A novel hybrid BPSO–SCA approach for feature selection

J. Kennedy and R. Eberhart propose Particle swarm optimization (PSO) (Kennedy 1995; Kennedy and Eberhart 1997) in 1995. The PSO is a population-based stochastic approach for solving continuous and discrete optimization problems. The PSO optimization algorithm is inspired by the flocking and schooling patterns of birds and fish. Here, each particle is considered as a potential solution. Every particle is associated with a fitness value, which is used to assess worthiness of the solution compared to others. The particle own best position is named as particle best solution (pbest). The best-fitted solution of the entire swarm is named as a global best solution (gbest). The movement of particle is controlled by its own pbest position and the gbest position of the swarm. Equations 1 and 2 show the velocity and position update of the ith particle.where \(d\) is the dimension of particle, \(t\) is the iteration, \(r_{1}\) and \(r_{2}\) are random numbers in the interim (0, 1), and \(e_{1}\) and \(e_{2}\) are positive learning constants.

Originally, PSO is developed for continuous problem while numerous optimization problems exist which are discrete/binary in nature, for example, demand side management system (Agneessens et al. 2011), feature selection (Behjat et al. 2014), Knapsack Problems (Bansal and Deep 2012). Kennedy (1995) extend the concept of PSO and develop a binary version of PSO named as binary particle swarm optimization (BPSO) to solve discrete/binary problem. Here, a sigmoidal function \(\left( {Sig\left( {{\text{vel}}_{\text{i}}^{\text{d}} } \right)} \right)\) is used to convert position (refer Eq. 2) in binary search space (refer Eq. 4) for BPSO. The sigmoidal function is a mathematical function whose curve looks same as the shape of English letter ‘S’. The sigmoid function is a kind of function that is real-valued and differentiable. It is defined on all real inputs and returns a positive derivative at every point of it. Position and velocity of ith particle in case of BPSO algorithm are computed by:where \(w\) is inertia weight, \(rand\) is an arbitrary number chose in interim (0, 1), \(pos_{i}^{d}\) is location of \(i{th}\) particle at dimension \(d\) and \(\left( {Sig\left( {vel_{i}^{d} } \right)} \right)\) indicates sigmoidal function. The BPSO algorithm is good at exploitation, however, poor at exploration.

Seyedali Mirjalili proposes a new nature-inspired algorithm named as Moth Flame Optimization (MFO) (Mirjalili 2015) in 2015 which is inspired by the navigation behavior of moths known as transverse orientation, i.e., keeping a fixed angle on a distant source of light for orientation. It is a population-based swarm intelligence technique which simulates the behaviour of moths at night. Moths attract towards the moonlight or counterfeit lights made by people. Here, a moth flies by using as a settled point with respect to the moon which is an extremely powerful instrument for traveling long separations in a straight way. This system ensures that moths are flying in a straight line in the light. When moths see a man-made light, they maintain a similar angle with the light to fly in a straight line causes a helical route for moths because such man-made lights are very close contrasted with the moon. It might be watched that the moth unsurprisingly meets towards the light. Fundamental segments of MFO calculation are moths and flames. In the MFO algorithm, it is acknowledged that the candidate solutions are moths and the issue’s factors are the location of moths in the space. The moths are real hunt specialists that move around the search space. However, flames are in the best position of moths that gets up until this point. As specified over that this advancement calculation is propelled by exceptional route technique, i.e., transverse introduction. The position of every moth which is refreshed with respect to a flame is mathematically defined as follows:where \(Moth^{i}\) shows the \(i{th}\) moth, \(Flame^{j}\) demonstrates the \(j{th}\) flame, and \(SP\) is the spiral function. The logarithmic spiral function for the MFO algorithm is defined as follows:where \(Dis^{i}\) demonstrates separation of the \(i{th}\) moth for the \(j{th}\) flame (refer Eq. (8)), \(b\) is a constant for characterizing shape of the logarithmic spiral, and \(I\) is a random number in [− 1, 1].where \(Moth^{i}\) indicates the \(i{th}\) moth, \(Flame^{j}\) indicates \(j{th}\) flame and \(Dis^{i}\) indicates separation between \(Moth^{i}\) and \(Flame^{j}\).

The parameter I [refer Eq. (7)] decides route of moth around the flame (I = − 1 indicates nearest location to the flame, while I = 1 demonstrates the uttermost). Exploration and exploitation of the binary space can be ensured due to the spiral equation which allows a moth to fly around a flame. I is defined as a random number in [r, 1], where r linearly decreased from − 1 to − 2 over the course of iterations. According to Eq. (7), each moth is constrained to move towards a flame that may incite adjacent perfect stagnation. Flames must be arranged according to their fitness values. The moths by then invigorate their situations with respect to their relating flames. A versatile mechanism for the number of flames (\(f^{no}\)) has been proposed as in the following formula due to the degradation in exploitation of the best promising solutions:where \(t\) is the present number of iteration, \(N^{f}\) is the maximum number of flames, and \(T\) is the total number of iterations.

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.

In binary search space, the location of moths is confined to twofold factor, i.e., 0 or 1. Here, 1 shows the presence of corresponding feature and 0 demonstrates the absence of that feature. Sigmoidal function (Reddy et al. 2017) \(Sig\left( {Moth\left( {t + 1} \right)} \right)\) is used to change the location of each moth which is invigorated in regards to a flame (refer Eq. (6)) in continuous search space into the new position in binary search space (refer Eq. (10)) for BMFOA.where \(Moth\left( {t + 1} \right)\) is the location update of the moth for \(\left( {t + 1} \right){th}\) iteration.

In binary space, the position of every moth with respect to a flame is computed (in binary search space) by the given Eq.:

Dragonfly algorithm (DFA) is introduced in 2015 by Mirjalili (2016b) inspired from dragonflies (small hunters that eat almost all other small insects in nature). Dragonfly has two main swarming behaviours which are the main inspiration of DFA and these behaviours are: static and dynamic swarming behaviours. Exploration and exploitation are two important components of any NIA. In DFA, mathematically these behaviours are expressed by:

Overall artificial dragonflies consist of five behaviours which are described above. In a continuous space, two vectors are computed to update movement, and location of these dragonflies, i.e., step (∆pos) and position (pos). The step vector and the position vector are calculated as defined in Eqs. (17) and (18).where \(u\) is the division weight, \(U_{i}\) demonstrates partition of \(ith\) individual, \(o\) is an alignment weight, \(O_{i}\) is separation of \(ith\) individual, \(q\) is the cohesion weight, \(Q_{i}\) is the cohesion of \(ith\) individual, \(v\) is the sustenance factor, \(V_{i}\) is the food source of the \(ith\) individual, \(z\) is the adversary factor, \(Z_{i}\) is the location of enemy of \(ith\) individual, \(w\) is an inertia weight, and \(t\) is the iteration counter.where \(pos\) denotes the current location of an individual and \(t\) is the current iteration.

Positions of DA are updated by adding step vectors (similar to velocity vector of PSO) to the position vectors in continuous searching space. In binary space, location of BDFA (Mafarja et al. 2017) is represented in binary form, i.e., 1 or 0 only without using step vector. A transfer function is used according to Mirjalili and Lewis (2013) and Saremi et al. (2015), to convert continuous space to a binary search space without changing the structure of swarm intelligence algorithm. Transfer function accepts velocity (step) values as an input and returns a value in [0, 1], which determine likelihood of changing positions. The transfer function is given below:where \(r\) is a random number in the interval of [0, 1].

According to Mirjalili and Lewis (2013) and Saremi et al. (2015), the algorithm was well-appointed 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.

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 bubble-net attacking method which is an exploitation phase.

Movement of whale around a prey is depicted in the exploitation stage. Mathematical formulation of this step is given by:where t indicates current number of iteration, \(M\) shows distance between whale and prey, \(S^{*}\) demonstrates the best solutions acquired up until this point, and \(S\) shows the current solution. \(H\) and \(G\) represent the coefficient vectors which is defined by:where \(c\) gradually decreases from 2 to 0 and \(g\) is a random vector in [0, 1], \(t\) is the current iteration, and \(T\) is the maximum number of iterations.

The distance between the solution (\(S\)) and the best solution (\(S^{*}\)) causes the spiral shaped path which is computed by:where \(b\) is the helix’s form of the spiral, and \(I\) is an irregular number in [− 1, 1].

Shrinking encircling model or spiral model is simulated by Eq. 5:where \(rand\) is a random number in uniform dissemination.

For exploration phase, the above approach can be utilized to hunt for prey by variation of H vector. A vector H with random values (greater than 1 or less than − 1) is utilized to constrain a solution to move far away from the best-known searched agent. It is computed by:where \(S_{randm}\) is a random whale taken from the present population.

For BWOA (Mafarja and Mirjalili 2017), the sigmoidal function is used to map continuous value into a binary value.

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 re-arranged 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.

In SCA, the position of solution is updated using Eq. 32:where \(pos_{i}^{t}\) vector is the current position at \(t{th}\) iteration in \(i{th}\) dimension, \(r_{3}\), \(r_{4}\), \(r_{5}\) are the random numbers, \(r_{6}\) are the random number in [0, 1] and \(L_{i}\) is targeted global optimal solution.where \(t\) is the current iteration, \(T\) is the total number of iterations, and \(a\) is a constant.

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.

This Subsection discusses the very first step of the proposed method, i.e., initialization of the swarm. Here, a particle is represented in binary form of size \(n\) (as shown in Fig. 3), where \(n\) represents the total number of features in the dataset. Here, binary value 1 shows presence of corresponding feature and 0 depicts its absence. We work with the population size of \(n\). For this, the first swarm is randomly initialized with the 2 \(n\) particles. Next, top \(n\) particles are selected from the 2 \(n\) particles with the assistance of fitness value. A pictorial representation of a particle with size n is presented in Fig. 3.

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 K-mean partitional clustering method. K-means 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. K-means 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.

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.

Consider a data point \(x_{i}\) lies in a cluster, where \(p\left( {x_{i} } \right)\) be the normal disparity of \(x_{i}\) with every other datum inside a similar group and \(f\left( {x_{i} } \right)\) be the most reduced normal difference of \(x_{i}\) to some other group. Silhouette index \(Sil\left( {x_{i} } \right)\) for a particular data point \(x_{i}\) is numerically calculated as follows:

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.

As mentioned in Sect. 1, the proposed HBPSOSCA algorithm is a nature-inspired wrapper feature selection algorithm. In this section, we present a detailed description of the proposed method. Firstly, best \(n\) particles are selected from the 2 \(n\) randomly generated particles based on fitness value (for more details refer, Sect. 3.1). Each particle is the potential solution in the search space. Next, K-means clustering is applied to create clusters of data points on the selected features subspace. Here, the particle’s quality is evaluated using fitness function (refer Sect. 3.3). Swarm gbest (best solution of the entire swarm) and pbest (individual best solution) particles are selected from the current population based on their fitness. During the iteration, some particle gets stuck at a local optimum solution. We use a replacement strategy to deal with this problem. This strategy replaces a particle with the opposite solution if a particle does not improve for the defined number of iterations. Flowchart and algorithmic flow of the proposed method are demonstrated in Fig. 4 and Algorithm 1, respectively.

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.

In this paper, we use a V-shaped function to change a real-valued swarm intelligence system to a binary algorithm without altering the knowledge of swarm. As per Saremi et al. (2015), the V-shaped transfer functions [refer Eq. (35)] are superior to S-shaped transfer functions since they don’t constrain particles to take estimations of 0 or 1.

For proposed HBPSOSCA, the accompanying new technique for updating particle’s velocity and position is utilized, respectively to refresh the position of particles in twofold search spaces. Mathematical formulation of these steps is given in Eqs. 36–39.where \(r_{1}\), \(r_{2}\) are random numbers in the interim (0, 1), \(r_{3}\) and \(r_{4}\) are also the random numbers, \(pos_{i}^{d}\) is position of \(i{th}\) particle at dimension \(d\), and \(w\) is the inertia weight (Jain et al. 2017; Bansal et al. 2011). The mathematical formulation of inertia weight is presented in Eq. 40.

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).

The parameter \(r_{3}\) shows the direction of movement that can be either outside or space in between the solution and destination. It is defined as follows:where \(t\) is the current iteration, \(T\) is the total number of iterations, and \(a\) is a constant.

The parameter \(r_{4}\) is an irregular number in [0, 2π] demonstrates how far the development ought to be towards or outwards the goal.where \(pos_{i}^{d}\) is the position of \(i{th}\) particle at dimension \(d\), \(rand\) is a random number, and \(TF\) is the V-shaped transfer function. The mathematical formula is given in Eq. 35.

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.

The effectiveness of the proposed method is demonstrated using several experiments conducted on seven real-life scientific datasets. The datasets are taken arbitrarily from the UCI archive of machine learning and GEMS. The brief description of the datasets is presented below. The statistical details of used datasets are presented in Table 1 (Frank 2010; http://​www.​gems-system.​org).

The parameters have a large impact on optimizing performance. In BPSO, particle’s position and velocity are initialized randomly. Here, the position is in the form of 1 or 0 and velocity is in the range of − 6 to 6, and parameters e₁ (\(0 \le e_{1}\)) and e₂ (\(e_{2} \le 2\)) are the weighting (acceleration/learning) coefficients for the personal best and global best positions respectively. In Chaotic-BPSO, tent map is preferable to logistics map according to Kennedy and Eberhart (1997). Therefore, the inertia weight parameter is set at to 0.48 for CBPSO. In BMFO, parameter \(b\) is a constant value which is used to define shape of the logarithmic spiral when position (refer Eq. (7)) of moth is updated. In BDFA, parameter \(w\) is the inertia weight which is used in updating the movement of the dragonfly (refer Eq. (17)) and its value varies from 0.9 to 0.2. In BWOA, parameter \(b\) is a constant value which is used to define shape of the logarithmic spiral when position of the whale is updated (refer Eq. (26)). In SCA, parameter a (refer Eq. (33)) is used to compute direction of movement which could be either in the space between the arrangement and objective or outside it. In artificial bee colony, the number of bees is the main factor that affects the speed and quality of the algorithm. In HBPSOSCA, parameters e₁ and e₂ are the acceleration coefficients used in BPSO, parameters \(w_{min}\) and \(w_{max}\) are the initial and final values of the inertia weight used in finding the linearly increasing inertia weight, i.e., \(w\) [refer Eq. (40)]. The parameter max_count is used to re-initialize the particular particle randomly or by taking the complement of that particle if the particle does not change its position for defined number of iteration. The parameter \(\xi\) varies in \(0 \le \xi \le 1\). It is used to control portion of iterations where inertia weight increments straightly from \(w_{min}\) and \(w_{max}\). Here, ξ = 0.9 [refer Eq. (40)] is used to achieve stronger exploration in initial iterations and high exploitation in later iterations, and parameter a [refer Eq. (41)] is used to compute direction of movement which could be either in the space between the arrangement and objective or outside it. Tabulated summary of the parameters adopted in this paper for eight different algorithms namely, BPSO, BMFO, BDFA, BWOA, and HBPSOSCA is presented in Table 2.

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.

DI is proposed in 1974 by Desgraupes (2013) and Bezdek and Pal (1995) that measure maximum cluster diameter and relate it to the minimum cluster distance to judge the clustering performance. DI is calculated by dividing minimum distance between clusters by maximum size of clusters. Therefore, high value of distances between clusters and low value of cluster sizes indicates high value of DI, leading to increased clustering efficiency. DI for \(m\) clusters is denoted as follows:where \(\delta \left( {C_{i} ,C_{j} } \right)\) is the distance among clusters \(i\) and \(j\) and \(\Delta_{k}\) denotes the maximum distance between clusters.

DBI is an evaluation scheme introduced in 1979 by Desgraupes (2013) and Davies and Bouldin (1979). It is the ratio between the within cluster distances and the between cluster distances and the average overall the clusters. The value of DB varies between 0 and 1. The lower value of DBI denotes best partitions of the data point. It is calculated as follows:where \(m\) is the number of clusters and \(D_{i}\) is defined as:where \(R_{ij}\) denotes clustering efficiency (refer Eq. (46)), distinction between \(i{th}\) and \(j{th}\) clusters is \(l_{ij}\) (refer Eq. (47)), within cluster distribution of \(i{th}\) cluster is denoted as \(P_{i}\) (refer Eq. (48)), \(EU\left( {x,y} \right)\) is the Euclidean separation amongst \(x\) and \(y\), center of cluster \(C_{i}\) is \(CI_{i}\) and \(||C_{i}||\) is the convention for \(C_{i}\).

The mean and standard deviations of function values obtained by original BPSO, C-BPSO, BMFO, BDFA, BWOA, SCA, Binary-ABC, and HBPSOSCA for 150 iterations for 10 independent runs are given in Table 3. It is evident from Table 3 that the HBPSOSCA method attains better performance compared to competitive methods in most of the cases.

Graphical summary of the proposed algorithm in comparison to the competitive methods BPSO, C-BPSO, BMFO, BDFA, BWOA, SCA, and Binary ABC is presented in Fig. 5. Here, the fitness value recorded over the number of runs for each algorithm is reported in Fig. 5. The Fig. 5 demonstrates that the proposed method is consistent over the number of run in comparison to competitive methods in most of the case. This behavior depicts stability of the proposed method in comparison to competitive methods.

In this subsection, we perform comparative analysis of all models for feature selection problem. The higher value of SI and DI represents better performance whereas lower value of DBI specifies better performance. The statistical results of 10 independent runs, obtained by HBPSOSCA and competitive methods are presented in Tables 4, 5 and 6. Furthermore, we measured the Wilcoxon test for all the algorithms and the results are presented in Tables 7, 8, 9, 10, 11, 12 and 13.

The data obtained from 10 runs is plotted in Fig. 6 to analyze the consistency and overall performance of HBPSOSCA. For the ionosphere dataset, the performance of proposed HBPSOSCA method is higher than the competitive methods with 92.7% average accuracy as shown in Fig. 6a. For BCW dataset, the performance of proposed method is higher than BPSO, BMFO, BDFA, SCA, and binary ABC methods as shown in Fig. 6b and also proposed method achieves a little bit better average accuracy with 87.86% than an average accuracy of C-BPSO and BWOA, i.e., 87.49% and 87.01% respectively. For CB dataset, the proposed method performs best as compared to other methods with 90.41% average accuracy. For Statlog dataset, the performance of BWOA and HBPSOSCA is nearly same up to four iterations. After 4th iterations, performance of the proposed method is higher than BWOA and other method with 84.61% average accuracy as shown in Fig. 6d. For Parkinson dataset, SCA performed better with average accuracy 95.43% with respect to other methods, but the proposed method achieves better average accuracy as shown in Fig. 6e as compared to stated methods with 96.13% average accuracy. Figure 6f shows performance of the proposed method for the 9_Tumors dataset with other methods used in this thesis. This figure describes that proposed method achieves higher performance as compared to other methods with 80.06% average accuracy. For Leukemia2 dataset, the proposed method achieves 90.73% average accuracy which is comparatively higher than the competitive method as shown in Fig. 6g. A possible reason of this improvement is good exploration and exploitation of the given feature space.

We can see that the score obtained by HBPSOSCA is significantly better compared to other stated methods in Tables 7, 8, 9, 10, 11, 12 and 13. The comparative performance of BPSO, C-BPSO, BMFO, BWOA, SCA, binary ABC, and HBPSOSCA in term of number of selected features is shown in Fig. 7. Here, the proposed method HBPSOSCA selects 6.5882%, 22.2222%, 4.715%, 19.794%, 9.693%, 3.2949%, and 8.3091% of features for Ionosphere, BCW, CB, Vehicle, Parkinson, 9_Tumors, and Leukemia2 datasets, respectively, which is comparatively less than the other stated methods. It is clear from the conducted analysis that the HBPSOSCA outperforms the other methods in most of cases.