A novel hybrid BPSO–SCA approach for feature selection
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Abstract
Nature is a great source of inspiration for solving complex problems in realworld. In this paper, a hybrid natureinspired algorithm is proposed for feature selection problem. Traditionally, the realworld datasets contain all kinds of features informative as well as noninformative. These features not only increase computational complexity of the underlying algorithm but also deteriorate its performance. Hence, there an urgent need of feature selection method that select an informative subset of features from high dimensional without compromising the performance of the underlying algorithm. In this paper, we select an informative subset of features and perform cluster analysis by employing a cross breed approach of binary particle swarm optimization (BPSO) and sine cosine algorithm (SCA) named as hybrid binary particle swarm optimization and sine cosine algorithm (HBPSOSCA). Here, we employ a Vshaped transfer function to compute the likelihood of changing position for all particles. First, the effectiveness of the proposed method is tested on ten benchmark test functions. Second, the HBPSOSCA is used for data clustering problem on seven reallife datasets taken from the UCI machine learning store and gene expression model selector. The performance of proposed method is tested in comparison to original BPSO, modified BPSO with chaotic inertia weight (CBPSO), binary moth flame optimization algorithm, binary dragonfly algorithm, binary whale optimization algorithm, SCA, and binary artificial bee colony algorithm. The conducted analysis demonstrates that the proposed method HBPSOSCA attain better performance in comparison to the competitive methods in most of the cases.
Keywords
Binary artificial bee colony algorithm Binary particle swarm optimization Binary dragonfly algorithm Binary moth flame optimization Binary whale optimization algorithm Clustering indices Feature selection Sine cosine algorithm1 Introduction
The high dimensionality of the feature space is a major concern in today’s day. Usually, there are so many irrelevant and redundant features in the datasets. These features not only increase computational complexity but also deteriorate performance of the underlying algorithms. Therefore, feature selection is necessary to improve the clustering performance, especially for data sets having very large dimensions. With respect to different selection strategies, feature selection methods are broadly categorized into two categories: Filter Methods and wrapper Methods.
The selection of an informative subset of features can be considered as a global combinatorial optimization problem in which the optimum features subset is selected from a high dimensional feature space. Nature inspired algorithms (NIA) gain attention for optimization problem (Agarwal and Mehta 2014). NIA is a mathematical formulation of the living beings present in the environment. Researchers have explored NIA such as genetic algorithm (GA) (Yang and Honavar 1998), particle swarm optimization (PSO) (Xue et al. 2013; Yang 2014), ant colony optimization (ACO) (Yang 2014; Blum 2005; Ali et al. 2017; Ahmed 2005), simulated annealing (SA) (Yang 2014), differential evolution (DE) (Yang 2014; Ali et al. 2017), and bacterial foraging optimization (BFO) (Chen et al. 2017) for feature selection problem. As they consider interaction of the learning algorithm for feature selection, they come under the wrapper method.
Originally, PSO is proposed for continuous problem (Kennedy 1995). Later, it is extended to solve discrete problem (Kennedy and Eberhart 1997). The discrete version of the PSO is named as binary particle swarm optimization. The BPSO is used to solve a wide variety of problems including feature selection (Cervante et al. 2012), cryptography algorithms (Jadon et al. 2011), optimum switching law of inverter (Wu et al. 2010), and classification (Cervantes et al. 2005). Several algorithms have also been developed to improve the performance of BPSO that includes modified BPSO which adopts concepts of the genotype–phenotype representation and the mutation operator of genetic algorithms (Lee et al. 2008), mutationbased binary particle swarm optimization (MBPSO) for multiple sequence alignment solving (Long et al. 2009), improved binary particle swarm optimization to select the small subset of informative genes (Mohamad et al. 2011), densitybased particle swarm optimization algorithm for data clustering (Alswaitti et al. 2018), Particle Swarm Clustering Fitness Evaluation with Computational Centroids (Raitoharju et al. 2017), hybrid binary version of bat and enhanced particle swarm optimization algorithm to solve feature selection problems (Tawhid and Dsouza 2018), hybrid improved BPSO and cuckoo search for review spam detection (Rajamohana and Umamaheswari 2017), and hybrid PSO with grey wolf optimizer (HPSOGWO) (Singh and Singh 2017). In any case, original BPSO (Kennedy 1995) effectively stick into neighborhood optima because of single directional data sharing system by the global best particle in the swarm. Chuang et al. (2008), enhance BPSO by embedding two sorts of chaotic maps, logistic maps and tent maps to evacuate superfluous features and select an informative subset of features. Here, they use a chaotic map to update the value of inertia weight over the number of iterations. This step helps the algorithm to avoid stagnation of the solution at a local optimum solution. Selection of an informative subset of features subset from high dimensional feature space and improvement in the searching capability of the existing NIA is still a great challenge in the area of optimization.
Mirjalili (2016a) was offered sine cosine algorithm (SCA) which is a novel population based optimization technique simply based on Sine and Cosine function. SCA applied for exploitation and exploration phases in global optimization functions. The sine cosine algorithm (SCA) generates different initial random agent solutions using a mathematical model based on sine and cosine functions and requires them to fluctuate outwards or towards the best possible solution.
Several new modified and hybrid variants of SCA algorithm are developed after motivated of this metaheuristics by the researchers of different areas to improve the convergence performance of SCA algorithm including SCA integrated with differential evolution (Bureerat and Pholdee 2017), Improved SCA based on levy flight (Li N and Deng ZL 2017), Hybrid SCA with multiorthogonal search strategy (RizkAllah 2017), and hybrid back tracking search with sine cosine algorithm (SCA) (Turgut 2017). The researchers are solved numerous real life problems with the help of SCA algorithm including a novel sine cosine algorithm for the feature selection (Hafez et al. 2016), solution of unit commitment problems (Kaur and Prashar 2016), the gear train design problem (RizkAllah 2017), Welded beam design (RizkAllah 2017), Pressure vessel design problem (RizkAllah 2017), Structural Damage Detection (Bureerat and Pholdee 2017), and many other biomedical and mechanical engineering problems.
In this study, we explore capability of NIA for feature selection problem by introducing a hybrid NIA with the combination of BPSO and SCA named as HBPSOSCA. The proposed model HBPSOSCA integrates the exploration capability of the SCA and exploitation capability of the PSO to select an informative subset of features. Here, the Vshaped transfer function is used to convert continuous swarm intelligence technique to binary search space. Next, the Kmeans algorithm is used to create clusters of data points. The Silhouette Index (SI), Dunn Index (DI) and Davies–Bouldin Index (DBI) are used for cluster assessment. Seven reallife scientific datasets are taken from the UCI machine learning archive and gene expression model selector (GEMS) to test effectiveness of the proposed method compared to other competitive methods. The comparative analysis is performed in terms of (1) number of selected feature subsets and, (2) clustering accuracy measured in terms of SI, DI, and DBI. The comparative analysis of the HBPSOSCA method is compared with state of the arts algorithm BPSO, modified BPSO with chaotic inertia weight (CBPSO), binary moth flame optimization algorithm (BMFOA), binary dragonfly algorithm (BDA), binary whale optimization algorithm (BWOA), sine cosine algorithm (SCA), and binary artificial bee colony algorithm (BinaryABC).
The rest of the paper is organized as follows: Sect. 2 describes the foundation of algorithms used as a part of this paper. The detailed descriptions of proposed approach are given in Sect. 3. The experimental results are presented in Sect. 4 while the conclusions and future are presented in Sect. 5.
2 Algorithms background
This section provides a background concerning the optimization algorithms.
2.1 Binary particle swarm optimization (BPSO)
2.2 Binary moth flame optimization algorithm (BMFOA)
According to Mirjalili (2015), it can be determined that local optima is high in MFO since MFO utilizes a population of moths to perform optimization. Decreasing the quantity of flames adjusts exploration and exploitation of the search space. The convergence of the MFO algorithm is ensured on the grounds that the moths always tend to update their positions with respect to flames.
2.3 Binary dragonfly algorithm (BDFA)
 Separation states to avoid the static collision from another sub group of dragonflies. Mathematically, this behaviour is defined by:where \(pos\) represents current location of an individual, \(pos_{j}\) indicates \(j{th}\) neighbouring individual’s location, and \(N\) is total quantity of neighbouring individuals.$$U_{i} =  \mathop \sum \limits_{j = 1}^{N} pos  pos_{j}$$(12)
 Alignment demonstrates matching of an individual’s velocity to other neighbourhood individuals. Mathematically, this behaviour is simulated by:where \(vel_{j}\) indicates velocity of \(j{th}\) neighbouring individual.$$O_{i} = \frac{{\mathop \sum \nolimits_{j = 1}^{N} vel_{j} }}{N}$$(13)
 The property of an individual to incline towards neighbouring mass center is known as Cohesion. It is computed by:where \(N\) is total number of neighbouring individuals, \(pos\) denotes location of current individual, and \(pos_{j}\) indicates location of \(j{th}\) neighbouring dragonfly.$$Q_{i} = \frac{{\mathop \sum \nolimits_{j = 1}^{N} pos_{j} }}{N}  pos$$(14)
 Attraction behaviour of dragonfly in the direction of food source is computed as:where \(pos_{ + }\) represents position of food source, and \(pos\) represents position of current individual.$$V_{i} = pos_{ + }  pos$$(15)
 Diversion behaviour from opponent is find as:where \(pos_{  }\) represents location of enemy, and \(pos\) represents location of current individual.$$Z_{i} = pos_{  } + pos$$(16)
According to Mirjalili and Lewis (2013) and Saremi et al. (2015), the algorithm was wellappointed with five parameters to control cohesion, alignment, separation, attraction (towards food sources), and diversion (outwards enemies) of individuals in the swarm. The convergence of the artificial dragonflies towards optimal solutions in continuous and binary search spaces was also observed and confirmed, which are because of the dynamic swarming design behaviour.
2.4 Binary whale optimization algorithm (BWOA)
Whale Optimization Algorithm (WOA) (Mirjalili and Lewis 2016) is introduced in 2016 by Mirjalili and Lewis. Whales are an elegant living being. Humpback whales (one of the largest whale) chase swarms of krill or little fishes near the surface. The foraging is performed by creating distinctive bubbles along a circle. However, Goldbogen et al. (2013) examine this conduct using label sensors. WOA has two stages: the first stage is scanning arbitrarily for prey which is exploration stage and the second stage is enclosing a prey and spiral bubblenet attacking method which is an exploitation phase.
2.5 Sine cosine algorithm (SCA)
SCA is proposed in 2016 by Mirjalili (2016a) and Meshkat and Parhizgar (2017). It is based on the functions of Sine and Cosine for exploration and exploitation phases, respectively. SCA starts with random solutions to swing near or away the best possible solution using mathematical expressions defined in Eq. 32. The cyclic condition of sine and cosine enables a solution to be rearranged around other solution, which facilitates exploitation. In the exploitation phase, however, there are gradual changes in the random solutions, and random variations are considerably less than those in the exploration phase.
The parameter \(r_{3}\) demonstrates the direction of movement which could be either in the space between the solution and goal or outside it. The parameter \(r_{4}\) is a random number in [0, 2π] which exhibits how far the development ought to be towards or outwards the objective and parameter \(r_{5}\) shows an unpredictable weight for the objective to emphasize (\(r_{5}\) > 1) or deemphasize (\(r_{5}\) < 1) the irregular effect of objective in describing the partition. Finally, the parameter \(r_{6}\) switches between the sine and cosine segments in Eq. 32.
According to Mirjalili (2016a) and Meshkat and Parhizgar (2017), it can be concluded that the mathematical model of position update equation update the solutions outwards or towards the goal point to ensure exploration and exploitation of the search space, respectively. SCA can be a very appropriate option compared to the current algorithms for solving different optimization problems.
3 Proposed method
In this paper, we explore capability of NIA for feature selection problem. Traditionally, realworld datasets include a large number of features. However, some of these features are noninformative, redundant and noisy. These features not only increase computational complexity of the underlying algorithm but deteriorate its performance. In literature different natureinspired algorithms are proposed for feature selection task (Diao and Shen 2015; Kumar 2018). In this paper, a new hybrid method HBPSOSCA is proposed using BPSO and SCA to select an informative subset of features. Here, the Vshaped transfer function is used to convert continuous nature of an algorithm to binary. The movement of a particle in the BPSO algorithm is improved using the sine–cosine algorithm. A detailed description of each of the step is presented in the subsequent sections.
3.1 Initialization of swarm
3.2 Clustering algorithm
Clustering is a process of grouping data points into clusters based on defined criteria. Traditionally, clustering methods are categorized into two categories; hierarchical clustering (Xu and Tian 2015), partitional clustering (Xu and Tian 2015). Hierarchical clustering creates clusters either top to bottom, known as divisive clustering or bottom to top known as agglomerative clustering. Initially, a divisive method considers each data point as a part of one cluster and recursively divides the clusters based on defined criteria. On the other hand, the agglomerative clustering method considers each data point as individual clusters and recursively merge the cluster based on defined criteria. On the contrary, partitional clustering creates clusters of the data point at one level.
In this paper, we use the Kmean partitional clustering method. Kmeans clustering (Xu and Tian 2015; Jain and Dubes 1988; Prakash and Singh 2012) is one of the most commonly used partitional clustering techniques. Here, K is the number of clusters. Kmeans works iteratively by assigning each data point to one of the K clusters based on some predefined measure. Traditionally, the Euclidian distance measure is used for this purpose. The Euclidian distance is calculated between the cluster center and data point. The data point is assigned to the cluster from which data point has a minimum distance. Next, each data point is assigned to the cluster to which it has smaller Euclidian distance from the cluster center. Assignment of data points to the respective clusters and refinement of cluster centers based on the average of the points assigned to the respective clusters are repeated until the termination criterion is met.
3.3 Evaluation criteria/fitness function
In this subsection, a detailed description of Silhouette Index is presented which is used to evaluate quality of potential solution. The silhouette index is proposed by Kaufman and Rousseeuw (2009) and Desgraupes (2013), which show the similarity of a data point to its own cluster as compared to its similarity with other clusters.
The values of Silhouette index lies in the range [− 1, 1]. Here, high value shows that the data point well matched with own and compared to other clusters.
3.4 Proposed algorithm
In this paper, an amalgamation of the BPSO with SCA helps the algorithm to expand its searching capability and locate the near global optimum solution. In SCA, if functions of sine and cosine generate a value that is higher than 1 or smaller than − 1, then it signifies the exploration of different areas within the search space. Likewise, if sine and cosine functions generate a value within the range between − 1 and 1 then it demonstrates that proficient areas of search space are exploited. The SCA algorithm effectively travels from exploration to exploitation utilizing versatile range in the sine and cosine functions. In the hybrid approach, the movement of a particle in the BPSO is improved using the sine–cosine algorithm.
Here, the inertia weight is increased linearly in every iteration and regulates the velocity of all particles. Here, \(w_{min}\) is the initial value and \(w_{max}\) is final value of the inertia weight, \(t\) represents current iteration, \(T\) is the total number of iterations, and \(\xi\) is used to control the fraction of iterations when \(w\) increases linearly from \(w_{min}\) to \(w_{max}\). Here, \(\xi = 0.9\) is used to achieve stronger exploration in initial iterations and high exploitation in later iterations (Jain et al. 2017).
4 Experimental results and discussions
4.1 Benchmark functions
Experiments are carried out over a set of 10 widely used benchmark test functions listed in Appendix A. These functions are taken from (Yao et al. 1999) each with different characteristics. Among these benchmarks, F1 to F3 are unimodal functions, F4 is the Rosenbrock function which is a unimodal function for D = 2 and multimodal function for D > 3 (Shang and Qiu 2006), F5 is a step function, F6 is a noisy quartic function, F7 to F10 are multimodal functions.
4.2 Datasets and parameters setup
Datasets description
Dataset  Number of cluster  Number of features  Number of samples 

Ionosphere  2  34  351 
Breast Cancer Wisconsin (BCW)  2  9  699 
Connectionist Bench (Sonar, Mines vs. Rocks)  2  60  208 
Statlog (Vehicle Silhoettes)  4  18  946 
Parkinson  2  23  195 
9_Tumors  9  5726  60 
Leukemia2  3  11,225  72 

Ionosphere: This dataset is drawn from the Johns Hopkins University. It comprises of 34 number of features, and 351 number of occurrences belonging to two distinct categories either good or bad. Out of 351 instances, 225 are good and rest 126 are bad occasions.

Breast Cancer Wisconsin: It is acquired from the University of Wisconsin Hospitals Madison, Wisconsin, USA. It consists of 9 features, and each one has ten different values. There is 699 number of instances in which 458 instances belong to the benign class, and the remaining 241 instances belong to the malignant class.

Connectionist Bench: This dataset is developed in AlliedSignal Aerospace Technology Center. There are 60 number of features in this dataset and 208 number of instances which belongs to two different classes (rock and mines). In this, 97 instances belong to rock, and remaining instances belong to mines.

Statlog: It has 18 features and 946 instances which is divided into four classes. The 226 instances belong to van, 240 instances belong to saab, 240 instances belong to bus, and the remaining 240 instances belong to Opel.

Parkinson: It is acquired from the Max Little of the University of Oxford. It has 23 features and 195 instances. The 147 instances belong to Parkinson category, and the remaining 48 belongs healthy.

9_Tumors: The dataset comes from a study of 9 human tumor types: NSCLC, colon, breast, ovary, leukemia, renal, melanoma, prostate, and CNS. There are 60 samples, each of which contains 5726 genes.

Leukemia2: The dataset has 72 samples, each of which contains 11,225 features. These samples categorized into AML, ALL, and mixedlineage leukemia (MLL).
4.3 Parameter setting
Parameter settings
Algorithm  Parameters  Value 

BPSO  Population size (swarm size)  50 
e_{1} and e_{2}  1.5  
[vel_{min}, vel_{max}]  [− 6, 6]  
T (number of iterations)  150  
ChaoticBPSO  Population size (swarm size)  50 
T (number of iterations)  150  
e_{1} and e_{2}  1.5  
Initial inertia weight  0.48  
BMFO  No. of search agents (moths)  50 
b (spiral’s shape)  1.0  
T (number of iterations)  150  
BDFA  Population  50 
T (number of iterations)  150  
\(w\)  0.9 to 0.2  
BWOA  Population  50 
T (number of iterations)  150  
b (spiral’s shape)  1.0  
SCA  Population  50 
T (number of iterations)  150  
A  2  
Binary ABC  Population  50 
T (number of iterations)  150  
HBPSOSCA  Population size (swarm size)  50 
e_{1} and e_{2}  1.5  
w _{ max}  0.9  
w _{ min}  0.4  
\(T\) (number of iterations)  150  
max_count  10  
\(\xi\)  0.9  
a  2 
4.4 Dunn Index and Davies–Bouldin Index
Silhouette Index (SI), Dunn Index (DI) and Davies–Bouldin Index (DBI) are used to evaluate the quality of created clusters. The higher value of SI (refer Sect. 3.3) and DI, and a lower value of DBI represents better quality of cluster.
4.5 Simulation results on benchmark functions
The comparative analysis of BPSO, CBPSO, BMFO, BDFA, BWOA, SCA, BinaryABC, and HBPSOSCA on Benchmark Functions F1–F10
Function  BPSO  CBPSO  BMFO  BDFA  BWOA  SCA  BinaryABC  HBPSOSCA 

F1  
Mean  9.5411  10.1450  23.8187  13.6173  5.2400  2.4120  22.0660  2.1473 
SD  0.4096  0.3703  0.4758  2.4929  1.1288  0.1705  1.1375  0.2203 
F2  
Mean  9.2974  10.2464  23.9813  11.2940  6.4200  2.2407  21.6920  2.1341 
SD  0.4253  0.3157  0.3652  2.9564  0.8354  0.1741  0.8724  0.3216 
F3  
Mean  160.9358  180.8801  483.4647  269.79  155.3867  37.2620  438.0480  33.1313 
SD  6.5366  8.7939  8.5697  38.0267  29.6846  3.3939  20.0694  7.9122 
F4  
Mean  1.1141 \(e^{ + 03}\)  1.1739 \(e^{ + 03}\)  1.7128 \(e^{ + 03}\)  1.4531 \(e^{ + 03}\)  217.1800  416.9060  1.9848 \(e^{ + 03}\)  156.3193 
SD  29.9715  30.3347  42.2886  192.1704  0  41.2802  83.0103  26.6875 
F5  
Mean  9.4483  10.0278  23.9440  12.3187  8.1467  2.3607  22.2707  1.9175 
SD  0.3503  0.3381  0.5597  2.8085  1.8724 \(e^{  15}\)  0.1384  0.9200  0.2485 
F6  
Mean  161.7552  182.3258  481.7741  41.2212  118.9141  36.0881  433.6432  6.1027 
SD  11.3762  5.5510  7.3281  113.1499  1.1579  3.1750  25.4729  1.5730 
F7  
Mean  1.6734 \(e^{ + 04}\)  1.6734 \(e^{ + 04}\)  1.6736 \(e^{ + 04}\)  1.6738 \(e^{ + 04}\)  1.6727 \(\varvec{e}^{ + 04}\)  1.6736 \(e^{ + 04}\)  1.6747 \(e^{ + 04}\)  1.6733 \(e^{ + 04}\) 
SD  0.3419  0.2780  0.3743  1.5369  22.4397  0.7677  0.4836  0.2019 
F8  
Mean  9.4702  10.1709  24.0533  13.2473  7.1200  2.3547  21.8020  2.1327 
SD  0.4195  0.3226  0.4479  2.8708  0.1232  0.1213  0.7693  0.2340 
F9  
Mean  1.8526  1.8979  2.8684  2.1322  1.7424  0.5544  2.7508  0.1391 
SD  0.0455  0.0380  0.0153  0.1501  1.0314  0.0261  0.0948  0.0304 
F10  
Mean  0.2661  0.2727  0.7199  0.2547  0.1668  0.0596  0.5834  0.0506 
SD  0.0134  0.0089  0.0123  0.0702  0.2440  0.0043  0.0260  0.0085 
4.6 Simulation results on reallife datasets
The results recorded with the BPSO, CBPSO, BMFO, BDFA, BWOA, SCA, Binary ABC, and HBPSOSCA with 10 independent runs on seven reallife benchmark datasets in term of SI
Algorithms  Dataset  Average length of selected features  Average SI  Best SI  Worst SI  Standard deviation 

BPSO  Ionosphere  4.0333  0.6102  0.6132  0.6067  0.0020 
BCW  2.0000  0.8581  0.8669  0.8464  0.0073  
CB  18.1722  0.5697  0.5832  0.5626  0.0069  
Vehicle  8.1325  0.8111  08147  0.8059  0.0026  
Parkinson  2.6093  0.9441  0.9590  0.9354  0.0089  
9_Tumors  2.8890e^{+03}  0.2143  0.2196  0.2116  0.0028  
Leukemia2  5.6196e^{+03}  0.6256  0.6596  0.5979  0.0165  
Chaotic BPSO  Ionosphere  7.5667  0.5856  0.5912  0.5828  0.0026 
BCW  2.0000  0.8749  0.8760  0.8737  9.9195e^{−04}  
CB  21.7351  0.5338  0.5432  0.5183  0.0072  
Vehicle  6.6821  0.8082  0.8108  0.8041  0.0022  
Parkinson  3.3113  0.9459  0.9459  0.9459  1.0937e^{−05}  
9_Tumors  2.8666e^{+03}  0.3670  0.3762  0.3588  0.0056  
Leukemia2  5.5675e^{+03}  0.8591  0.8596  0.8584  4.3881e^{−04}  
BMFO  Ionosphere  21.8267  0.4075  0.3637  0.3536  0.0257 
BCW  5.3800  0.7579  0.7601  0.7532  0.0021  
CB  38.4867  0.3579  0.3637  0.3536  0.0234  
Vehicle  11.4400  0.6109  0.4156  0.4020  0.0296  
Parkinson  13.0800  0.7823  0.7941  0.7715  0.0087  
9_Tumors  3.6465e^{+03}  0.1415  0.1466  0.1376  0.0030  
Leukemia2  7.2229e^{+03}  0.3205  0.3280  0.3130  0.0051  
BDFA  Ionosphere  17.7733  0.2064  0.2649  0.1964  0.0263 
BCW  2.3000  0.5871  0.6202  0.5697  0.0118  
CB  28.0933  0.2405  0.2649  0.1964  0.0123  
Vehicle  6.6067  0.2463  0.2277  0.1817  0.0286  
Parkinson  11.3400  0.5073  0.5161  0.4998  0.0055  
9_Tumors  2833  0.0944  0.1102  0.0717  0.0101  
Leukemia2  5604  0.0345  0.2833  − 0.0624  0.1030  
BWOA  Ionosphere  28.1800  0.5783  0.6099  0.5097  0.0213 
BCW  2.0267  0.8701  0.8769  0.8615  0.0058  
CB  22.2800  0.5364  0.6099  0.5097  0  
Vehicle  8.3133  0.8372  0.6086  0.5605  0.0080  
Parkinson  5.8067  0.9349  0.9459  0.9159  0.0129  
9_Tumors  2.9149e^{+03}  0.2213  0.2213  0.2213  5.8514e^{−17}  
Leukemia2  6.0829e^{+03}  0.8029  0.8029  0.8029  0  
SCA  Ionosphere  4.2067  0.8378  0.8591  0.7990  0.0213 
BCW  2.0067  0.8409  0.8782  0.8175  0.0232  
CB  9.3733  0.7825  0.7980  0.7505  0.0140  
Vehicle  4.0267  0.8273  0.8659  0.8207  0.0136  
Parkinson  3.0000  0.9543  0.9572  0.9523  0.0020  
9_Tumors  1.6526e^{+03}  0.2665  0.2859  0.2487  0.0106  
Leukemia2  3.6262e^{+03}  0.7599  0.8152  0.6209  0.0807  
Binary ABC  Ionosphere  17.5733  0.4073  0.4221  0.3857  0.0118 
BCW  4.9000  0.6677  0.7471  0.5011  0.0701  
CB  30.8267  0.3920  0.4017  0.3816  0.0079  
Vehicle  7.6533  0.6292  0.6606  0.5989  0.0189  
Parkinson  15.9267  0.8698  0.9209  0.8280  0.0278  
9_Tumors  2.8651e^{+03}  0.0603  0.0654  0.0528  0.0043  
Leukemia2  5.6533e^{+03}  0.2461  0.3354  0.1724  0.0670  
HBPSOSCA  Ionosphere  2.2400  0.9270  0.9366  0.8809  0 
BCW  2.0000  0.8786  0.8800  0.8771  0.0010  
CB  2.5629  0.9041  0.9366  0.8809  0.0237  
Vehicle  3.5629  0.8461  0.9751  0.8303  0.0013  
Parkinson  2.1325  0.9613  0.9647  0.9589  0.0020  
9_Tumors  188.6644  0.8006  0.8556  0.7421  0.0330  
Leukemia2  932.6980  0.9073  0.9781  0.8237  0.0486 
The results recorded with the BPSO, CBPSO, BMFO, BDFA, BWOA, SCA, Binary ABC, HBPSOSCA with 10 independent runs on seven real life benchmark datasets in term of DI
Algorithms  Dataset  Average length of selected features  Average DI  Best DI  Worst DI  Standard deviation 

BPSO  Ionosphere  25.0397  0.1379  0.3105  0.1119  0.0162 
BCW  6.0132  0.1770  0.1770  0.1770  0.0376  
CB  30.8940  0.3480  0.3703  0.3272  0.0689  
Vehicle  10.8609  0.5012  0.8443  0.2011  0.2175  
Parkinson  19.5033  0.3409  0.3409  0.3409  0.1472  
9_Tumors  2.8893e^{+03}  0.5510  0.5728  0.5425  0.0108  
Leukemia2  6.1238e^{+03}  0.7044  0.8235  0.6171  0.0632  
Chaotic BPSO  Ionosphere  18.6689  0.0975  0.0993  0.0949  0.0015 
BCW  6.9404  0.1741  0.1746  0.1735  3.6457e^{−04}  
CB  31.4437  0.2790  0.2838  0.2695  0.0040  
Vehicle  9.6490  0.2950  0.3184  0.2751  0.0127  
Parkinson  2.5430  0.3513  0.3513  0.3513  5.2872e^{−08}  
9_Tumors  2.8133e^{+03}  0.4950  0.5036  0.4824  0.0065  
Leukemia2  5.5576e^{+03}  1.5373  1.5427  1.5317  0.0030  
BMFO  Ionosphere  23.0067  0.0939  0.0980  0.0910  0.0026 
BCW  5.7867  0.1729  0.1770  0.1628  0.0053  
CB  38.8200  0.2662  0.2900  0.2450  0.0092  
Vehicle  11.1933  0.2058  0.2442  0.1713  0.4827  
Parkinson  14.8067  0.3409  0.3409  0.3409  1.0790 e^{−06}  
9_Tumors  3.6822e^{+03}  0.4084  0.4159  0.4037  0.0034  
Leukemia2  7.1904e^{+03}  0.4107  0.4287  0.3973  0.0085  
BDFA  Ionosphere  21.2733  0.0296  0.0357  0.0239  0.0068 
BCW  4.2000  0.0516  0.0570  0.0486  0.0032  
CB  26.0533  0.0607  0.0703  0.0467  0.0074  
Vehicle  2.9600  0.0334  0.0405  0.0244  0.0048  
Parkinson  14.8133  0.0108  0.0130  0.0081  0.0016  
9_Tumors  2834  0.3221  0.3400  0.3026  0.0136  
Leukemia2  5662  0.2215  0.3412  0.1872  0.0462  
BWOA  Ionosphere  27.6133  0.1046  0.1046  0.1046  1.4628 e^{−17} 
BCW  6.0867  0.1770  0.1770  0.1770  2.9257 e^{−17}  
CB  32.2600  0.2985  0.3071  0.2876  5.5814 e^{−17}  
Vehicle  12.2200  0.2647  0.2786  0.2483  5.8514 e^{−17}  
Parkinson  19.9667  0.3409  0.3409  0.3409  0  
9_Tumors  3.2932e^{+03}  0.5400  0.5400  0.5400  0  
Leukemia2  4.0762e^{+03}  0.7654  0.7654  0.7654  1.1703e^{−16}  
SCA  Ionosphere  13.2400  0.1040  0.1055  0.1004  0.0016 
BCW  6.2867  0.1692  0.1739  0.1684  0.0017  
CB  21.3067  0.2917  0.3035  0.2801  0.0065  
Vehicle  7.8133  0.2284  0.2496  0.2184  0.0091  
Parkinson  13.2933  0.3409  0.3409  0.3409  3.5692 e^{−07}  
9_Tumors  2.2772e^{+03}  0.5143  0.5258  0.5051  0.0068  
Leukemia2  3.5778e^{+03}  0.7537  0.7890  0.7104  0.0260  
Binary ABC  Ionosphere  20.9667  0.0648  0.0709  0.0614  0.0028 
BCW  4.1533  0.0753  0.0805  0.0709  0.0031  
CB  29.0800  0.1320  0.1611  0.1150  0.0139  
Vehicle  13.4200  0.0727  0.0896  0.0529  0.0107  
Parkinson  4.3600  0.2322  0.3215  0.0977  0.0836  
9_Tumors  2.9178e^{+03}  0.3663  0.3732  0.3596  0.0053  
Leukemia2  5.5821e^{+03}  0.3111  0.3357  0.2839  0.0186  
HBPSOSCA  Ionosphere  12.2450  0.1436  0.2131  0.1089  0.0309 
BCW  5.7067  0.1770  0.1770  0.1770  2.9257 e^{−17}  
CB  11.7947  0.5342  0.6705  0.3158  0.0922  
Vehicle  2.2318  1.9029  3.0612  0.3531  0.9707  
Parkinson  14.2367  0.3409  0.3409  0.3409  5.6413 e^{−10}  
9_Tumors  2.2312e^{+03}  0.6019  0.6878  0.5061  0.0863  
Leukemia2  6.0090e^{+03}  0.7923  0.8866  0.6828  0.0835 
The results recorded with BPSO, CBPSO, BMFO, BDFA, BWOA, SCA, and Binary ABC with 10 independent runs on seven reallife benchmark datasets in term of DBI
Algorithms  Dataset  Average length of selected features  Average DBI  Best DBI  Worst DBI  Standard deviation 

BPSO  Ionosphere  13.7550  0.4076  0.2960  0.4790  0.4843 
BCW  2.0000  0.4046  0.4046  0.4046  0.1357  
CB  8.2914  0.3612  0.2575  0.3996  0.5335  
Vehicle  9.2318  0.2689  0.2584  0.3582  0.2311  
Parkinson  2.6358  0.3513  0.3513  0.3513  0.1865  
9_Tumors  2.8437e^{+03}  1.2174  1.3379  1.1403  0.0639  
Leukemia2  5.5800e^{+03}  0.4942  0.4852  0.5044  0.0066  
Chaotic BPSO  Ionosphere  8.2980  0.8413  0.6997  0.8981  0.0627 
BCW  1.9868  0.4057  0.4046  0.4067  6.8361e^{−04}  
CB  21.8013  0.9725  0.9725  0.9725  0  
Vehicle  9.6490  0.2950  0.3184  0.2751  0.0127  
Parkinson  2.5430  0.3513  0.3513  0.3513  5.2872e^{−08}  
9_Tumors  2.9017e^{+03}  1.0080  0.4935  1.7777  0.4577  
Leukemia2  5.5783e^{+03}  0.4946  0.4918  0.4971  0.0017  
BMFO  Ionosphere  21.4267  0.2967  0.2521  0.3237  0.0368 
BCW  5.1933  0.3134  0.2881  0.3237  0.0486  
CB  38.0200  1.3745  1.3131  1.4308  0.0361  
Vehicle  11.2867  0.4014  0.3970  0.4094  0.0044  
Parkinson  13.6400  0.3513  0.3513  0.3513  2.8570e^{−06}  
9_Tumors  3.6878e^{+03}  1.9514  1.9111  1.9819  0.0222  
Leukemia2  7.2183e^{+03}  1.5994  1.5648  1.6283  0.0163  
BDFA  Ionosphere  16.8867  22.375  14.742  36.383  7.2495 
BCW  2.0067  0.4046  0.4046  0.4046  0  
CB  20.5400  0.4648  0.4001  0.5826  0.0732  
Vehicle  8.8667  0.2918  0.2637  0.3291  0.0212  
Parkinson  5.0467  0.3513  0.3513  0.3513  2.5809e^{−10}  
9_Tumors  2.8659e^{+03}  2.4811  2.3381  2.5535  0.0551  
Leukemia2  6.1308e^{+03}  4.8836  4.7844  5.0287  0.0630  
BWOA  Ionosphere  16.0467  1.9234  1.7631  2.1573  2.3460e^{−16} 
BCW  4.0000  0.8782  0.8192  0.9075  1.1703e^{−16}  
CB  37.6600  1.9696  1.8445  2.1628  0  
Vehicle  9.0867  1.2479  1.1077  1.4317  2.3406e^{−16}  
Parkinson  12.8533  0.9786  0.9171  0.9947  0  
9_Tumors  4.2436e^{+03}  2.5855  2.5059  2.7404  0.0690  
Leukemia2  8.3635e^{+03}  2.9038  2.7576  3.2958  0.1813  
SCA  Ionosphere  4.4400  0.4178  0.3574  0.4780  0.0478 
BCW  2.0000  0.4046  0.4046  0.4083  0.0012  
CB  8.9133  1.3552  1.3107  1.4451  0.0437  
Vehicle  7.1258  0.3659  0.3627  0.3707  0.0022  
Parkinson  2.9333  0.3513  0.3513  0.3513  3.4508e^{−06}  
9_Tumors  1.5022e^{+03}  1.3520  1.2649  1.4017  0.0468  
Leukemia2  3.6995e^{+03}  0.5701  0.5166  0.6938  0.0633  
Binary ABC  Ionosphere  17.1133  0.3694  0.2311  0.4223  0.0691 
BCW  2.0000  0.4032  0.3692  0.4094  0.0120  
CB  4.5800  0.5338  0.4805  0.5850  0.0366  
Vehicle  2.8133  0.2755  0.2636  0.2982  0.0107  
Parkinson  2.3533  0.3513  0.3513  0.3513  2.6403e^{−07}  
9_Tumors  2.8847e^{+03}  1.2358  1.1981  1.3040  0.0334  
Leukemia2  5.6059e^{+03}  0.5365  0.5320  0.5405  0.0030  
HBPSOSCA  Ionosphere  3.8800  0.2960  0.2960  0.2960  0.0894 
BCW  2.0000  0.3046  0.3046  0.3046  0  
CB  1.1000  0.2614  0.2575  0.2922  0.0466  
Vehicle  6.5828  0.2596  0.2584  0.2629  0.0157  
Parkinson  2.0000  0.3513  0.3513  0.3513  5.8514e^{−17}  
9_Tumors  21.3067  0.3425  0.2717  0.3929  0.0403  
Leukemia2  92.4000  0.3243  0.2160  0.4293  0.0716 
Wilcoxon test for ionosphere dataset
Optimizers  Wilcoxon test value  Comment 

HBPSOSCA—BPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—CBPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—BMFO  1.8267 e^{−04}  Significant 
HBPSOSCA—BDFA  1.8267 e^{−04}  Significant 
HBPSOSCA—BWOA  1.8267 e^{−04}  Significant 
HBPSOSCA—SCA  0.0013  Significant 
HBPSOSCA—BinaryABC  1.8267 e^{−04}  Significant 
Wilcoxon test for BCW dataset
Optimizers  Wilcoxon test value  Comment 

HBPSOSCA—BPSO  1.8165 e^{−04}  Significant 
HBPSOSCA—CBPSO  1.8165 e^{−04}  Significant 
HBPSOSCA—BMFO  1.8165 e^{−04}  Significant 
HBPSOSCA—BDFA  1.8063 e^{−04}  Significant 
HBPSOSCA—BWOA  1.8165 e^{−04}  Significant 
HBPSOSCA—SCA  4.2880 e^{−04}  Significant 
HBPSOSCA—BinaryABC  1.8165 e^{−04}  Significant 
Wilcoxon test for CB dataset
Optimizers  Wilcoxon test value  Comment 

HBPSOSCA—BPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—CBPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—BMFO  1.8267 e^{−04}  Significant 
HBPSOSCA—BDFA  1.8267 e^{−04}  Significant 
HBPSOSCA—BWOA  1.8267 e^{−04}  Significant 
HBPSOSCA—SCA  1.8267 e^{−04}  Significant 
HBPSOSCA—BinaryABC  1.8267 e^{−04}  Significant 
Wilcoxon test for vehicle dataset
Optimizers  Wilcoxon test value  Comment 

HBPSOSCA—BPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—CBPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—BMFO  1.8267 e^{−04}  Significant 
HBPSOSCA—BDFA  1.8267 e^{−04}  Significant 
HBPSOSCA—BWOA  0.0539 e^{−04}  Insignificant 
HBPSOSCA—SCA  5.8284 e^{−04}  Significant 
HBPSOSCA—BinaryABC  1.8267 e^{−04}  Significant 
Wilcoxon test for Parkinson dataset
Optimizers  Wilcoxon test value  Comment 

HBPSOSCA—BPSO  2.4480 e^{−04}  Significant 
HBPSOSCA—CBPSO  8.7450 e^{−05}  Significant 
HBPSOSCA—BMFO  1.8267 e^{−04}  Significant 
HBPSOSCA—BDFA  1.8267 e^{−04}  Significant 
HBPSOSCA—BWOA  1.6118 e^{−04}  Significant 
HBPSOSCA—SCA  1.8267 e^{−04}  Significant 
HBPSOSCA—BinaryABC  1.8267 e^{−04}  Significant 
Wilcoxon test for 9_Tumors dataset
Optimizers  Wilcoxon test value  Comment 

HBPSOSCA—BPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—CBPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—BMFO  1.8267 e^{−04}  Significant 
HBPSOSCA—BDFA  1.8267 e^{−04}  Significant 
HBPSOSCA—BWOA  6.3864 e^{−05}  Significant 
HBPSOSCA—SCA  1.8267 e^{−04}  Significant 
HBPSOSCA—BinaryABC  1.8165 e^{−04}  Significant 
Wilcoxon test for Leukemia2 dataset
Optimizers  Wilcoxon test value  Comment 

HBPSOSCA—BPSO  1.8267 e^{−04}  Significant 
HBPSOSCA—CBPSO  0.0257  Significant 
HBPSOSCA—BMFO  1.8267 e^{−04}  Significant 
HBPSOSCA—BDFA  1.8267 e^{−04}  Significant 
HBPSOSCA—BWOA  6.3864 e^{−05}  Significant 
HBPSOSCA—SCA  1.8267 e^{−04}  Significant 
HBPSOSCA—BinaryABC  1.8267 e^{−04}  Significant 
5 Conclusions and future directions
A feature selection method is used to select an informative subset of features from high dimensional irrelevant, redundant, and noisy feature space. The irrelevant, redundant, and noisy feature not only increases computational complexity but also deteriorate performance of the underlying algorithms. In this paper, a natureinspired algorithm is used to select an informative subset of features from given feature space. Each algorithm has its own advantages and disadvantage. We introduce a new hybrid method to take advantage of one method and lessen the disadvantage of others for feature selection task. Here, we integrate the BPSO with the SCA, named as HBPSOSCA for this task. The integration of the two algorithms provides global search ability and local exploitation ability to the HBPSOSCA by improving the movement of a particle in the BPSO with the SCA. The proposed algorithm is tested on ten benchmark test functions and seven wellknown scientific datasets. Experimental results show that the proposed algorithm obtains the near global minimum compared to other competitive natureinspired algorithms for most of the test functions. Moreover, it achieves better clustering accuracy compare to the competitive methods for all reallife datasets. The Wilcoxon Test confirms that the results obtained by the HBPSOSCA are significantly better than the competitive methods.
In the future, we intend to combine other feature selection methods with the natureinspired algorithm to select an informative subset of features from high dimensional space without much increasing computational complexity of the algorithm. As the parameters’ value significantly affects performance of the underlying algorithm, we plan to develop a model to adaptively set the parameter’s values.
Notes
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