shows poor ability in global search and easily gets trapped
into local optima. A Quantum GSO algorithm based on
CMs was proposed [45] in order to solve such problems.
First of all, a chaotic sequence was generated to initialize
the population. This process results in higher probability to
cover more local optimal areas, and provides the ground for
further optimization and tuning. Next, the quantum behavior
was applied to the elite population, which made it possible
for individuals to locate any position of the solution space
randomly with a certain probability. This greatly enhanced
the capability of the algorithm in global search and avoid-
ing local optima. Finally, it adopted the single dimension
loop swimming instead of the original fixed-step movement
mode. This not only improved the solution precision and
convergence speed, but also solved GSO problems that were
too sensitive to the step-size, and indirectly enhanced the
robustness of the algorithm. The simulation results indicated
that the proposed method was feasible and effective.
The Fruit Fly Algorithm (FOA) has recently been pro-
posed as a metaheuristic technique, and is inspired by the
behavior of fruit flies. Mitic etal. [46] improved the stand-
ard FOA through introducing the novel parameter in com-
bination with chaos. The performance of this chaotic FOA
(CFOA) was studied on ten famous benchmark problems
using 10 different CMs. In addition, comparison studies with
the basic FOA, FOA with Levy flight distribution, and other
recently-published chaotic algorithms were made. Statistical
findings on each optimization task showed that the CFOA
results in a very high convergence rate. In addition, CFOA
is compared with recently developed chaos enhanced algo-
rithms such as chaotic bat algorithm, chaotic-accelerated
PSO, chaotic FA, chaotic ABC, and chaotic CSA. Research
findings generally indicate that FOA with Chebyshev map
show superiority to the similar methods in terms of the reli-
ability of global optimality and the algorithm success rate.
In addition, Gandomi et al.[47] proposed a chaos-
enhanced version of the accelerated PSO. Some other
instances of chaos-enhanced metaheuristic algorithms
include the chaotic Genetic Algorithm [48], Chaotic PSO
[49, 50], Chaotic Salp Swarm Algorithm [51], Chaotic Ele-
phant Herding Optimization (EHO) algorithm [52], Chaotic
Bat Algorithm[53], Chaotic FOA[46], Chaotic GSO Algo-
rithm [45, 54], Chaotic Black Hole algorithm [55], Cha-
otic Simulated Annealing PSO Algorithm (CSAPSO) [56],
Chaotic Social Spider Optimization Algorithm[57], Cha-
otic Bean Optimization Algorithm[58], Chaotic Quantum
CSA [59], Chaotic Antlion Algorithm[60], Chaotic Hybrid
Cognitive Optimization Algorithm[61], Chaotic Simulated
Annealing [62], Chaotic Based Quantum Genetic Algorithm
[63], Chaotic Teaching Learning Algorithm[64], Chaotic
DE algorithm [65], Chaotic Grey Wolf Optimization Algo-
rithm[66], Chaotic Fractal Search[67], Chaotic Brain Storm
Optimization Algorithm [68], Multi-Objective CCSA [69],
Chaotic Grasshopper Optimization Algorithm [70], Chaotic
Krill Herd [21, 71, 72], Chaotic DE[73], Chaotic Firefly
Algorithm [74, 75], Chaotic Starling PSO Algorithm[76],
Chaotic CCSA [77], Chaotic Grey Wolf Optimization Algo-
rithm [78] and etc. Table1 shows a comparison of different
models of meta-heuristic algorithms based on chaotic map.
3 Vortex search algorithm
The VSA is a recent metaheuristic optimization algorithm
that changes into the stimulated mode by the vertical flow of
the stirred fluids. Its processes consist of the simplified gen-
eration phases similar to other single-solution algorithms.
The generation of VSA populations is modified to any gen-
erations with the aid of the usage of values completely shape
the modern single solution. Furthermore, the performance
of every update and seek of iteration pass at the seek space
is an essential section in rendering single-solution. Inside
the proposed VSA, this stability is performed with the aid
of using a vortex-like search pattern. The strategies of vortex
sample are simulated through some of the nested circles. The
info of VSA techniques may be in brief defined in 4 steps
as follow [79].
3.1 Generating theinitial solution
The preliminary procedure initials ‘center’ μ0 and ‘radius’
r0. In this phase, the initial center (μ0) can be calculated
using Eq.(1).
where
upperlimit
and
lowerlimit
are the bound constraints of
the problem, which can be defined in vector of d × 1 dimen-
sional-space. In addition, σ0 is the initial radius r0 generated
with Eq.(2).
(1)
𝜇
0=upperlimit +lowerlimit
2
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3.2 Generating thecandidate solutions
The procedure of producing candidate solutions is
applied for the purpose of rendering the generation of
(2)
𝜎
0=
max (upperlimit)−min(lowerlimit)
2
populations
Ct(s)
in any iterations, where t is the t-th
iteration. The VSA is randomly generated around the
initial center μ0 by using a Gaussian distribution, where
C0
(s)=
{
s
1
,s
2
,…,s
m}
m=1, 2, 3, …,
n
represents the
solution and n is the overall number of candidate solu-
tions. The equation of multivariate Gaussian distribution
has been shown in Eq.(3).
Table 1 A Comparison of Different Models of Meta-heuristic Algorithms based on Chaotic Maps
Refs Models Application Chaotic map Improved
[38] Chaotic MSA Optimization problem Sinusoidal *Convergence speed
[39] CCSA Solving optimization problems Gaussian map *Exploration and exploitation
[40] CCSA Optimization problems Circle map *Obtaining the global optimum
*accelerate the convergence
performance
[41] Chaotic artificial bee colony and
chaotic simulated annealing
Solving optimization problems Sinusoidal map *Faster convergence *better
exploration
[42] Adaptive chaotic BFO (ACBFO) Solving optimization problems Logistic map *Convergence speed
[43] DE algorithm based on chaotic
local search
20 benchmark functions Logistic chaotic function *Convergence performance *solu-
tion accuracy
[44] Opposition based Chaotic DE
(OCDE)
Benchmark function Logistic chaotic function *Obtaining the global optimum
*accelerate the convergence
performance
[45] Quantum GSO algorithm
based on Chaotic Sequence
(QCSGSO)
Solving optimization problems Logistic chaotic function *Avoid from prematurity
[46] Chaotic FOA Multi-mode functions Logistic chaos mapping *Convergence speed
[47] Chaos-enhanced accelerated PSO *Global optimization *three
engineering problems
*Sinusoidal map *Singer map *Global optimality *convergence
speed
[48] Chaos-Genetic Algorithm Grid scheduling Logistic map *Convergence performance *solu-
tion accuracy
[49] An improved chaotic PSO algo-
rithm based on adaptive inertia
weight (AIWCPSO)
Benchmark functions Logistic map *Convergence accuracy
[50] Chaotic PSO algorithm (CS-
PSO)
Recommendation system Chaos methods *Global search capability; *avoid-
ing the premature convergence;
[51] Chaotic Salp Swarm Algorithm
(CSSA)
*Global optimization; *feature
selection;
Logistic map *Minimizing the number of
selected features; *maximizes
the classification accuracy;
[52] Chaotic EHO algorithm 15 benchmark functions from
CEC 2013
Tent map *Convergence speed
[53] Chaotic bat algorithm Robust global optimization Thirteen different chaotic maps *Global search capability; *avoid-
ing the premature convergence;
[54] Chaotic GSO algorithm Eight standard test functions Logistic map *Convergence performance *solu-
tion accuracy
[55] Chaotic Inertia Weight Black
Hole Algorithm (CIWBH)
Twenty-three benchmark func-
tions
Logistic map *Enhance the global search
*trade-off between exploita-
tion and exploration *better
convergence
[56] Chaotic Simulated Annealing
PSO Algorithm
Complex optimization problems Logistic map *Global search ability; *conver-
gence precision;
[57] Chaotic Social Spider Optimiza-
tion Algorithm
Robust Clustering Logistic map *Convergence speed
[58] Chaotic bean optimization algo-
rithm (CBOA)
CEC2014 benchmark functions Logistic map *Global optimization
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In Eq.(3) d indicates the dimension, while x is the d × 1
vector of a random variable, μ indicates the d × 1 vector of
the sample mean (i.e., center), and Σ indicates the covariance
matrix. Equation(4) indicates that when the diagonal elements
(i.e., variances) of the Σ values are equal and the off-diago-
nal elements (i.e., covariance) equal zero (uncorrelated), the
resulting shape of the distribution will be spherical. Thus, the
value of Σ is computed through utilizing equal variances with
zero covariance.
where the representation in Eq.(4), σ2 is the variance of the
distribution, I represent the
d×d
identity matrix and σ0 is
the initial radius (r0) as can see in Eq.(2).
3.3 Replacement ofthecurrent solution
The replacement of the current solution is conducted for the
selection process. A solution (which is the best one)
s∈C0(s)
is selected and memorized from
C0(s)
for the purpose of
replacing the current circle center (μ0). Before the selection
process, it must be made sure that the candidate solutions are
inside the search spaces (Eq.(5)).
where
k=1, 2, …,n
and
i=1, 2, …,d
, and rand indicates
a random number that is distributed uniformly. VSA uses
s
as a new center, and reduces the vortex size using Eq.(3) to
(3)
p
(x
𝜇,Σ)=1
(
2
𝜋)d
Σ
exp
−1
2(x−𝜇)T
−1
(x−𝜇)
(4)
∑
=𝜎2.[I]
d
×
d
(5)
s
i
k=
rand.
upperlimiti−lowerlimiti
+lowerlimiti
,si
k<lowerlimit
i
lowerlimiti≤si
k≤upperlimiti
rand.upperlimiti−lowerlimiti+lowerlimiti
,si
k
>upperlimit
i
select the next solutions. Thus, the new set of solutions
C1(s)
can be generated. If the chosen solution is better than the
best solution, it can be determined as the new best solution
and was memorized.
3.4 The radius decrement process
In the VSA, the inverse incomplete gamma function is
applied for the purpose of decreasing the radius value
during each iteration pass. The incomplete gamma func-
tion provided in Eq.(6) often arises in probability the-
ory, especially in applications that involve the chi-square
distribution.
where a > 0 is the shape parameter while x ≥ 0 is a random
variable. Similar to the incomplete gamma function, its com-
plementary
Γ(x,a)
is usually also introduced (Eq.(7)). In
Eq.(7),
Γ(a)
is a (1).
Table2 describes pseudocode of VSA algorithm.
4 Proposed methods
In this section, the hybrid form of VSA and CMs will be
explained. The simple shape of the VSA consists of impor-
tant keys that can be center and radius. First, the center is
a current position from which the VSA may be evaluated
(6)
𝛾
(x,a)=
x
∫
0
e−tta−1dta >
0
(7)
Γ
(x,a)=
∞
∫
0
e−tta−1dta >
0
Table 2 A description of the
VSA algorithm Initializing step
Algorithm parameters: Input the population size, the lower and upper bounds
Fitness of best solution
Center of the circle(µ0), Eq. (1)
Radius of the circle (0), Eq. (2)
Repeat
Create candidate solutions within the circle by Eq. (3)
If Exceeded, then shift values into the boundaries by Eq. (5)
Select best solution to replace the current center
Decrease the standard deviation (radius) for the next iterationbyEq. (7)
End
Output
Best solution found so far
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based on the problem search space where iterations skip.
With respect to exploration for a premier solution that has
been carried out up to now, VSA used this function to
identify the ‘center’ with the purpose of replacing a new
position of the populations. Secondly, ‘radius’ is a method
that is utilized to simplify the issues-creating a massive-
radius problem to grow to be a small-radius problem. In
extra, a Gaussian distribution is a VSA which is used to
stability the exploration and exploitation at every itera-
tion skip. However, the VSA used best a single center this
is referred to as the single strategy to generate candidate
solutions around the current great answer. However, the
disadvantages of the VSA can be not noted from the local
factor whilst it suffers from the issues that have numer-
ous neighborhood minimal values. in the equal time, the
radius used to update the pleasant solution have been capa-
ble of decrease the new release skip by way of the usage
of a Gaussian distribution, making it less complicated
to trap the VSA in neighborhood optima. This explains
some drawbacks of the VSA. The presented have a look
at specializes in hybridizing the VSA with the CMs. This
hybridization is referred to as the chaotic VSA which 10
CMs have been used. These 10 maps have been used in
three different locations of the VSA [74]. Figure1 shows
flowchart of proposed method. In the first step is done
the initialization of the parameters. In the second step,
the VSA Eqs.(9, 10, and 11) are optimized based on the
chaotic maps in order to FS. In the third step, the samples
are classified and at the end, the accuracy percentage is
displayed.
Fig. 1 Flowchart of Proposed Method
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In the proposed model, we combine the formulas of CMs
based on Table3 with Eqs.(3), (5), and (6). The goal is to
find the best CMs to optimize VSA. These places can be
expressed as follows:
State 1 the production of candidate solutions inside the
search circle [Eq.(9)].
State 2 If the solution is out of range, these mappings are
used to move to the desired range. (Eq.10).
State 3 Reduced search radius using reverse gamma func-
tion and CMs [Eq.(11)].
In Table3, the CMs formulas and methods are shown.
The optimization of the VSA based on three methods
(State1, State2, and State3) has been done. In each method
have been used 10 CMs. So, in each run, there are 30 differ-
ent modes for a given dataset.
Chaos is described as a phenomenon. Any exchange
of its preliminary scenario might also purpose non-linear
change for future behavior. Chaos optimization is one of
the optimization models for search algorithms. The primary
idea behind it is too seriously change parameters/variables
from the chaos to the solution area. It relies upon for look-
ing out of the global optimum on chaotic motion properties
including ergodicity, regularity, and stochastic properties.
The major advantages of chaos are speedy convergence and
functionality for warding off local minima. CMs have a form
of determinate in which no random factors are applied. In
this paper, 10 distinguished non-invertible unidimensional
maps were adopted to attain chaotic sets. The adopted CMs
have been defined in Table3, where q denotes the index
of the chaotic sequence p, and
pq
is the
qth
number in the
chaotic sequence. The remaining parameters including d, c,
and μ are considered as the control parameters, determining
the chaotic behavior of the dynamic system. The initial point
p0 was set to 0.7 for all CMs, as the initial values for CMs
may have a great influence of fluctuation patterns on CMs.
In this paper, ten different CMs were applied for the opti-
mization process. These maps are Chebyshev, circle, gauss/
mouse, iterative, logistic, piecewise, sine, singer, sinusoidal,
and tent [74].
Descriptions of State 1, State 2, and State 3 are as follows:
State 1 VSA generates candidate solutions using just a
single ‘center’ (μ). The generation of ‘center’ is
then transformed to new center when iterations
pass through the limitation of upper and lower
bound of problems. This mechanism has some
problems. One of such problems is that VSA tends
to be trapped in local minima when suffering from
a local point of minimum problems. To overcome
this, the CMs of candidate solution VSA was
proposed.
In this method, chaos maps are used to generate candidate
solutions. Several neighbor solutions
Ct(s)
, (t indicates the
iteration index and is t = 0 at initial stages) were generated
randomly around the initial center µ0 in the d-dimensional
Table 3 CMs and proposed methods
ID Map Definition Range State1 State2 State3
1 Chebyshev map p
q+
1=cos
(
qcos
−1(
p
q))
(− 1,1) C11 C12 C13
2 Circle map pq+1=mod
pq+d−
c
2
𝜋
sin
2𝜋pq
,1
,c=0.5 and d =
0.2
(0,1) C21 C22 C23
3 Gauss/mouse map
pq+1=
{
1, pq=0
1
mod
(
p
q
,1
)
,
otherwise
(0,1) C31 C32 C33
4 Iterative map pq+1=sin
(
c𝜋
p
q)
,c=
0.7
(− 1,1) C41 C42 C43
5 Logistic map p
q+
1=cp
q(
1−p
q)
,c=
4
(0,1) C51 C52 C53
6 Piecewise map
pq+1=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
p
q
l,0 ≤pq<l
pq−1
0.5−l,l≤pq<0.5
1−l−pq
0.5−l, 0.5 ≤pq<1−l,l=
0.4
1−pq
l
,1−l≤pq<1
(0,1) C61 C62 C63
7 Sine map pq+1=
c
4
sin
(
𝜋pq
)
,c=
4
(0,1) C71 C72 C73
8 Singer map pq+1=𝜇(7.86pq−23.31p
2
q
+28.75p
3
q
−13.302875p
4
q
,𝜇=
1.07
(0,1) C81 C82 C83
9 Sinusoidal map pq+1=cp
2
qsin
(
𝜋pq
)
,c=
2.3
(0,1) C91 C92 C93
10 Tent map
pq+1=
{p
q
0.7 ,pq<0.7
10
3
(
1−pq
)
,pq≥
0.7
(0,1) C101 C102 C103
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1 3
space by using a Gaussian distribution and CMs. Here,
C0
(s)=
{
s
1
,s
2
,…,s
m}
m=1, 2, 3, …,
n
represents the solu-
tions, and n represents the total number of candidate solutions.
In Eq.(9), the formula of the proposed method is given.
where d represents the dimension, cm is the
d×1
vector of a
CMs variable, µ is the d × 1 vector of sample mean (center),
and Σ is the covariance matrix.
State 2 If the solution is out of range, these mappings
are used to move to the desired range. During
the selection phase, a solution (i.e., the best one),
s∈C0(s)
is selected and memorized from C0(s) for
the purpose of replacing the current circle center
µ0. Before the selection phase, it must be made
sure that the candidate solutions are inside the
search boundaries. To attain this goal, the solu-
tions that exceed the boundaries are shifted into
the boundaries, as in Eq.(10). The VSA combined
with chaotic sequences is described in Eq.(10).
In Eq.(10),
Cm(i)
is the obtained value of chaotic
map at
jth
iteration.
State 3 Reduced search radius using reverse gamma func-
tion and CMs. In the VSA, the inverse incomplete
gamma function is used for the purpose of decreas-
ing the value of the radius during each iteration
pass. The incomplete gamma function has been
given in Eq.(11).
where a > 0 is known as the shape parameter and cm ≥ 0 is
a CMs variable.
In the current study, the chaotic VSA has been imple-
mented as an FS algorithm based on the wrapper method.
In VSA, a chaotic sequence is embedded in the search
iterations, and the optimal feature subset that describes the
dataset is selected using VSA. The FS strategy is aimed at
(9)
p
(x
𝜇)=1
2𝜋
d
exp
−1
2(cm −𝜇)T
−1
(cm −𝜇)
(10)
s
i
k=
Cmi∗
upperlimiti−lowerlimiti
+lowerlimiti,si
k<lowerlimit
i
lowerlimiti≤si
k≤upperlimiti
Cmi∗upperlimiti−lowerlimiti+lowerlimiti,si
k>upperlimit
i
(11)
𝛾
(x.a)=
cm
∫
0
e−tta−1dta >
0
improving the classification efficiency, reducing the length
of feature subset, and reducing the computational costs.
4.1 Fitness function
At each iteration, every point position is evaluated the
use of a special fitness function fit. The data are ran-
domly divided into extraordinary components, especially
training and testing datasets by using the m-fold tech-
niques. Goal standards are used for assessment, which
are classification accuracy and the number of selected
features. The followed fitness function equation hybrids
the two standards into one by means of setting a weight
factor as in Eq.(12). a is the class accuracy calculated
with the aid of dividing the variety of efficiently labeled
instances over the full variety of instances. K-nearest
neighbor (KNN) [80] is the used classifier in which k
equals to three with suggesting absolute distance. KNN
is one in every of supervised learning algorithms which
rely on classifying new instance based totally on distance
from the new sample to the training samples. The KNN
classifier predicts the class of the testing sample through
calculating and sorting the distances between the testing
sample and each one of the training samples. Such a pro-
cess is repeated until each datum in the dataset has been
selected once as the testing sample. What is meant by the
classification accuracy of a feature subset is the ratio of
the number of samples that have been predicted correctly
to that of all the samples. In this paper, KNN has been
used for determining the fitness of the selected features.
The selection of K and distance method was decided
based on trial and error. Ls is the length of the selected
feature subset, Ln is the total number of features, and β is
the weighted factor which has value in [0, 1]. β is used to
control the importance of classification accuracy and the
number of selected features. Since improving accuracy
is the primary goal for any classifier, the weight factor is
usually set to values near 1 [81]. In this paper, β was set
to 0.8. The best solution is maximizes the classification
accuracy and minimizes the number of selected features
[81].
5 Result anddiscussion
In this section, first a summary of the main characteristics
of the implemented datasets will be discussed. Second,
the proposed methods (State1, State2 and State3) using
different CMs will be investigated. Third, comparisons
(12)
Fit
=maximize
(
a+𝛽×
(
1−
Ln
L
s)).
Evolutionary Intelligence
1 3
will be made between VSA and the proposed method
based on FS. Finally, to emphasize the advantages of the
proposed method compared to other algorithms, different
experiments will be described and the obtained results
will thoroughly be discussed.
5.1 Datasets description
Twenty-four benchmark datasets from different types
including medical/biology and business were used in the
experiments. Four datasets (21, 22, 23, and 24) were related
to the identification and classification of the text author.
The datasets were collected from the UCI machine learn-
ing repository [82]. A short description of each one of the
adopted datasets has been presented in Table4. As it can be
observed, the used datasets involve missing values in some
records. In the current study, all such missing values were
replaced by the median value of all known values of a given
feature class. The mathematical definition of the median
method has been defined in Eq.(13).
Si,j
parameter is the
missing value for
jth
feature of a given
ith
class W. For miss-
ing categorical values, the most appeared value for a feature
given class is replaced with the missing value.
Four different statistical measurements—including the
worst, the best, the mean fitness value, and the standard devi-
ation (SD) were adopted. In the current study, this test was
used to evaluate the performance of each CM and determine
the best one. The worst, the best, the mean fitness value, and
the SD are mathematically defined as follows:
BS is the best score gained so far for each iteration.
(13)
s
i,j
=median
i∶s
i,j
∈W
r
S
i,j
(14)
Best
=max
tMax
i=1
BS
i
(15)
Wors t
=min
tMax
i=1
BS
i
(16)
Mean
=
1
tMax
tMax
∑
i=1
BS
i
(17)
SD
=
�∑
M
i=1(BSi−𝜇)2
tMax
Table 4 Dataset description
ID Dataset Missing values No. of features No. of classes No. of instances Type
Dataset1 Chess No 36 2 3196 Game
Dataset2 Poker hand No 10 10 25,010 Game
Dataset3 Germen credit No 24 2 1000 Business
Dataset4 Credit approval Yes 15 2 690 Business
Dataset5 Cylinder bands Yes 40 2 512 Physical
Dataset6 Abalone No 8 29 4177 Life
Dataset7 Glass identification No 10 6 214 Physical
Dataset8 Letter recognition No 17 26 20,000 Computer
Dataset9 Waveform No 21 3 5000 Physical
Dataset10 Zoo No 18 2 101 Life
Dataset11 Wisconsin Diagnosis Breast Cancer (WBCD) No 32 2 596 Clinical
Dataset12 Mice Protein Expression Data set (MPED) Yes 82 8 1080 Clinical
Dataset13 Parkinson’s Disease Detection Data set (PDD) No 23 2 197 Clinical
Dataset14 Cardiotocography No 23 3 2126 Clinical
Dataset15 Hepatitis Yes 19 2 155 Clinical
Dataset16 Lung cancer Yes 56 3 32 Clinical
Dataset17 Single proton emission computed tomography
(SPECT)
No 44 2 267 Clinical
Dataset18 Thoracic surgery No 17 2 470 Clinical
Dataset19 Statlog (heart) No 13 2 270 Clinical
Dataset20 Indian Liver Patient Dataset No 10 2 583 Clinical
Dataset21 WebKB No 2350 4 4199 Computer
Dataset22 Cade12 No 5340 12 40,983 Computer
Dataset23 Reuters 21,578 – R8 No 3343 8 7674 Computer
Dataset24 Reuter_50_50 No 2340 50 5000 Computer
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5.2 Analysis anddiscussion
For evaluation of methods on different datasets of four cri-
teria (worst, best, mean and SD) have been used. In Table5,
30 modes and the VSA with the mentioned criteria are inves-
tigated. Proposed Method (State1) is equal to VSAC11 to
VSAC101 modes, Proposed Method (State2) is equal to
VSAC12 to VSAC102 modes, and Proposed Method (State3)
is equal to VSAC13 to VSAC103 modes. With regard to the
results, it can be stated that Proposed Method (State2) has
better results. Proposed Method (State2) with VSAC101
mode offers best result than other modes using Tent map.
The main target of this test is to evaluate the efficiency of
VSA with different chaotic maps and define the optimal cha-
otic map (Tables6, 7, 8).
5.3 Comparisons betweenVSA andproposed
method based onFS
In Table9 and Fig.2, the results of the VSA and the Pro-
posed Method are shown based on the FS. We chose the
Proposed Method in order to FS because it had a high per-
centage of accuracy. Based on the results, it can be said that
the Proposed Method in 19 datasets is better than the VSA.
5.4 Comparison andevaluation
Comparison of the Proposed Method with GA, PSO, ABC,
BOA, IWO, FPA, and FA algorithms has been performed to
evaluate the efficiency. In Table10, the control parameters
of the algorithms are expressed.
The comparison of the Proposed Method with the PSO,
ABC, BOA, IWO, GA, FA, FPA, and VSA was performed
according to the worst criteria. According to Table11 and
Fig.3, it is clear that the results of other algorithms are
worse than the Proposed Method.
In Table12, the comparison of the Proposed Method with
PSO, ABC, BOA, IWO, GA, FA, FPA, and VSA was per-
formed based on the best criteria. According to Table12 and
Fig.3, it is clear that the results of the Proposed Method are
better than other algorithms.
In Table13, the comparison of the Proposed Method with
PSO, ABC, BOA, IWO, GA, FA, FPA, and VSA was per-
formed based on the mean criteria. According to Table13
and Fig.3, it is clear that the results of the Proposed Method
are better than other algorithms.
To sum up, the results and discussion of this paper dem-
onstrate that integrating CMs to the VSA is definitely benefi-
cial. The reason why that the Proposed Method outperforms
all the other algorithms is that the Tent chaotic map assists
this algorithm to highly emphasize exploration in the initial
steps of optimization and reduced search radius.
6 Real application: author identication
Author identification, is a
stylometric
problem that tries to
identify a copied text belonging to an original author [85,
86]. With ever-increasing volume of documents uploaded
to the internet, new methods for analyzing and extracting
data and knowledge are needed. In order to prevent plagia-
rism and copying copyrighted materials, the best solution
is to use authorship identification. Every writer has his/her
own writing style in manuscripts that he/she writes, and the
writer’s style can be identified in other papers [87]. Author-
ship identification is one of the up-to-date problems in the
field of natural language processing. Author identification, is
an effort to show the writer’s personal characteristics, based
on a piece of linguistic information [88] such that various
manuscripts written by various authors can be distinguished.
Humans possess certain writing patterns for using a lan-
guage in their writings, which act like figure prints of the
writer (writer print); these patterns are specific to the writ-
ers [89].
Authors in [90] have proposed an approach known as the
stylometric
approach to deal with the problem of Author
Identification. There are four different steps in this approach:
• Calculation of word frequencies to find the most frequent
words in the entire corpus.
• Calculation of normalized frequency. This is done by
dividing the frequency of the most frequent word in that
document to the total number of words in entire corpus.
• Using Z-score method.
• Calculation of distance table by finding distance between
two matrices.
Therefore, since the text is converted into numeric repre-
sentation (feature extraction), classification, and clustering
techniques of machine learning can be implemented on it.
The Reuter_50_50 data set is used for experiments. There
are 50 authors and 50 documents per each author in this
dataset. Thus, both training corpus and test corpus contains
2500 texts. These corpuses do not overlap with each other.
By applying
stylometry
approach and n-gram features to the
author identification problem an accuracy of about 85% of
that of SVM classifier is achieved which is a higher accuracy
in comparison to Delta and KNN classifier.
Dissimilarity Counter Method (DCM), DCM-Voting,
and DCM-Classifier have been applied in [91] to the
problem of Author Identification. Once the representation
spaces are selected, similarity measures such as Euclidean
distance, correlation coefficient, and Cosine can be used to
compare the documents and then, the document author can
be identified using one of the above-mentioned approaches
(DCM, DCM-voting, or DCM-Classifier). DCM only uses
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Table 5 Comparison of results of methods with VSA
Methods Dataset1 Dataset2 Dataset3
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 1.0976 1.1776 1.3368 0.0394 VSA 0.8055 1.1195 1.0782 0.0793 VSA 0.9830 1.4678 1.3695 0.0993
Proposed method (state1) VSAC11 1.0393 1.3076 1.3048 0.0658 VSAC11 0.7894 1.0633 0.9814 0.0548 VSAC11 0.9766 1.4506 1.3541 0.0945
VSAC21 1.0889 1.4099 1.2423 0.0711 VSAC21 0.7765 1.0660 1.0233 0.0676 VSAC21 1.0311 1.4668 1.3387 0.0728
VSAC31 1.0593 1.2748 1.3172 0.0529 VSAC31 0.7732 1.0566 1.0201 0.0659 VSAC31 0.9761 1.4632 1.3333 0.0960
VSAC41 1.0116 1.3581 1.2463 0.0844 VSAC41 0.7901 1.1190 1.0133 0.0771 VSAC41 0.9957 1.4124 1.3662 0.0765
VSAC51 1.0452 1.3552 1.2530 0.0690 VSAC51 0.7989 1.0896 0.9964 0.0612 VSAC51 1.0332 1.4214 1.3989 0.0679
VSAC61 1.0631 1.3285 1.2346 0.0499 VSAC61 0.7839 1.0976 0.9859 0.0698 VSAC61 1.0205 1.4478 1.3981 0.0808
VSAC71 1.0792 1.3860 1.2341 0.0653 VSAC71 0.7733 1.1161 1.0379 0.0867 VSAC71 1.0017 1.4035 1.3894 0.0762
VSAC81 1.0559 1.2231 1.3089 0.0451 VSAC81 0.7896 1.0840 1.0707 0.0757 VSAC81 0.9730 1.4676 1.3427 0.1000
VSAC91 1.0044 1.2229 1.3067 0.0674 VSAC91 0.7710 1.0815 0.9993 0.0713 VSAC91 0.9725 1.4393 1.3897 0.0994
VSAC101 1.0876 1.2742 1.2768 0.0286 VSAC101 0.8021 1.1128 1.0435 0.0732 VSAC101 1.0290 1.4433 1.3449 0.0667
Proposed method (state2) VSAC12 1.0529 1.3255 1.3045 0.0638 VSAC12 0.8056 1.0884 0.9876 0.0571 VSAC12 0.9721 1.4164 1.3905 0.0936
VSAC22 1.0143 1.2019 1.2335 0.0368 VSAC22 0.7981 1.0509 1.0391 0.0565 VSAC22 0.9730 1.4143 1.3357 0.0822
VSAC32 1.0957 1.2719 1.2366 0.0161 VSAC32 0.7941 1.0908 0.9889 0.0631 VSAC32 1.0060 1.4569 1.3820 0.0873
VSAC42 1.0308 1.1706 1.2720 0.0389 VSAC42 0.7932 1.0744 1.0606 0.0694 VSAC42 0.9840 1.4357 1.3943 0.0939
VSAC52 1.0021 1.1868 1.3132 0.0677 VSAC52 0.7975 1.0971 1.0658 0.0744 VSAC52 1.0301 1.4388 1.3899 0.0723
VSAC62 1.0717 1.3209 1.2547 0.0454 VSAC62 0.7822 1.1272 0.9832 0.0815 VSAC62 1.0105 1.4248 1.3607 0.0721
VSAC72 1.0866 1.2734 1.3173 0.0400 VSAC72 0.8063 1.1066 1.0752 0.0747 VSAC72 0.9759 1.4203 1.3721 0.0891
VSAC82 1.0315 1.3233 1.2861 0.0697 VSAC82 0.7939 1.0659 1.0045 0.0565 VSAC82 0.9845 1.4172 1.3819 0.0862
VSAC92 1.0091 1.2543 1.2962 0.0666 VSAC92 0.7982 1.1023 0.9993 0.0663 VSAC92 0.9728 1.4507 1.3435 0.0947
VSAC102 1.0389 1.2108 1.2746 0.0395 VSAC102 0.7977 1.1233 1.0319 0.0771 VSAC102 0.9941 1.4224 1.3632 0.0795
Proposed method (state3) VSAC13 1.0100 1.2235 1.2112 0.0379 VSAC13 0.8058 1.0891 1.0382 0.0633 VSAC13 1.0375 1.4241 1.3660 0.0602
VSAC23 1.0807 1.3786 1.2443 0.0618 VSAC23 0.7862 1.1007 0.9722 0.0691 VSAC23 0.9806 1.4586 1.3366 0.0928
VSAC33 1.0246 1.2044 1.2127 0.0268 VSAC33 0.7987 1.0708 1.0361 0.0609 VSAC33 0.9746 1.4617 1.3871 0.1042
VSAC43 1.0654 1.4037 1.3171 0.0835 VSAC43 0.8038 1.0880 1.0316 0.0629 VSAC43 0.9925 1.4351 1.3494 0.0817
VSAC53 1.0581 1.3064 1.2760 0.0506 VSAC53 0.8100 1.0656 1.0628 0.0598 VSAC53 0.9870 1.4028 1.3497 0.0748
VSAC63 1.0844 1.2271 1.3428 0.0457 VSAC63 0.7987 1.0646 1.0441 0.0608 VSAC63 0.9880 1.4462 1.3796 0.0921
VSAC73 1.0181 1.2147 1.2440 0.0403 VSAC73 0.7880 1.1148 1.0455 0.0806 VSAC73 1.0125 1.4290 1.3646 0.0730
VSAC83 1.0375 1.3182 1.2657 0.0618 VSAC83 0.7752 1.1201 0.9816 0.0817 VSAC83 0.9941 1.4543 1.4039 0.0961
VSAC93 1.0261 1.2238 1.2300 0.0347 VSAC93 0.8083 1.0643 1.0528 0.0581 VSAC93 1.0280 1.4174 1.4057 0.0709
VSAC103 1.0195 1.1869 1.2116 0.0254 VSAC103 0.7738 1.0790 1.0458 0.0767 VSAC103 0.9906 1.4521 1.3675 0.0906
Methods Dataset4 Dataset5 Dataset6
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 1.0587 1.3144 1.2876 0.0548 VSA 1.0114 1.3799 1.3384 0.0748 VSA 0.5416 0.6599 0.8384 0.0420
Evolutionary Intelligence
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Table 5 (continued)
Methods Dataset4 Dataset5 Dataset6
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
Proposed
Method
(State1)
VSAC11 1.0880 1.3770 1.2946 0.0616 VSAC11 1.0138 1.3721 1.3292 0.0698 VSAC11 0.4663 0.6720 0.8419 0.0736
VSAC21 1.0712 1.3161 1.3080 0.0536 VSAC21 1.0112 1.3921 1.3328 0.0773 VSAC21 0.5123 0.8086 0.8081 0.0595
VSAC31 1.1361 1.3954 1.2624 0.0459 VSAC31 1.0169 1.4002 1.3189 0.0749 VSAC31 0.5026 0.7927 0.8480 0.0715
VSAC41 1.0834 1.2983 1.2900 0.0394 VSAC41 1.0137 1.3826 1.3106 0.0697 VSAC41 0.4878 0.5965 0.8528 0.0730
VSAC51 1.0942 1.3998 1.3078 0.0680 VSAC51 1.0137 1.4198 1.3033 0.0807 VSAC51 0.5451 0.6770 0.8684 0.0527
VSAC61 1.0741 1.3500 1.2901 0.0585 VSAC61 1.0170 1.3781 1.3388 0.0718 VSAC61 0.5194 0.7853 0.8184 0.0538
VSAC71 1.1019 1.3126 1.3021 0.0370 VSAC71 1.0142 1.3869 1.3070 0.0702 VSAC71 0.4786 0.7875 0.8698 0.0884
VSAC81 1.0968 1.3949 1.3050 0.0648 VSAC81 1.0189 1.3773 1.3061 0.0649 VSAC81 0.5494 0.8144 0.8144 0.0449
VSAC91 1.1381 1.3435 1.2693 0.0249 VSAC91 1.0120 1.4163 1.3212 0.0826 VSAC91 0.5451 0.7079 0.8144 0.0307
VSAC101 1.1095 1.3773 1.3092 0.0537 VSAC101 1.0112 1.4444 1.3252 0.0927 VSAC101 0.5253 0.6853 0.8626 0.0578
Proposed
Method
(State2)
VSAC12 1.1262 1.3242 1.2510 0.0218 VSAC12 1.0135 1.3774 1.3370 0.0728 VSAC12 0.5178 0.8738 0.8303 0.0786
VSAC22 1.0578 1.3752 1.3054 0.0762 VSAC22 1.0194 1.4010 1.3332 0.0762 VSAC22 0.4797 0.8377 0.7928 0.0792
VSAC32 1.0881 1.3501 1.2761 0.0503 VSAC32 1.0106 1.3900 1.3131 0.0738 VSAC32 0.5330 0.7060 0.8127 0.0353
VSAC42 1.0537 1.2853 1.2986 0.0524 VSAC42 1.0194 1.4036 1.3164 0.0744 VSAC42 0.5134 0.6942 0.8297 0.0496
VSAC52 1.1042 1.3998 1.2805 0.0614 VSAC52 1.0175 1.3704 1.3133 0.0647 VSAC52 0.4875 0.7062 0.8177 0.0571
VSAC62 1.1158 1.3116 1.2496 0.0217 VSAC62 1.0162 1.4367 1.3220 0.0875 VSAC62 0.4753 0.8537 0.8308 0.0932
VSAC72 1.1114 1.3633 1.2641 0.0436 VSAC72 1.0159 1.3933 1.3083 0.0717 VSAC72 0.5008 0.8408 0.8633 0.0858
VSAC82 1.0555 1.2932 1.2426 0.0422 VSAC82 1.0183 1.4367 1.3218 0.0865 VSAC82 0.4679 0.7445 0.7921 0.0630
VSAC92 1.1109 1.3618 1.2823 0.0447 VSAC92 1.0106 1.4437 1.3042 0.0905 VSAC92 0.4674 0.6364 0.7936 0.0532
VSAC102 1.0776 1.3493 1.2812 0.0554 VSAC102 1.0187 1.3993 1.3371 0.0767 VSAC102 0.4943 0.6033 0.8131 0.0523
Proposed
Method
(State3)
VSAC13 1.0813 1.2764 1.2555 0.0274 VSAC13 1.0124 1.4059 1.3146 0.0781 VSAC13 0.4681 0.6582 0.8086 0.0593
VSAC23 1.1154 1.2914 1.2469 0.0147 VSAC23 1.0136 1.3882 1.3077 0.0710 VSAC23 0.4696 0.6789 0.8111 0.0606
VSAC33 1.0596 1.2843 1.2631 0.0413 VSAC33 1.0169 1.4456 1.3272 0.0907 VSAC33 0.5189 0.5296 0.8083 0.0540
VSAC43 1.0727 1.2788 1.2980 0.0420 VSAC43 1.0188 1.4463 1.3186 0.0891 VSAC43 0.4603 0.6987 0.8460 0.0789
VSAC53 1.0895 1.3832 1.2876 0.0623 VSAC53 1.0104 1.4034 1.3054 0.0770 VSAC53 0.4699 0.7904 0.7909 0.0712
VSAC63 1.1028 1.3547 1.2910 0.0469 VSAC63 1.0118 1.3891 1.3186 0.0738 VSAC63 0.5173 0.8254 0.7994 0.0596
VSAC73 1.0776 1.3072 1.2938 0.0452 VSAC73 1.0185 1.4352 1.3083 0.0844 VSAC73 0.4795 0.8182 0.7962 0.0747
VSAC83 1.0892 1.3187 1.2608 0.0375 VSAC83 1.0161 1.4071 1.3166 0.0771 VSAC83 0.5205 0.7337 0.8562 0.0587
VSAC93 1.0779 1.2916 1.2883 0.0400 VSAC93 1.0154 1.4041 1.3047 0.1049 VSAC93 0.5359 0.6846 0.8695 0.0564
VSAC103 1.1324 1.3713 1.2418 0.0376 VSAC103 1.0126 1.4016 1.3223 0.1078 VSAC103 0.4834 0.5521 0.7961 0.0542
Bold values is to show the best-obtained value in the comparisons
Evolutionary Intelligence
1 3
Table 6 Comparison of results of methods with VSA (continuance)
Methods Dataset7 Dataset8 Dataset9
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 1.0327 1.7105 1.5810 0.0938 VSA 1.0813 1.3577 1.2658 0.0549 VSA 1.0725 1.3744 1.2617 0.0446
Proposed Method (State1) VSAC11 1.0622 1.7039 1.5924 0.0799 VSAC11 1.0842 1.3458 1.2703 0.0499 VSAC11 1.0763 1.3764 1.2678 0.0441
VSAC21 1.1233 1.7201 1.5845 0.0554 VSAC21 1.0974 1.3345 1.2662 0.0397 VSAC21 1.0600 1.3357 1.2679 0.0373
VSAC31 1.0912 1.7133 1.5996 0.0704 VSAC31 1.1245 1.3479 1.2598 0.0319 VSAC31 1.0716 1.3087 1.3014 0.0301
VSAC41 1.0180 1.7291 1.6070 0.1105 VSAC41 1.1043 1.3529 1.2628 0.0428 VSAC41 1.0726 1.3331 1.3121 0.0382
VSAC51 1.0277 1.7234 1.6277 0.1079 VSAC51 1.1059 1.3438 1.2547 0.0381 VSAC51 1.0629 1.3375 1.3166 0.0448
VSAC61 1.0476 1.7064 1.6000 0.0888 VSAC61 1.1266 1.3489 1.2777 0.0327 VSAC61 1.0785 1.3285 1.3007 0.0319
VSAC71 1.1099 1.7213 1.5909 0.0629 VSAC71 1.1106 1.3372 1.2722 0.0353 VSAC71 1.0553 1.3474 1.2740 0.0441
VSAC81 1.0484 1.7283 1.5891 0.0932 VSAC81 1.0897 1.3369 1.2594 0.0432 VSAC81 1.0789 1.3591 1.2761 0.0375
VSAC91 1.1205 1.7090 1.6188 0.0588 VSAC91 1.0907 1.3496 1.2670 0.0480 VSAC91 1.0720 1.3313 1.2798 0.0321
VSAC101 1.0627 1.7109 1.5934 0.0820 VSAC101 1.1240 1.3310 1.2703 0.0269 VSAC101 1.0709 1.3294 1.3060 0.0367
Proposed Method (State2) VSAC12 1.0749 1.7106 1.5961 0.0766 VSAC12 1.1094 1.3493 1.2786 0.0407 VSAC12 1.0601 1.3193 1.2913 0.0361
VSAC22 1.0502 1.7035 1.5814 0.0836 VSAC22 1.0901 1.3542 1.2707 0.0502 VSAC22 1.0612 1.3613 1.3001 0.0495
VSAC32 1.0779 1.7251 1.6094 0.0818 VSAC32 1.1174 1.3573 1.2672 0.0389 VSAC32 1.0586 1.3569 1.2616 0.0444
VSAC42 1.1263 1.7100 1.6274 0.0579 VSAC42 1.1006 1.3506 1.2615 0.0434 VSAC42 1.0717 1.3782 1.2670 0.0467
VSAC52 1.0769 1.7185 1.6142 0.0811 VSAC52 1.1282 1.3312 1.2727 0.0254 VSAC52 1.0501 1.3567 1.2594 0.0479
VSAC62 1.0740 1.7092 1.6143 0.0798 VSAC62 1.0959 1.3408 1.2547 0.0414 VSAC62 1.0710 1.3517 1.2908 0.0406
VSAC72 1.1288 1.7209 1.6054 0.0563 VSAC72 1.0887 1.3538 1.2535 0.0493 VSAC72 1.0650 1.3280 1.2680 0.0326
VSAC82 1.0588 1.7105 1.6272 0.0896 VSAC82 1.0950 1.3500 1.2522 0.0450 VSAC82 1.0760 1.3314 1.3151 0.0367
VSAC92 1.0938 1.7192 1.6253 0.0754 VSAC92 1.1124 1.3497 1.2631 0.0380 VSAC92 1.0531 1.3195 1.2529 0.0332
VSAC102 1.0309 1.7026 1.6180 0.0987 VSAC102 1.1055 1.3332 1.2709 0.0360 VSAC102 1.0729 1.3093 1.2719 0.0238
Proposed Method (State3) VSAC13 1.0201 1.7183 1.5909 0.1036 VSAC13 1.1158 1.3449 1.2569 0.0344 VSAC13 1.0780 1.3084 1.3018 0.0271
VSAC23 1.0979 1.7097 1.5866 0.0642 VSAC23 1.1167 1.3431 1.2648 0.0339 VSAC23 1.0690 1.3738 1.2685 0.0464
VSAC33 1.0945 1.7074 1.5988 0.0670 VSAC33 1.0918 1.3363 1.2616 0.0423 VSAC33 1.0503 1.3379 1.2865 0.0452
VSAC43 1.1189 1.7240 1.6134 0.0631 VSAC43 1.1008 1.3495 1.2594 0.0428 VSAC43 1.0736 1.3764 1.2590 0.0446
VSAC53 1.0429 1.7097 1.6164 0.0948 VSAC53 1.1031 1.3534 1.2775 0.0448 VSAC53 1.0714 1.3163 1.2974 0.0313
VSAC63 1.0439 1.7063 1.6284 0.0956 VSAC63 1.0805 1.3353 1.2586 0.0467 VSAC63 1.0649 1.3548 1.2807 0.0430
VSAC73 1.0555 1.7082 1.5865 0.0834 VSAC73 1.0915 1.3577 1.2710 0.0508 VSAC73 1.0611 1.3612 1.3028 0.0499
VSAC83 1.0849 1.7247 1.6176 0.0798 VSAC83 1.1007 1.3575 1.2730 0.0469 VSAC83 1.0682 1.3586 1.3106 0.0471
VSAC93 1.0543 1.7269 1.5807 0.0888 VSAC93 1.0905 1.3507 1.2705 0.0488 VSAC93 1.0596 1.3686 1.3034 0.0530
VSAC103 1.0679 1.7055 1.6236 0.0832 VSAC103 1.1026 1.3343 1.2715 0.0379 VSAC103 1.0761 1.3645 1.2945 0.0428
Methods Dataset4 Dataset11 Dataset12
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 1.1862 1.5481 1.3965 0.0884 VSA 1.2308 1.6919 1.6400 0.0762 VSA 1.2996 1.7581 1.6363 0.0839
Evolutionary Intelligence
1 3
Table 6 (continued)
Methods Dataset4 Dataset11 Dataset12
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
Proposed
Method
(State1)
VSAC11 1.1992 1.5405 1.4010 0.0801 VSAC11 1.2294 1.6963 1.5527 0.0653 VSAC11 1.2860 1.7431 1.6426 0.0861
VSAC21 1.1862 1.5494 1.3920 0.0887 VSAC21 1.1812 1.6990 1.5806 0.0915 VSAC21 1.2751 1.7532 1.5976 0.0891
VSAC31 1.1948 1.5347 1.4500 0.0845 VSAC31 1.2143 1.6811 1.5910 0.0722 VSAC31 1.2730 1.7468 1.6656 0.0969
VSAC41 1.1829 1.5137 1.4000 0.0773 VSAC41 1.1760 1.6916 1.5470 0.0872 VSAC41 1.2848 1.7570 1.6617 0.0939
VSAC51 1.1891 1.5379 1.4067 0.0838 VSAC51 1.2027 1.6881 1.6216 0.0848 VSAC51 1.2748 1.7562 1.6665 0.0990
VSAC61 1.1908 1.5153 1.4068 0.0749 VSAC61 1.2408 1.6897 1.6102 0.0656 VSAC61 1.2830 1.7360 1.5938 0.0792
VSAC71 1.1969 1.5441 1.4278 0.0843 VSAC71 1.2456 1.6821 1.5549 0.0533 VSAC71 1.2768 1.7524 1.6126 0.0896
VSAC81 1.1949 1.5376 1.4331 0.0834 VSAC81 1.1655 1.6963 1.5902 0.0993 VSAC81 1.2906 1.7450 1.6614 0.0875
VSAC91 1.1922 1.5333 1.4525 0.0855 VSAC91 1.1947 1.6981 1.5973 0.0875 VSAC91 1.2968 1.7443 1.6201 0.0786
VSAC101 1.1807 1.5416 1.4281 0.0907 VSAC101 1.1607 1.6989 1.6173 0.1068 VSAC101 1.2792 1.7594 1.6602 0.0970
Proposed
Method
(State2)
VSAC12 1.1896 1.5193 1.4497 0.0819 VSAC12 1.2071 1.6990 1.5745 0.0788 VSAC12 1.2777 1.7337 1.6396 0.0865
VSAC22 1.1976 1.5403 1.4394 0.0838 VSAC22 1.1616 1.6863 1.5449 0.0917 VSAC22 1.2881 1.7462 1.5954 0.0806
VSAC32 1.1817 1.5146 1.4334 0.0817 VSAC32 1.1985 1.6905 1.6434 0.0917 VSAC32 1.2896 1.7306 1.6276 0.0784
VSAC42 1.1909 1.5381 1.3953 0.0825 VSAC42 1.2291 1.6815 1.6307 0.0723 VSAC42 1.2984 1.7429 1.6667 0.0841
VSAC52 1.1819 1.5424 1.4465 0.0924 VSAC52 1.2395 1.6918 1.5861 0.0632 VSAC52 1.2888 1.7338 1.6233 0.0792
VSAC62 1.1942 1.5210 1.4141 0.0761 VSAC62 1.2383 1.6826 1.5341 0.0547 VSAC62 1.2890 1.7528 1.6634 0.0909
VSAC72 1.1844 1.5117 1.4324 0.0794 VSAC72 1.1832 1.6858 1.6093 0.0911 VSAC72 1.2915 1.7565 1.6725 0.0923
VSAC82 1.1989 1.5139 1.4465 0.0754 VSAC82 1.2376 1.6812 1.6132 0.0651 VSAC82 1.2726 1.7555 1.6607 0.0989
VSAC92 1.1957 1.5220 1.4578 0.0811 VSAC92 1.2332 1.6904 1.6416 0.0750 VSAC92 1.2802 1.7582 1.6326 0.0923
VSAC102 1.1845 1.5175 1.4039 0.0782 VSAC102 1.2143 1.6913 1.6421 0.0842 VSAC102 1.2872 1.7540 1.6200 0.0862
Proposed
Method
(State3)
VSAC13 1.1861 1.5153 1.4381 0.0806 VSAC13 1.2198 1.6898 1.5747 0.0700 VSAC13 1.2812 1.7340 1.6387 0.0849
VSAC23 1.1818 1.5145 1.4077 0.0787 VSAC23 1.2087 1.6895 1.5412 0.0710 VSAC23 1.2820 1.7369 1.6606 0.0889
VSAC33 1.1878 1.5408 1.4187 0.0864 VSAC33 1.2407 1.6906 1.6048 0.0650 VSAC33 1.2706 1.7320 1.6602 0.0927
VSAC43 1.1891 1.5249 1.4392 0.0825 VSAC43 1.2347 1.6808 1.6047 0.0649 VSAC43 1.2771 1.7576 1.5999 0.0900
VSAC53 1.1849 1.5402 1.4029 0.0863 VSAC53 1.1923 1.6829 1.6446 0.0928 VSAC53 1.2784 1.7579 1.6529 0.0958
VSAC63 1.1825 1.5323 1.3975 0.0841 VSAC63 1.1791 1.6939 1.5991 0.0937 VSAC63 1.2736 1.7307 1.6575 0.0905
VSAC73 1.1821 1.5187 1.4187 0.0811 VSAC73 1.2165 1.6856 1.5374 0.0658 VSAC73 1.2997 1.7443 1.5993 0.0752
VSAC83 1.1829 1.5214 1.4029 0.0802 VSAC83 1.2224 1.6829 1.6467 0.0791 VSAC83 1.2829 1.7364 1.6692 0.0899
VSAC93 1.1863 1.5402 1.4211 0.0870 VSAC93 1.1692 1.6938 1.6277 0.1033 VSAC93 1.2709 1.7507 1.6201 0.0926
VSAC103 1.1918 1.5205 1.4030 0.0760 VSAC103 1.1848 1.6969 1.5420 0.0844 VSAC103 1.2840 1.7564 1.6505 0.0924
Bold values is to show the best-obtained value in the comparisons
Evolutionary Intelligence
1 3
Table 7 Comparison of results of methods with VSA (continuance)
Methods Dataset13 Dataset14 Dataset15
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 1.1441 1.6251 1.4886 0.0924 VSA 1.1812 1.5273 1.5207 0.0716 VSA 1.1460 1.7857 1.6785 0.0997
Proposed method (state1) VSAC11 1.0892 1.5685 1.4332 0.0918 VSAC11 1.1704 1.5441 1.5232 0.0814 VSAC11 1.1630 1.7513 1.6713 0.0805
VSAC21 1.1195 1.5472 1.4824 0.0782 VSAC21 1.1874 1.5341 1.4535 0.0582 VSAC21 1.1570 1.7418 1.6000 0.0690
VSAC31 1.1091 1.5705 1.5483 0.1025 VSAC31 1.1790 1.5162 1.5298 0.0722 VSAC31 1.1493 1.7299 1.6540 0.0777
VSAC41 1.0870 1.5579 1.5286 0.1054 VSAC41 1.1727 1.5294 1.5127 0.0743 VSAC41 1.1584 1.7621 1.6856 0.0884
VSAC51 1.1146 1.5887 1.5776 0.1110 VSAC51 1.1763 1.5168 1.4998 0.0666 VSAC51 1.1406 1.7398 1.5210 0.0676
VSAC61 1.1436 1.6068 1.5190 0.0909 VSAC61 1.1893 1.5752 1.4953 0.0763 VSAC61 1.1758 1.7855 1.5634 0.0720
VSAC71 1.1567 1.5226 1.5409 0.0669 VSAC71 1.1895 1.5488 1.4515 0.0617 VSAC71 1.1680 1.7983 1.5230 0.0780
VSAC81 1.1112 1.5242 1.5405 0.0887 VSAC81 1.1812 1.5464 1.4671 0.0668 VSAC81 1.1648 1.7897 1.5283 0.0762
VSAC91 1.1511 1.5526 1.5002 0.0682 VSAC91 1.1747 1.5269 1.4733 0.0650 VSAC91 1.1611 1.7312 1.6098 0.0652
VSAC101 1.0727 1.5530 1.5720 0.1210 VSAC101 1.1700 1.5292 1.4665 0.0667 VSAC101 1.1711 1.7363 1.5221 0.0530
Proposed method (state2) VSAC12 1.1623 1.5626 1.5780 0.0825 VSAC12 1.1849 1.5092 1.5295 0.0679 VSAC12 1.1672 1.7850 1.5450 0.0743
VSAC22 1.0926 1.6297 1.4349 0.1120 VSAC22 1.1723 1.5311 1.4913 0.0706 VSAC22 1.1709 1.7863 1.6375 0.0821
VSAC32 1.0783 1.5767 1.5770 0.1250 VSAC32 1.1890 1.5308 1.4701 0.0589 VSAC32 1.1791 1.7460 1.5282 0.0535
VSAC42 1.1287 1.5378 1.5334 0.0818 VSAC42 1.1834 1.5098 1.5199 0.0663 VSAC42 1.1794 1.7680 1.6707 0.0776
VSAC52 1.1125 1.6148 1.4404 0.0982 VSAC52 1.1754 1.5187 1.5000 0.0676 VSAC52 1.1486 1.7252 1.6701 0.0798
VSAC62 1.1348 1.5408 1.5033 0.0732 VSAC62 1.1800 1.5200 1.4650 0.0590 VSAC62 1.1793 1.7619 1.6899 0.0794
VSAC72 1.0734 1.6152 1.4171 0.1139 VSAC72 1.1791 1.5708 1.5295 0.0857 VSAC72 1.1439 1.7678 1.6385 0.0889
VSAC82 1.1081 1.5829 1.4854 0.0947 VSAC82 1.1855 1.5637 1.5266 0.0802 VSAC82 1.1721 1.7848 1.6272 0.0798
VSAC92 1.0951 1.6040 1.5084 0.1108 VSAC92 1.1875 1.5707 1.4886 0.0747 VSAC92 1.1556 1.7730 1.6695 0.0900
VSAC102 1.1343 1.5986 1.5142 0.0919 VSAC102 1.1803 1.5127 1.5118 0.0665 VSAC102 1.1591 1.7647 1.5690 0.0723
Proposed method (state3) VSAC13 1.1636 1.5286 1.5780 0.0748 VSAC13 1.1725 1.5495 1.4963 0.0766 VSAC13 1.1517 1.7890 1.5272 0.0815
VSAC23 1.1256 1.6364 1.4539 0.1013 VSAC23 1.1772 1.5437 1.5003 0.0735 VSAC23 1.1659 1.7216 1.6352 0.0642
VSAC33 1.0866 1.5536 1.4532 0.0907 VSAC33 1.1810 1.5741 1.4962 0.0799 VSAC33 1.1496 1.7648 1.5559 0.0754
VSAC43 1.1054 1.5949 1.4492 0.0952 VSAC43 1.1873 1.5780 1.4658 0.0743 VSAC43 1.1598 1.7357 1.5758 0.0627
VSAC53 1.1309 1.5694 1.4590 0.0762 VSAC53 1.1872 1.5031 1.4708 0.0519 VSAC53 1.1494 1.7346 1.6714 0.0822
VSAC63 1.1253 1.6004 1.5405 0.1013 VSAC63 1.1703 1.5673 1.4430 0.0758 VSAC63 1.1655 1.7815 1.5669 0.0753
VSAC73 1.1628 1.5362 1.5881 0.0794 VSAC73 1.1706 1.5053 1.4890 0.0641 VSAC73 1.1430 1.7487 1.5400 0.0712
VSAC83 1.1032 1.5374 1.5235 0.0915 VSAC83 1.1814 1.5611 1.4473 0.0691 VSAC83 1.1440 1.7794 1.6899 0.1008
VSAC93 1.1575 1.5742 1.4197 0.0620 VSAC93 1.1749 1.5740 1.4582 0.0776 VSAC93 1.1579 1.7901 1.6727 0.0945
VSAC103 1.1055 1.6348 1.4488 0.1092 VSAC103 1.1788 1.5262 1.4693 0.0621 VSAC103 1.1592 1.7674 1.6270 0.0800
Methods Dataset4 Dataset17 Dataset18
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 0.7088 1.8566 1.6167 0.1044 VSA 1.0689 1.7354 1.6695 0.0899 VSA 1.1120 1.6863 1.6872 0.1109
Evolutionary Intelligence
1 3
Table 7 (continued)
Methods Dataset4 Dataset17 Dataset18
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
Proposed
method
(state1)
VSAC11 0.6902 1.8539 1.5967 0.1091 VSAC11 1.0864 1.7759 1.6687 0.0930 VSAC11 1.1723 1.7033 1.6068 0.0710
VSAC21 0.8725 1.8518 1.5915 0.0242 VSAC21 1.0813 1.7860 1.6392 0.0936 VSAC21 1.1095 1.7584 1.5863 0.1145
VSAC31 0.8720 1.8505 1.6129 0.0267 VSAC31 1.1189 1.7389 1.6394 0.0619 VSAC31 1.1338 1.7174 1.6306 0.0971
VSAC41 0.8873 1.8525 1.5597 0.0141 VSAC41 1.0986 1.8086 1.5121 0.0812 VSAC41 1.1430 1.7571 1.5692 0.0969
VSAC51 0.7766 1.8590 1.5251 0.0626 VSAC51 1.0884 1.7982 1.6546 0.0964 VSAC51 1.1540 1.6816 1.5372 0.0626
VSAC61 0.8194 1.8598 1.5882 0.0506 VSAC61 1.0562 1.7333 1.5936 0.0819 VSAC61 1.1528 1.6619 1.5289 0.0556
VSAC71 0.8537 1.8548 1.5435 0.0283 VSAC71 1.0740 1.7528 1.6620 0.0909 VSAC71 1.1517 1.6910 1.5192 0.0649
VSAC81 0.8656 1.8514 1.5442 0.0219 VSAC81 1.0919 1.7855 1.5151 0.0754 VSAC81 1.1644 1.6820 1.5290 0.0571
VSAC91 0.8687 1.8551 1.5206 0.0196 VSAC91 1.0547 1.7438 1.5134 0.0764 VSAC91 1.0928 1.7002 1.6988 0.1260
VSAC101 0.7156 1.8599 1.6146 0.1019 VSAC101 1.0918 1.7454 1.6640 0.0808 VSAC101 1.1239 1.7315 1.6920 0.1176
Proposed
method
(state2)
VSAC12 0.7711 1.8576 1.6005 0.0736 VSAC12 1.1128 1.7373 1.5244 0.0492 VSAC12 1.1395 1.6854 1.5345 0.0702
VSAC22 0.7731 1.8597 1.5485 0.0669 VSAC22 1.0669 1.7586 1.6877 0.1007 VSAC22 1.1702 1.6906 1.6156 0.0697
VSAC32 0.7652 1.8593 1.5492 0.0704 VSAC32 1.0431 1.7597 1.5528 0.0912 VSAC32 1.1796 1.7157 1.6159 0.0728
VSAC42 0.8710 1.8524 1.5246 0.0179 VSAC42 1.1024 1.8005 1.5231 0.0770 VSAC42 1.1012 1.7181 1.6372 0.1138
VSAC52 0.7723 1.8510 1.5716 0.0671 VSAC52 1.1161 1.7365 1.6164 0.0586 VSAC52 1.1537 1.7449 1.5215 0.0837
VSAC62 0.7172 1.8541 1.5670 0.0927 VSAC62 1.1157 1.7601 1.6432 0.0703 VSAC62 1.1446 1.7060 1.5646 0.0784
VSAC72 0.8708 1.8513 1.5962 0.0254 VSAC72 1.1381 1.7894 1.6688 0.0729 VSAC72 1.1835 1.7404 1.5216 0.0691
VSAC82 0.7680 1.8575 1.5988 0.0748 VSAC82 1.0417 1.7339 1.5291 0.0803 VSAC82 1.1049 1.6851 1.5991 0.0956
VSAC92 0.7155 1.8535 1.5680 0.0934 VSAC92 1.0597 1.8063 1.5647 0.1010 VSAC92 1.1621 1.6884 1.6959 0.0899
VSAC102 0.8549 1.8545 1.5769 0.0313 VSAC102 1.0837 1.8083 1.5419 0.0893 VSAC102 1.0988 1.6896 1.5441 0.0913
Proposed
method
(state3)
VSAC13 0.7810 1.8520 1.5818 0.0648 VSAC13 1.1011 1.8030 1.6219 0.0875 VSAC13 1.1840 1.6864 1.6454 0.0678
VSAC23 0.8029 1.8574 1.5613 0.0541 VSAC23 1.0747 1.7686 1.6550 0.0939 VSAC23 1.1284 1.6943 1.5352 0.0783
VSAC33 0.7254 1.8520 1.6354 0.0981 VSAC33 1.0406 1.7397 1.6881 0.1081 VSAC33 1.1336 1.6709 1.5366 0.0683
VSAC43 0.8986 1.8577 1.6172 0.0174 VSAC43 1.0686 1.7778 1.6737 0.1027 VSAC43 1.1232 1.7109 1.5954 0.0942
VSAC53 0.7473 1.8524 1.5286 0.0739 VSAC53 1.1298 1.7651 1.6274 0.0629 VSAC53 1.1565 1.7258 1.6131 0.0862
VSAC63 0.7880 1.8561 1.5700 0.0615 VSAC63 1.0953 1.7621 1.6328 0.0788 VSAC63 1.1862 1.6637 1.5385 0.0422
VSAC73 0.8155 1.8535 1.6382 0.0573 VSAC73 1.0707 1.8040 1.5938 0.0983 VSAC73 1.1614 1.7475 1.6339 0.0938
VSAC83 0.7385 1.8572 1.6048 0.0891 VSAC83 1.1067 1.7787 1.6165 0.0763 VSAC83 1.0969 1.7391 1.6547 0.1250
VSAC93 0.8227 1.8512 1.6346 0.0527 VSAC93 1.0478 1.7816 1.5119 0.0930 VSAC93 1.1100 1.7445 1.6903 0.1272
VSAC103 0.8314 1.8518 1.5446 0.0374 VSAC103 1.0768 1.7904 1.6597 0.1002 VSAC103 1.1569 1.6792 1.6744 0.0851
Bold values is to show the best-obtained value in the comparisons
Evolutionary Intelligence
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Table 8 Comparison of Results of methods with VSA (continuance)
Methods Dataset19 Dataset20 Dataset21
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 0.9269 1.4273 1.3141 0.0743 VSA 0.9335 1.3484 1.3985 0.0784 VSA 1.0266 1.3506 1.2821 0.0594
Proposed method (state1) VSAC11 0.9138 1.3929 1.3797 0.0828 VSAC11 0.9271 1.3953 1.3572 0.0823 VSAC11 1.0539 1.3512 1.2407 0.0427
VSAC21 0.9192 1.4345 1.3057 0.0790 VSAC21 0.9327 1.3936 1.3533 0.0785 VSAC21 1.0210 1.3840 1.2459 0.0696
VSAC31 0.9218 1.4036 1.3280 0.0716 VSAC31 0.9293 1.3813 1.3138 0.0691 VSAC31 1.0013 1.3876 1.2364 0.0789
VSAC41 0.9331 1.4302 1.3624 0.0801 VSAC41 0.9343 1.3927 1.3533 0.0774 VSAC41 1.0586 1.3805 1.2804 0.0545
VSAC51 0.9117 1.3751 1.3125 0.0653 VSAC51 0.9185 1.4397 1.3324 0.0947 VSAC51 1.0258 1.3886 1.2647 0.0706
VSAC61 0.9198 1.4048 1.3046 0.0691 VSAC61 0.9368 1.3669 1.3234 0.0633 VSAC61 1.0251 1.3697 1.2523 0.0631
VSAC71 0.9354 1.3796 1.3286 0.0585 VSAC71 0.8805 1.4118 1.3516 0.1075 VSAC71 1.0254 1.3722 1.2542 0.0639
VSAC81 0.9280 1.3706 1.3393 0.0617 VSAC81 0.8644 1.3749 1.2917 0.0936 VSAC81 1.0009 1.3761 1.2723 0.0782
VSAC91 0.9411 1.3743 1.3765 0.0647 VSAC91 0.9223 1.4006 1.3328 0.0813 VSAC91 1.0429 1.3889 1.2431 0.0618
VSAC101 0.9398 1.4117 1.3012 0.0616 VSAC101 0.8920 1.3797 1.2962 0.0830 VSAC101 1.0517 1.3861 1.2671 0.0584
Proposed method (state2) VSAC12 0.9206 1.4400 1.3166 0.0816 VSAC12 0.9051 1.3541 1.3658 0.0845 VSAC12 1.0380 1.3510 1.2309 0.0490
VSAC22 0.9374 1.4449 1.3079 0.0744 VSAC22 0.9261 1.3785 1.2897 0.0657 VSAC22 1.0620 1.3730 1.2777 0.0501
VSAC32 0.9331 1.4191 1.3284 0.0710 VSAC32 0.9244 1.4068 1.3796 0.0913 VSAC32 1.0449 1.3792 1.2432 0.0572
VSAC42 0.9201 1.4053 1.3627 0.0794 VSAC42 0.9249 1.3893 1.3939 0.0900 VSAC42 1.0150 1.3561 1.2393 0.0615
VSAC52 0.9359 1.3762 1.3241 0.0565 VSAC52 0.8656 1.3387 1.3980 0.1082 VSAC52 1.0486 1.3520 1.2623 0.0472
VSAC62 0.9321 1.3753 1.3651 0.0665 VSAC62 0.9008 1.3425 1.3884 0.0898 VSAC62 1.0189 1.3714 1.2493 0.0662
VSAC72 0.9486 1.3840 1.3346 0.0546 VSAC72 0.8763 1.3433 1.2977 0.0802 VSAC72 1.0248 1.3516 1.2422 0.0558
VSAC82 0.9209 1.4353 1.3635 0.0875 VSAC82 0.9252 1.3333 1.2750 0.0502 VSAC82 1.0112 1.3700 1.2759 0.0719
VSAC92 0.9251 1.4186 1.3434 0.0771 VSAC92 0.9218 1.3572 1.3824 0.0815 VSAC92 1.0336 1.3584 1.2449 0.0546
VSAC102 0.9450 1.3770 1.3235 0.0523 VSAC102 0.8720 1.4383 1.3011 0.1112 VSAC102 1.0433 1.3690 1.2348 0.0537
Proposed method (state3) VSAC13 0.9414 1.4073 1.3230 0.0627 VSAC13 0.9005 1.3308 1.4063 0.0928 VSAC13 1.0267 1.3762 1.2303 0.0633
VSAC23 0.9377 1.4139 1.3537 0.0717 VSAC23 0.8611 1.3899 1.3941 0.1202 VSAC23 1.0277 1.3520 1.2370 0.0542
VSAC33 0.9488 1.4272 1.3755 0.0744 VSAC33 0.9262 1.3827 1.3839 0.0855 VSAC33 1.0325 1.3843 1.2598 0.0657
VSAC43 0.9419 1.4002 1.3880 0.0732 VSAC43 0.8640 1.4302 1.2804 0.1096 VSAC43 1.0189 1.3516 1.2896 0.0645
VSAC53 0.9473 1.4355 1.3805 0.0783 VSAC53 0.9094 1.3536 1.3330 0.0747 VSAC53 1.0437 1.3756 1.2891 0.0606
VSAC63 0.9500 1.3758 1.3061 0.0465 VSAC63 0.9042 1.3611 1.4056 0.0966 VSAC63 1.0615 1.3703 1.2842 0.0501
VSAC73 0.9245 1.3929 1.3346 0.0684 VSAC73 0.9112 1.3800 1.2873 0.0727 VSAC73 1.0443 1.3731 1.2428 0.0552
VSAC83 0.9179 1.3834 1.3330 0.0686 VSAC83 0.9284 1.3581 1.2770 0.0564 VSAC83 1.0132 1.3852 1.2568 0.0742
VSAC93 0.9342 1.4103 1.3894 0.0797 VSAC93 0.8747 1.3644 1.3424 0.0959 VSAC93 1.0072 1.3574 1.2468 0.0662
VSAC103 0.9395 1.3729 1.3707 0.0638 VSAC103 0.9255 1.3728 1.3494 0.0756 VSAC103 1.0457 1.3581 1.2763 0.0522
Methods Dataset4 Dataset23 Dataset24
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
VSA VSA 1.1897 1.5564 1.5208 0.0851 VSA 1.0158 1.3750 1.3256 0.0690 VSA 0.9870 1.4522 1.3572 0.0407
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Table 8 (continued)
Methods Dataset4 Dataset23 Dataset24
Modes Worst Best Mean SD Modes Worst Best Mean SD Modes Worst Best Mean SD
Proposed
method
(state1)
VSAC11 1.1876 1.5374 1.5285 0.0829 VSAC11 1.0145 1.3895 1.3149 0.0721 VSAC11 1.0012 1.4425 1.3945 0.0377
VSAC21 1.1773 1.5394 1.4289 0.0715 VSAC21 1.0175 1.4069 1.3059 0.0750 VSAC21 1.0067 1.4529 1.3599 0.0322
VSAC31 1.1897 1.5401 1.4896 0.0746 VSAC31 1.0174 1.3841 1.3352 0.0725 VSAC31 1.0213 1.4292 1.3603 0.0183
VSAC41 1.1712 1.5071 1.4836 0.0731 VSAC41 1.0179 1.4491 1.3330 0.0922 VSAC41 1.0305 1.4663 1.3747 0.0276
VSAC51 1.1700 1.5216 1.4667 0.0744 VSAC51 1.0142 1.3903 1.3020 0.0706 VSAC51 1.0346 1.4278 1.3573 0.0112
VSAC61 1.1824 1.5080 1.5095 0.0738 VSAC61 1.0116 1.4418 1.3009 0.0890 VSAC61 0.9807 1.4672 1.3742 0.0509
VSAC71 1.1721 1.5156 1.5276 0.0848 VSAC71 1.0115 1.4461 1.3299 0.0937 VSAC71 1.0164 1.4096 1.3419 0.0116
VSAC81 1.1782 1.5442 1.4576 0.0762 VSAC81 1.0136 1.4351 1.3286 0.0890 VSAC81 0.9928 1.4460 1.3534 0.0355
VSAC91 1.1733 1.5612 1.4795 0.0870 VSAC91 1.0101 1.4122 1.3065 0.0802 VSAC91 1.0327 1.4621 1.3757 0.0255
VSAC101 1.1804 1.5746 1.5080 0.0923 VSAC101 1.0102 1.4492 1.3079 0.0930 VSAC101 0.9989 1.4285 1.3604 0.0285
Proposed
method
(state2)
VSAC12 1.1888 1.5573 1.5084 0.0834 VSAC12 1.0132 1.3813 1.3252 0.0720 VSAC12 1.0320 1.4259 1.3587 0.0121
VSAC22 1.1808 1.5455 1.4537 0.0749 VSAC22 1.0121 1.4121 1.3255 0.0818 VSAC22 1.0225 1.4153 1.3390 0.0100
VSAC32 1.1836 1.5436 1.4478 0.0722 VSAC32 1.0165 1.4380 1.3291 0.0886 VSAC32 1.0110 1.4184 1.3305 0.0151
VSAC42 1.1718 1.5017 1.4702 0.0686 VSAC42 1.0122 1.4304 1.3296 0.0882 VSAC42 0.9899 1.4146 1.3517 0.0271
VSAC52 1.1720 1.5058 1.4722 0.0700 VSAC52 1.0195 1.4455 1.3002 0.0868 VSAC52 0.9791 1.4246 1.3695 0.0383
VSAC62 1.1770 1.5322 1.4853 0.0776 VSAC62 1.0165 1.4336 1.3381 0.0884 VSAC62 0.9829 1.4062 1.4061 0.0395
VSAC72 1.1851 1.5333 1.4687 0.0713 VSAC72 1.0164 1.4128 1.3129 0.0783 VSAC72 1.0237 1.4276 1.4013 0.0245
VSAC82 1.1853 1.5710 1.4496 0.0810 VSAC82 1.0187 1.4131 1.3199 0.0783 VSAC82 1.0158 1.4184 1.3413 0.0145
VSAC92 1.1790 1.5481 1.4977 0.0834 VSAC92 1.0102 1.4219 1.3355 0.0873 VSAC92 0.9728 1.4201 1.3367 0.0342
VSAC102 1.1821 1.5046 1.4437 0.0599 VSAC102 1.0158 1.4147 1.3215 0.0804 VSAC102 1.0189 1.4217 1.3563 0.0165
Proposed
method
(state3)
VSAC13 1.1749 1.5629 1.5217 0.0940 VSAC13 1.0159 1.4236 1.3113 0.0819 VSAC13 1.0292 1.4056 1.3730 0.0103
VSAC23 1.1821 1.5664 1.4781 0.0843 VSAC23 1.0122 1.4042 1.3016 0.0760 VSAC23 0.9879 1.4021 1.3331 0.0212
VSAC33 1.1847 1.5522 1.4861 0.0800 VSAC33 1.0157 1.3706 1.3188 0.0666 VSAC33 1.0202 1.4470 1.4036 0.0318
VSAC43 1.1831 1.5304 1.4298 0.0659 VSAC43 1.0142 1.3926 1.3026 0.0714 VSAC43 1.0170 1.4634 1.3791 0.0336
VSAC53 1.1713 1.5451 1.4829 0.0835 VSAC53 1.0135 1.4211 1.3004 0.0810 VSAC53 1.0399 1.4135 1.4087 0.0150
VSAC63 1.1825 1.5467 1.4260 0.0715 VSAC63 1.0197 1.4042 1.3214 0.0752 VSAC63 1.0005 1.4222 1.3626 0.0263
VSAC73 1.1803 1.5572 1.4855 0.0834 VSAC73 1.0160 1.4196 1.3254 0.0824 VSAC73 1.0089 1.4579 1.3645 0.0334
VSAC83 1.1852 1.5076 1.5033 0.0710 VSAC83 1.0104 1.4202 1.3364 0.0868 VSAC83 1.0287 1.4363 1.4090 0.0261
VSAC93 1.1794 1.5254 1.4340 0.0664 VSAC93 1.0140 1.4161 1.3062 0.0797 VSAC93 0.9908 1.4042 1.3628 0.0259
VSAC103 1.1858 1.5717 1.4757 0.0840 VSAC103 1.0107 1.3905 1.3235 0.0755 VSAC103 1.0019 1.4603 1.3812 0.0401
Bold values is to show the best-obtained value in the comparisons
Evolutionary Intelligence
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the similarities between Victoria representations of docu-
ments in one space to solve a problem p of P. In the other
two DCM-based approaches, it is possible to hybrid dif-
ferent representation spaces. In the case of DCM-voting
approach, this is done using a voting technique and as
for the DCM-classifier, it can be performed through a
supervised learning method which requires the definition
of predictive features. During evaluation of the challenge
PAN-CLEF 2013, it is observed that DCM-classifier has
the best performance only on the Greek corpus with 85%,
and the two other approaches i.e. DCM-voting and DCM-
classifier obtain the best results or equivalent to the winner
of the competition for all evaluation measures (F1, preci-
sion and recall) on all the corpora.
The General Impostors Method (GENIM) which took
part in the PAN’13 authorship identification competition has
been evaluated in [92]. The basis of this model is the com-
parison made between the given documents and a number
of external (impostor) documents, and since there are two
stages in their method, the performance had to be meas-
ured and parameters needed to be optimized at each step.
25–33 percent of the training documents of each language
were used for measuring and optimizing IM, whereas the
rest were used for evaluation of GENIM. For the IM evalu-
ation set, 3 or 4 documents were used as seed documents
to retrieve the web impostor. The test accuracy is equal to
75.3%.
Blocks containing 140, 280, and 500 characters were
investigated. The feature set contains conventional features
like syntactic, lexical, application specific features, and
some new features that are extracted from n-gram analy-
sis. Moreover, the proposed approach has a mechanism
for handling issues related to unbalanced dataset. It also
uses Support Vector Machine (SVM) for data classification
and uses Information Gain and Mutual Information as a
FS strategy. The proposed approach was evaluated experi-
mentally using the Enron email and Twitter corpuses. The
results of this evaluation were very promising including
an Equal Error Rate (EER) changing between 9.98% and
21.45%, for different block sizes [93].
In [94], by using a cluster-based classification approach, a
model is presented for email authorship identification (EAI).
Contributions of this paper are as follows: a) Developing a
new model for email authorship identification. b) Evaluation
Table 9 Results of the VSA and
the proposed method based on
the FS
Bold values is to show the best-obtained value in the comparisons
Datasets VSA Proposed method
Feature
count-total
Accuracy (%) Time(sec) Feature
count-select
Accuracy (%) Time(sec)
Dataset1 36 44.2472 2.3244 25 56.5962 2.105
Dataset2 10 48.4139 1.5936 8 67.549 0.7746
Dataset3 24 91.6334 1.0279 11 87.8578 2.2247
Dataset4 15 63.7899 1.6667 8 64.9167 0.175
Dataset5 36 77.4904 1.7881 19 92.073 1.1341
Dataset6 8 60.3178 1.6666 6 62.4584 1.191
Dataset7 10 72.4666 1.941 6 81.1022 2.4077
Dataset8 16 71.3516 3.5829 8 96.3686 2.6418
Dataset9 21 87.4691 1.1463 9 96.3448 2.1726
Dataset10 16 82.3352 0.0883 7 90.1066 0.456
Dataset11 30 98.4868 2.5427 16 98.7271 0.3067
Dataset12 80 36.3746 0.1836 53 71.7682 0.9082
Dataset13 22 54.9418 2.5736 13 50.01 0.8355
Dataset14 22 35.7599 0.5076 17 59.9471 2.2448
Dataset15 20 88.6538 2.1595 8 80.2569 1.2598
Dataset16 57 36.5519 2.3174 13 43.957 0.5924
Dataset17 45 45.2476 3.066 23 93.6619 1.7498
Dataset18 17 60.1896 3.4182 6 58.0906 0.0865
Dataset19 13 69.6363 2.7172 7 92.9925 2.6428
Dataset20 10 82.1821 1.1213 6 83.4601 1.9857
Dataset21 2350 48.185 0.9042 1200 90.7128 1.4438
Dataset22 5340 84.1727 1.245 2341 83.7111 2.7264
Dataset23 3343 95.4847 3.8613 1845 99.7029 0.8341
Dataset24 2340 47.1271 0.6734 800 50.26194 1.3153
Evolutionary Intelligence
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of using additional features together with basic
stylometric
features for email authorship identification as well as content
features that are based on Info Gain FS. On the Enron data-
set, the proposed model achieved accuracies of 94, 89, and
81 percent for 10, 25, and 50 authors, respectively. Whereas,
on real email dataset constructed by authors, it attained an
accuracy of 89.5%.
A large number of researches only focus on enhanc-
ing predictive accuracy and do not pay much attention to
intrinsic value of the collected evidence. In this paper, a
customized associative classification approach, which is a
well-known data mining technique, is applied to the author-
ship attribution problem. This method models the features
of writing style which are unique to a person. Then, it meas-
ures the associativity level of these features and generates
an instinctive classifier. In this research, it is also concluded
that a more accurate write print can also be provided by
applying modifications on the rule pruning and ranking
system described in the popular Classification by Multiple
Association Rule (CMAR) algorithm. More convincing evi-
dences can be provided for a court of law by eliminating
patterns common amongst different authors since it leads
to fairly unique and easy-to-understand write prints. Since
this customized abandonment counter method is helpful in
solving the problem of the e-mail authorship attribution, it
can be used as a powerful tool against cybercrimes. The
Fig. 2 a Feature Count-Total. b Accuracy (%). c Time (sec)
Evolutionary Intelligence
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effectiveness of the presented approach is verified by the
results obtained through experiments [95].
An effort is made by authors in [96] to identify the author
of articles written in Arabic. They introduced a new data-
set which is composed of 12 features and 456 samples of 7
authors. Furthermore, to distinguish different authors from
each other, powerful classification techniques were hybrids
with the proposed dataset in their approach. The obtained
results revealed that the proposed dataset was very success-
ful and achieved a classification performance accuracy of
82% in the hold-out tests. They also conducted some exper-
iments with two well-known classifiers namely the SVM
and functional trees (FT) in order to show the efficiency
of the proposed feature set. The reported an accuracy of
82% with the FT approach and holdout testing which con-
firmed robustness of the proposed feature set. Moreover, an
accuracy of 100% has been achieved in one of the classes.
They also conducted some test on FT by using tenfold cross
validation and the proposed approach retained its accuracy
to some extent.
One of the classifiers which have been extensively used
for language processing is the Naive Bayes classifiers.
Nevertheless, the event model used which can remarkably
affect the classifier performance is not often mentioned.
So far, Naive Bayes (NB) classifiers have never been used
for authorship attribution in Arabic. Thus, they proposed
to apply these classifiers to this problem, taking into con-
sideration various event models such as simple NB, multi-
nomial NB (MNB), multi-variant Bernoulli NB (MBNB),
and Multi-variant Poisson NB (MPNB). The MBNB prob-
ability estimation is dependent on whether a feature exists
or not, whereas MNB and MPNB a probability estimation
is dependent on the frequency of the feature. The mean and
standard deviation of the features form the basis of probabil-
ity estimation in the NB model. The performances of these
models are evaluated using a large Arabic dataset taken from
books written by 10 different authors. Then, they are com-
pared with other methods. The obtained results reveal that
MBNB outperforms other techniques and is able to identify
the author of a text with an accuracy of 97.43%. In addition,
these results show that MNB and MBNB can be considered
as a good choice for authorship attribution [97].
In [98], authorship identification methods were applied
to messages of Arabic web forum. In this study, syntactic,
lexical, structural, and content-specific writing style features
were used to identify the authors. Some of the problematic
characteristics of Arabic language were addressed in order to
present a model with an acceptable degree of classification
accuracy for authorship identification. SVM had a better per-
formance than C4.5 and compared to English performance,
the overall accuracy for Arabic was lower. These results
were in consistence with previous researches. Finally, as
future work, the authors proposed to analyze the differences
between these two languages by evaluating the key features
as determined by decision trees. Highlighting the linguistic
differences between English and Arabic languages provides
further insight into possible technique for enhancing the per-
formance of authorship discrimination methodologies in an
online, multilingual setting. The results showed accuracies
of 85.43 and 81.03 for SVM and C4.5, respectively.
In [99], they developed an authorship visualization known
as Write prints which can be used for identification of indi-
viduals based on their writing style. Unique writing style
patterns are created through this visualization. These pat-
terns can be distinguished in a similar way that fingerprint
biometric systems work. Write prints provide an approach
which is based on component analysis and utilizes a dynamic
feature-based sliding window algorithm. This makes them
very suitable for visualizing authorship across larger groups
of messages. The performance of visualization across mes-
sages taken from three different Arabic and English forums
was evaluated and compared with the performance of
SVM. This comparison indicated that Write prints show
Table 10 Control parameters of the algorithms
Algorithms Description Parameters Value
GA [83] Crossover rate Pc 0.9
Generation Ng 200
Mutation rate Pm 0.01
PSO [84] Acceleration coefficients C1 1.5
Acceleration coefficients C2 1.5
Random number R1 0.5
Random number R2 0.5
Inertia weight (linearly
decreases)
(w) 0.6 to 0.3
ABC [9] Employed bees Ne 50
Onlooker bees No 50
Scout bee Ns 50
Random number ϕ 0.5
BOA [8] Flight distance Stepe0.05
Random number Rand 0.02
Linearly decreases a 2–0
IWO [7] Minimum number of seeds smin 0
Maximum number of seeds smax 5
Final value of standard devia-
tion
σfinal 0.001
Initial value of standard devia-
tion
σinitial 3
FPA [14] Step size scaling factor γ 0.01
Switch probability between
Local and Global pollination
P 0.4
Randomization α 0.2
Attractiveness of a firefly β 1
FA [11] Absorption coefficient γ 1
Evolutionary Intelligence
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an excellent classification performance and provide better
results than SVM in many instances. They also concluded
that visualization can be used to identify cyber criminals and
can help users authenticate fellow online members to prevent
cyber fraud. Accuracies of 68.92 and 87.00 were obtained
for Write prints and SVM, respectively.
In [100], they introduced approaches to deal with imbal-
anced multi-class textual datasets. The main idea behind
their approach is to divide the training texts into text sam-
ples based on the class size thus, a fairer classification model
could be generated. Therefore, it becomes possible to divide
majority classes into less and longer samples and minority
classes into many shorter samples. They used text sampling
techniques to form a training set based on a desirable dis-
tribution over the classes. By text sampling, they developed
new synthetic data that artificially caused the training size
of a class to increase. A series of authorship identification
experiments were conducted by these researchers on dif-
ferent multiclass imbalanced cases belonging to two text
corpora of two languages; newspaper reportage in Arabic
and newswire stories in English. Properties of the presented
techniques were revealed by the results obtained through
these experiments. They also tested four methods to deal
with the problem of class imbalance [100]:
• The first method: To under-sample majority classes based
on training texts. The same amount of text which was
equal to the base was used. No modification is applied to
the length of each text.
• The second method: To Under-sample majority classes
based on training text lines. All the training texts for a
particular author were merged to form a big text. Assum-
ing that xmin represents the size (in text lines) of the
shortest big file then, the first xmin text lines of each big
file were segmented into text samples of length a (in text
lines). It is worth noting that there was as least one com-
plete sentence in each text line in both corpora. It was
concluded that smaller values (such as 2 or 3) lead to
better results.
• The third method: Re-balancing the dataset by text sam-
ples of varying length. As was mention earlier in this
paper, one big file is generated for each author by concat-
enation of training texts. In other words, the length of text
sample is equal to xi/k (where, k is predefined param-
eter). Short text samples belong to minority authors and
long text samples belong to majority authors. Therefore,
a balanced dataset is generated which consists of k text
samples per class. Experiments were conducted for
k = 10, 20, and 50. It is noteworthy that each text line
Table 11 Comparison of the
proposed method with other
algorithms based on the worst
criterion
Bold values is to show the best-obtained value in the comparisons
Datasets PSO ABC BOA IWO GA FA FPA VSA Proposed Method
Dataset1 1.1652 1.3169 1.2795 1.2116 1.3078 1.0623 1.3221 1.151 1.3021
Dataset2 1.0134 0.8357 1.0904 0.7935 0.859 0.9388 0.9091 1.0718 1.0804
Dataset3 1.1403 0.9748 1.0289 1.0323 1.0699 1.0364 1.0189 1.0692 1.0699
Dataset4 1.2313 1.2398 1.0536 1.3387 1.0844 1.076 1.1111 1.1463 1.3387
Dataset5 1.1624 1.1992 1.0705 1.19 1.0613 1.1965 1.0762 1.1983 1.1083
Dataset6 0.5866 0.7323 0.7112 0.472 0.7894 0.6187 0.7379 0.502 0.7874
Dataset7 1.1565 1.2648 1.2797 1.4996 1.1722 1.4984 1.4836 1.0209 1.4984
Dataset8 1.2758 1.0955 1.0956 1.1194 1.2625 1.1351 1.1704 1.1696 1.2758
Dataset9 1.1248 1.0839 1.1775 1.1833 1.0772 1.2775 1.264 1.0373 1.2375
Dataset10 1.2866 1.1589 1.167 1.3825 1.2842 1.2472 1.2791 1.3497 1.2825
Dataset11 1.243 1.327 1.2921 1.3961 1.6147 1.3582 1.3734 1.2358 1.3961
Dataset12 1.4566 1.3633 1.4114 1.2155 1.2255 1.5031 1.2839 1.4008 1.2255
Dataset13 1.2436 1.1973 1.0512 1.2454 1.0363 1.2885 1.4906 1.3004 1.3204
Data set14 1.2653 1.1884 1.2394 1.3543 1.3115 1.2894 1.3822 1.3922 1.1644
Dataset15 1.1821 1.2728 1.3882 1.4679 1.4981 1.3427 1.4981 1.4827 1.3181
Dataset16 1.4262 1.0759 1.6764 1.2535 1.3716 1.5304 1.5123 0.8414 1.3264
Dataset17 1.2562 1.1611 1.1055 1.2166 1.031 1.0767 0.9641 1.369 1.0169
Dataset18 1.3983 1.4124 1.2653 1.303 1.0773 1.2703 1.0404 1.467 1.1167
Data set19 1.1328 0.9997 0.9467 1.0967 0.9658 0.9974 1.198 1.1236 1.1208
Dataset20 0.944 1.0662 1.2361 1.1814 0.9685 1.0653 1.1904 1.1242 1.0361
Dataset21 1.6487 1.4398 1.0724 1.2875 1.6143 1.631 1.1727 1.7802 1.1802
Dataset22 1.3458 1.6606 1.7973 1.1826 1.3905 1.0926 1.8201 1.1224 1.0201
Dataset23 1.154 1.0374 0.9335 1.9864 1.0774 0.8664 1.4506 1.2632 1.1206
Dataset24 1.0724 1.1301 1.1549 0.8827 0.7526 1.1974 1.8704 0.9771 1.1974
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of the training corpus is used exactly once in the text
samples.
• The fourth method: Re-balancing the dataset through text
re-sampling. A big file is generated for each author once
again. Assuming that x represents the text-length (in text
lines) of the
ith
author and xmax is the longest file then,
for each author, k + xmax/xi text samples are generated
each of which consisting of xi/k. Therefore, based on the
length of the big file, a variable number of text samples
are generated for each author. Nonetheless, the relation-
ship is inversed now. Longer text samples are generated
for the majority classes but a large number of short text
samples are generated for the minority classes.
Using a data set extracted from Arabic novels, they the
modified this to two sets of words AFW54 and AFW65,
with 11 words eliminated [101]. These two sets were used
convert several Arabic texts into frequency vectors. They
carried out a performance evaluation on these word sets
Fig. 3 Result of comparison a The worst criterion. b The best criterion. c The mean criterion
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through experiments which used a hybridization of an EA
and LDA to generate a classifier. Then, they fed unseen
data to that classifier in order to test it. The obtained per-
formance was apparently consistent with results of author-
ship attribution researches performed on other languages.
It is arguable that AFW54 is a more suitable choice nev-
ertheless; such a claim cannot be made with any statis-
tical significance. For the cases considered here, only a
small number of investigations are reported for evaluating
the appropriate ‘chunk’ size. In real-world applications
this will be probably dependent on several factors, but
they have identified at least about 1,000 characterization
of function word usage for Arabic authors. Through this
work, they have confirmed that the concept of function
words translates properly into the Arabic language. In
other words, various authors use this set of words in vari-
ous ways, and this enables us to recognize stylistic features
of individual authors and use them to distinguish between
different authors [101].
High dimensional datasets bring about more computa-
tional challenges. One of the problems with high dimen-
sional datasets is that in most cases, all features of data
are not crucial for the knowledge implicit in the data [85,
102]. Consequently, in most occasions, reduction in the
dimensions of data is a favored subject. Often, many of
candidate features for learning are irrelevant and super-
fluous and degrade the efficiency of the learning algo-
rithm [103, 104]. Learning accuracy and teaching speed
may be worsening with superfluous features. Therefore,
choosing the corresponding necessary features in preproc-
essing phase is essentially important. In this section, for
identifying the author, at the first stage, the frequency of
words is obtained using the method TF-IDF [105]. At the
second stage, each feature is weighted [106]. At the third
stage, using metaheuristic algorithms, FS is performed.
At the fourth stage, classification is performed via KNN
[106].
Furthermore, we used the accuracy as the evaluation
measure. This accuracy is calculated as:
In the case that TP represents the number of authors who
are in the positive class while, TN indicates the number of
authors who are in the negative class. Furthermore, FP is
the number of authors falsely was considered as positive
class by the model and FN is the number of authors falsely
(22)
Accuracy
=
TP +TN
TP +TN +FP +FN
∗
100
Table 12 Comparison of the
proposed method with other
algorithms based on the best
criterion
Bold values is to show the best-obtained value in the comparisons
Datasets PSO ABC BOA IWO GA FA FPA VSA Proposed Method
Dataset1 1.2178 1.2977 1.1785 1.3271 1.1248 1.3193 1.1703 1.3125 1.3425
Dataset2 1.0049 1.011 0.9677 1.0707 1.0007 1.0669 1.0093 1.0633 1.0833
Dataset3 1.3354 1.3319 1.2486 1.2042 1.271 1.2486 1.3494 1.3764 1.3494
Dataset4 1.4182 1.2193 1.3134 1.3666 1.3154 1.3457 1.3129 1.4819 1.4182
Dataset5 1.3251 1.3573 1.2022 1.2551 1.1989 1.1362 1.1206 1.3422 1.3673
Dataset6 0.8806 0.6019 0.8504 0.6873 0.6848 0.5194 0.7076 0.9001 0.9031
Dataset7 1.6409 1.4838 1.4203 1.6609 1.5315 1.6245 1.4666 1.7136 1.6609
Dataset8 1.2302 1.1222 1.3725 1.1426 1.2599 1.1481 1.3451 1.4027 1.4047
Dataset9 1.3527 1.3663 1.3565 1.3902 1.2539 1.3352 1.2761 1.3717 1.3902
Dataset10 1.5953 1.4125 1.4442 1.5566 1.4284 1.3015 1.2941 1.3567 1.5966
Dataset11 1.6458 1.4675 1.4298 1.5415 1.6417 1.276 1.7019 1.6149 1.7019
Dataset12 1.7133 1.7001 1.4547 1.4802 1.7619 1.2872 1.3965 1.7329 1.7610
Dataset13 1.6127 1.208 1.2482 1.2267 1.2807 1.3122 1.4292 1.3327 1.6227
Dataset14 1.4605 1.5531 1.5892 1.2713 1.4945 1.4966 1.4835 1.5378 1.5892
Dataset15 1.4354 1.8054 1.3415 1.5058 1.3346 1.6282 1.5713 1.5591 1.8154
Dataset16 1.7259 1.3034 1.5922 1.0983 1.7772 1.0201 1.4102 1.6328 1.8328
Dataset17 1.6531 1.7521 1.6948 1.1967 1.7978 1.673 1.8108 1.4214 1.8108
Dataset18 1.7742 1.3852 1.6935 1.7389 1.6448 1.5793 1.7353 1.5856 1.7742
Dataset19 1.3372 1.3511 1.4452 1.3507 1.4515 1.2678 1.1654 1.0718 1.6515
Dataset20 1.4412 1.4441 1.2213 1.1804 1.2992 1.4351 1.2057 1.2681 1.5441
Dataset21 1.0257 1.1663 1.4461 1.2206 1.3912 1.1613 1.4962 1.6151 1.6151
Dataset22 1.3227 1.0865 1.6166 1.44 1.2056 1.731 1.0597 1.7058 1.7031
Dataset23 1.1935 1.426 1.2313 1.2189 0.9236 1.01 0.9433 1.0665 1.4466
Dataset24 1.2898 0.9603 1.2427 1.3963 1.2943 1.1029 1.1937 1.1191 1.3963
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was considered as negative class by the model, even though
they were positive.
6.1 Reuter_50_50 dataset
In this subsection, the Proposed Method and other algo-
rithms are applied to Reuter_50_50 datasets. The dataset
contains 2500 documents and 50 writers (https:// archi ve.
ics. uci. edu/ ml/ datas ets/ reuter_ 50_ 50). The results from
the discussed algorithms and the results from other papers
are presented in Table14 and Fig.4. The results show that
the proposed method has a better identification accuracy
compared to other algorithms. Moreover, BOA and FPA
have also better identification accuracy compared to other
algorithms.
6.2 PAN dataset
These datasets consist of scientific documents in Greek,
English, and Spanish, and from 2011 until now, a new
dataset has been added to the existing ones every year
(https:// pan. webis. de). The results from the discussed
algorithms and the results from other papers on these
datasets are evaluated in Table15 and Fig.4. Identifica-
tion accuracy of proposed method for PAN11, PAN12,
PAN13, PAN14, PAN15, and PAN16 are 84%, 80.9%,
81.3%, 82.12%, 83.25%, and 81.79%, respectively.
Table 13 Comparison of the
Proposed Method with other
algorithms based on the mean
criterion
Bold values is to show the best-obtained value in the comparisons
Datasets PSO ABC BOA IWO GA FA FPA VSA Proposed Method
Data set1 1.3916 1.1978 1.3929 1.3897 1.2765 1.3521 1.1941 1.3898 1.3929
Data set2 0.9622 1.0635 0.9613 0.9978 1.0673 1.0438 1.0089 0.9741 1.0673
Data set3 1.1495 1.3674 1.1479 1.1516 1.2789 1.1939 1.3617 1.1515 1.3717
Data set4 1.5543 1.2753 1.2469 1.2985 1.4039 1.2656 1.2512 1.3383 1.4019
Data set5 1.0945 1.1002 1.23701 1.1344 1.3232 1.1146 1.3776 1.1253 1.3776
Data set6 0.9357 0.9499 0.6947 0.7346 0.8517 0.6554 0.9393 0.8642 0.9393
Data set7 1.5092 1.3172 1.3024 1.6366 1.4551 1.4772 1.3768 1.4806 1.6366
Data set8 1.1491 1.2448 1.1641 1.1609 1.2675 1.2762 1.2003 1.1581 1.2762
Data set9 1.2819 1.2329 1.3252 1.1997 1.2909 1.2488 1.2102 1.1581 1.3152
Data set10 1.2122 1.3358 1.4584 1.4552 1.4227 1.2478 1.4237 1.2327 1.4584
Data set11 1.6026 1.3283 1.5601 1.6014 1.5996 1.4536 1.5107 1.3743 1.6214
Data set12 1.2846 1.4074 1.4251 1.2414 1.5571 1.3214 1.3231 1.5242 1.5271
Data set13 1.5899 1.5719 1.5801 1.4036 1.3117 1.2544 1.5519 1.4592 1.5811
Data set14 1.3798 1.5262 1.4221 1.4781 1.2758 1.4304 1.3609 1.3481 1.5662
Data set15 1.4311 1.5769 1.4087 1.3936 1.3683 1.4963 1.6504 1.6188 1.6504
Data set16 0.9593 1.6964 1.1033 1.0707 1.0311 1.6315 1.3922 1.6179 1.6964
Data set17 1.2304 1.4441 1.1709 1.5863 1.5396 1.6017 1.6124 1.5419 1.6224
Data set18 1.4723 1.4846 1.6506 1.3603 1.4505 1.5661 1.4996 1.3536 1.6706
Data set19 1.0788 1.2331 1.2337 1.2852 1.0904 1.2745 1.3522 1.0931 1.3622
Data set20 1.3175 1.1659 1.2635 1.3272 1.1213 1.1434 1.2574 1.2245 1.3272
Data set21 0.8526 0.8718 1.1197 0.8701 1.3629 1.4002 1.2535 0.9912 1.4032
Data set22 0.8526 0.8718 1.1197 0.8701 1.3629 1.4002 1.2535 0.9912 1.4082
Data set23 1.3227 1.2502 0.9458 1.1709 1.2078 1.3314 0.8971 1.3211 1.3414
Data set24 0.9655 1.0916 1.2229 0.9476 0.9983 1.1743 0.9823 1.2059 1.2429
Table 14 Comparison of proposed method with other algorithms on
Reuter_50_50 datasets
# Algorithms Accuracy (%)
1GA 82
2 PSO 87.91
3 ABC 86
4BOA 88.13
5IWO 87.01
6FPA 88
7FA 85.6
8SVA 88.2
9 Delta [90] 67
10 KNN [90] 69
11 SVM [90] 85
12 Proposed method 89.3
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Fig. 4 Result of comparison a Reuter_50_50 dataset. b PAN Dataset. c Enron Email Dataset. d Arabic Scripts
Table 15 Comparison of
proposed method with other
algorithms on PAN datasets
# Dataset PAN11 PAN12 PAN13 PAN14 PAN15 PAN16
Method
1GA 73 72 75 74 76 69
2 PSO 81.5 78.3 79.5 80.11 79.78 78.01
3 ABC 80.1 78.8 77.6 80.1 79.9 79.32
4BOA 80.14 79.35 80.82 81.24 80.3 81
5IWO 76 74.3 79 80.1 81.2 79
6FPA 81.01 78.51 79.08 79.03 78.4 77
7FA 80 76.5 77.3 79 77.02 78.6
8SVA 83 78.3 76.9 78.1 83 80.9
9 DCM [91] – – 74.4 – – –
10 DCM-voting [91] – – 78.1 – – –
11 DCM-classifier [91] – – 76 – – –
12 Best result [92] – – 75.3 – – –
13 Proposed method 84 80.9 81.3 82.12 83.25 81.79
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Moreover, identification accuracies of DCM models are
less than other algorithms. The algorithms BOA, ABC
and IWO have better identification accuracies compared
to the algorithms GA, PSO, FPA, and FA.
6.3 Enron email dataset
This dataset is collected and prepared by CALO project (a
cognitive assistant that learns and organizes). This dataset
includes comments of 150 users who are CEOs of Enron
(https:// www. cs. cmu. edu/ ~enron/). The results from the
proposed method and the results from other papers on
Enron Email dataset are presented in Table16 and Fig.4.
The results show that the accuracy and error rate in pro-
posed method are 95.04 and 11.68, respectively. Accuracy
in the algorithms PSO, BOA and FPA are 91.02, 93.01,
and 90.78, respectively. The accuracy and error rate in
ABC algorithm are 90.02 and 15.2, respectively. Among
other models, the model CCM-10 has a better accuracy,
and the lowest accuracies are seen in the models Naïve
Bayes and Bayes Net.
6.4 Arabic scripts
This dataset consists of 30 documents from 10 authors. The
author was chosen from the website, (http:// www. alwar aq.
net) and their names are: Aljahedh, Alghazali, Alfarabi,
Almas3ody, Almeqrezi, Altabary, Altow7edy, Ibnaljawzy,
Ibnrshd, and Ibnsena. The results from the proposed method
and the results from other papers on these datasets are pre-
sented in Table17 and Fig.4. The identification accuracy
of proposed method model is 93.24%, which is better than
other models.
According to the experiments results, it is concluded that
the Proposed Method has a better performance than other
models in terms of identification accuracy. According to
Tables15, 16, 17 the proposed method in benchmark func-
tions is the closest to minimum compared to the algorithms
FPA, IWO, BOA, ABC, PSO, GA, and FA. Moreover, the
proposed method has a better accuracy in the author identifi-
cation problem. The rate of accuracy of ABC, BOA and pro-
posed method is indicated in Table17. The results revealed
that the proposed method outperformed to the other models
that is ABC and BOA models. The percentage of proposed
method is 93.24%. Consequently, the percentage of ABC is
91.00% and it is 92.51% for BOA model.
Table 16 Comparison of
proposed method with other
algorithms on enron email
dataset
# Algorithms Accuracy (%) Error rate
1GA 85.06 22
2 PSO 91.02 12
3 ABC 90.02 15.2
4BOA 93.01 12.3
5IWO 89.03 17.32
6FPA 90.78 13
7FA 88.05 20.3
8SVA 93.89 10.98
9 Linear(SVM) [93] – 18.86
10 Linear(SVM-LR) [93] – 15.34
11 Polynomial3(SVM-LR) [93] – 19.20
12 Polynomial5(SVM-LR) [93] – 28.47
13 Gaussian(SVM-LR) [93] – 46.01
14 CEAI: CCM-10 [94] 94 –
15 CEAI: CCM-25 [94] 89 –
16 CEAI: CCM-50 [94] 81 –
17 Author miner-AM [95] 68.19 –
18 Naïve bayes [95] 79.08 –
19 Bayes net [95] 79.56 –
20 Classification by multiple association rule-CMAR [95] 88.47 –
21 Classification by association-CBA [95] 84.18 –
22 J48 [95] 89.45 –
23 Classification by multiple association rule for authorship
attribution-CMARAA [95]
90.08 –
24 Proposed method 95.04 11.68
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7 Conclusion andfeature works
We proposed three State based on the hybrid of chaotic and
VSA in this paper for FS. State2 compared with Method1,
Method3, and VSA where it had better values. We also used
State2 to FS and text author identification. This paper is
accompanied by using 10 CMs to enhance the overall per-
formance and precision of the VSA. VSA is introduced to
one of the challenge problems, especially FS. The proposed
methods have been evaluated on 24 benchmark datasets.
Four precise evaluation standards are followed in this paper.
These standards are worst, best, mean, and SD. Similarly,
the performance of Proposed Method is compared with the
popular and maximum current other algorithms. These algo-
rithms are PSO, ABC, BOA, IWO, GA, FA, FPA, and VSA.
The experimental effects show that State2 outperforms the
other algorithms in terms of best and mean fitness.
Moreover, the outcomes displayed that the Proposed
Method (State2) with Tent map can drastically enhance
VSA in terms of classification overall performance, sta-
bility exceptional, number of FS, and convergence speed.
Moreover, the outcomes showed that Tent map turned into
the satisfactory map. Therefore, the following conclusion
can be drawn:
• The CMs improve the section of exploration because they
change the radius of value search, helping the trapped
masses to release themselves from local minima.
• The CMs are permitted to adaptively adjust exploration
and exploitation by the proposed method. As it were, the
Proposed Method (State1) encourages VSA to transit grad-
ually from the exploration stage to the exploitation stage.
Essential work on the integration of CMs with the addi-
tion of other metaheuristic algorithms will be considered.
VSA’s performance on more problematic science and
real-world engineering problems will be applied in future
verification.
Declarations
Conflict of interest The authors declare that they have no conflict of
interest.
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