A self-adaptive binary cat swarm optimization using new time-varying transfer function for gene selection in DNA microarray expression cancer data

Microarray technology is beneficial in terms of diagnosing various diseases, including cancer. Despite all DNA microarray benefits, the high number of genes versus the low number of samples has always been a crucial challenge for this technology. Accordingly, we need new optimization algorithms to select optimal genes for faster disease diagnosis. In this article, a new version of the binary cat optimization algorithm, named SBCSO, for gene selection in DNA microarray expression cancer data is presented. The main contributions in this paper are listed as follows: First, the opposition-based learning (OBL) mechanism is employed to improve the proposed algorithm's population members' diversity. Second, a time-varying V-shaped transfer function is employed to balance the two phases of exploration and extraction in the proposed algorithm. Third, the MR and λ parameters in the proposed algorithm are adapted over time, and finally, single-objective and multi-objective approaches are proposed to solve the gene selection problems. The 15 datasets pertinent to microarray data of various cancer types are employed to compare the proposed method with other well-known binary optimization algorithms. The experiments' results indicate that the proposed algorithm has a better capability to select the optimal genes for a faster disease diagnosis.


Introduction
Cancer is among the deadly and painful diseases with increasing yearly mortality rate. Therefore, cancer research is one of the main activities in bioinformatics. In this regard, major cancer diagnosis methods include photography, laboratory tests, tumor sampling, endoscopy, surgery, or genetic tests (Crosby et al. 2022;Pruthi 2019). Since cancer is caused by a gene disorder, investigating gene expression data can be a basic and essential method in genetic disorders (Oliveira et al. 2022). Microarray data are extremely useful in terms of diagnosis and prediction of various diseases such as cancer. These data are obtained based on individual's DNA and their genetic information. DNA microarray technology simultaneously investigates the expression of thousands of genes, so it is a significant breakthrough in diagnosing and treating various diseases, particularly cancer. Figure 1 illustrates the generating process of a microarray dataset. The figure compares gene expression in normal and cancer cells using microarray technology. According to Fig. 1, the results of the microarray data can be classified into four categories after the hybridization of normal and cancer samples as follows: (1) genes are not expressed in any cell, (2) genes are expressed in both healthy and cancer cell types, (3) genes are expressed only in healthy cells, and (4) genes are expressed only in cancer cells. In this respect, microarray datasets are generated based on gene expression data (Sörensen 2015). Moreover, as most of the generated datasets have many genes, useless and redundant genes should first be removed based on the feature selection methods. Accordingly, different types of cancer can be diagnosed based on a machine learning model (Fig. 2).
One of the ever-present, significant challenges in microarray data analysis is the selection of optimal genes. The high number of genes, in addition to the low number of samples in microarray data, will bring about some problems in terms of data analysis. Also, in some cases, the high proportion of genes to sample numbers increases the possibility of selecting an unsuitable gene. Nowadays, data analysis could be complicated without statistical analysis and intelligent algorithms (Hambali et al. 2020a). The traditional methods take the vast majority of genes into account as effective genes in diseases (Glazier 2002). Investigations indicate that most DNA microarray gene expression cancer data are not of informational value based on classification criteria (Golub et al. 1999). Thus, feature (gene) selection and diagnosis and elimination process of irrelevant genes play pivotal roles in the analysis of microarray data obtained from DNA. Hence, researchers are in the pursuit of introducing novel algorithms of feature selection (Oliveira et al. 2022).
To date, several metaheuristic optimization algorithms have been developed to solve problems in engineering, economics, medicine, etc. Most of these algorithms are based on nature and social behaviors (Molina et al. 2020;Swan et al. 2022;Dorigo and Stützle 2019;Nayar et al. 2021). One of the basic applications of metaheuristic algorithms is to solve feature selection problems. The most important artificial intelligence algorithms proposed in this field are the ant colony optimization (ACO) algorithm (Nayar et al. 2021;Sakri et al. 2018), genetic algorithm (GA) (Jahwar and Ahmed 2021), particle swarm optimization (PSO) (Wang et al. 2018;Sindhiya and Gunasundari 2014), differential evolution (DE) (Pant et al. 2020), cat swarm optimization (CSO) (Kumar and Singh 2018), etc.
During the last decades, various algorithms are introduced to solve the feature (gene) selection problem, including filter-based, wrapper-based, hybrid, and embedded algorithms, among which the filter-based and wrapperbased methods are more prevalent (Chandrashekar and Sahin 2014;Almugren and Alshamlan 2019). Wrapperbased methods take advantage of machine learning algorithms as a cost function to evaluate a subset of Fig. 1 The process of generating a microarray dataset Fig. 2 Microarray data analysis pathway features. Some of the wrapper-based algorithms introduced in recent decades for selecting a subset of features in various applications include ISSA (Tubishat et al. 2020), BWSSO (Kalaimani and Umagandhi 2020), QWOA (Agrawal et al. 2020), and TLBOSA (Shukla et al. 2019). Besides, the filter-based algorithms make use of statistical methods to select a subset of features (Lazar et al. 2012). The hybrid method employs the two wrapper and filter methods to improve the feature selection algorithm's effectiveness. In this regard, we can name WFA-COFS (Ghosh et al. 2019), IG-GA (Yang et al. 2010), and BDE-xRank (Apolloni et al. 2016) algorithms.
When comparing the wrapper-based and filter-based methods, it is indicated that wrapper-based algorithms have better results. However, the major drawback of these methods is numerous calculations. Accordingly, researchers have always pursued novel methods of evolutionary optimization algorithms to accelerate the selection process of a subset of optimal features by reducing the computational load (Chandrashekar and Sahin 2014).
In the literature review, several evolutionary optimization algorithms have been proposed in terms of gene selection in microarray data (Hambali et al. 2020b;Alhenawi et al. 2022;Tabakhi et al. 2015). Among these algorithms, the binary genetic algorithm (BGA) is one of the most popular ones (Durgam et al. 2022;Pashaei and Aydin 2017). In Sharbaf et al. (2016), an unsupervised gene selection was developed based on a new ant colony optimization algorithm for microarray data classification. This method employs a new fitness function to assess a set of selected genes.
A combined approach of ACO and cellular learning automata was developed for the gene selection process in microarray data (Prasad et al. 2018). This method uses Fisher's measure ranking to decrease the number of features. A recursive PSO method was proposed for gene selection, which was much faster than the standard PSO and its variants (Ramaswamy et al. 2021). Besides, an improved version of the binary PSO algorithm was developed for the gene selection in Alzheimer's gene expression data. In this method, crossover and mutation operators of the GA are employed to prevent the PSO algorithm from getting trapped in the local optimal trap (Maleki et al. 2021).
Binary black hole algorithm (BBHO) ) and binary gravitational search algorithm (BGSA) (Hussain et al. 2021) are some physics-based evolutionary algorithms in which gene selection is performed. Likewise, among examples of gene selection algorithms inspired by human behavior in society are the binary teaching learningbased optimization algorithm (Wang et al. 2016) and binary imperialist competition algorithm (BICA) (Li et al. 2008). Moreover, binary firefly algorithm [84], binary marine predators algorithm (BMPA) (Blum et al. 2011), binary competitive optimization algorithm (BCOOA) (Nasrabadi et al. 2016), binary whale optimization algorithm (BWOA) (Too et al. 2019), and binary Harris hawks optimization (BHHO) (Abd Elaziz et al. 2022) are some swarm intelligence developed in the diagnosis of many cancers based on microarray gene expression data.
In recent decades, researchers have proposed combining evolutionary algorithms in gene selection in microarray gene expression data (Pashaei et al. 2019). A hybrid approach of two sine-cosine optimization algorithm and Harris hawk's optimization (HHO) was developed to increase the power of the exploration phase in the HHO algorithm (Sazzed 2022). In addition, a hybrid of two artificial bee colony (ABC) algorithm and whale optimization algorithm (WOA) approach was developed for breast cancer detection (Kumari and Jagadesh 2022). Furthermore, a hybrid PSO-GA approach was proposed for gene selection in microarray gene expression data (Kowsari et al. 2022).
In addition to single-objective approaches in binary optimization algorithms for gene selection, many researchers have proposed multi-objective optimization algorithms for this purpose. A multi-objective approach of binary particle swarm optimization (BPSO) was described based on adaptive k-nearest neighbors (KNN) algorithm (Zhang et al. 2022). A multi-objective HHO algorithm was developed based on associative learning and chaotic local search to improve the gene selection process in microarray gene expression data (Xie et al. 2023).
The structure of all evolutionary algorithms comprises an initial population that is evolved in an evolutionary approach. Suitable diversity in the members of an initial population can play a pivotal part in population convergence toward optimal global solutions. Suppose the diversity of population members is not appropriate. In that case, the two exploitation and exploration phases will not be balanced well, and the evolutionary algorithms can get trapped in a local minimum. Various investigations have been conducted to improve the diversity of population members and the effectiveness of evolutionary algorithms.
The opposition-based learning (OBL) method is one of the most that has recently captured a variety of researchers' interests. OBL is a new concept in machine learning, which is inspired by the contrasting relationships between entities (Mahdavi et al. 2018). This investigation employs the OBL mechanism to improve the diversity of initial population members and the middle generation (Sharafi and Teshnehlab 2021;Nasrabadi et al. 2016;Abd-Elnaby et al. 2021).
The cat swarm optimization (CSO) algorithm is presented based on cats' group behavior in two phases of tracing and seeking (Kumar and Singh 2018). In the CSO algorithm, the percentage of cats' involvement in both phases is determined based on the mixture rate (MR) parameter's extent. In essence, the CSO is a continuous algorithm, and the feature selection is an optimization problem, the variables' spaces of which are binary. In evolutionary algorithms, some transfer functions, including linear, S-shaped, and V-shaped functions, are usually employed to convert the continuous space to binary space. Optimization algorithms employing S-shaped functions include BPSO (Hu et al. 2020), BGWO (Islam et al. 2017) and BCSO (Mafarja et al. 2018). Also, the BGSA (Beheshti 2020), BBA (Kahya et al. 2020), and BDA (Chu and Tsai 2007) algorithms take advantage of V-shaped functions for various applications. Recently, a new type of transfer function is introduced, the form of which varies by time. The PSO binary optimization algorithm is introduced based on time-varying transfer functions to strike a balance between exploitation and exploration (Mafarja et al. 2018). Also, in dragonfly optimization algorithm, the time-varying transfer functions are employed in feature selection (Beheshti 2020). A time-varying mirrored S-shaped transfer function is used for BPSO algorithms to solve feature selection problems (Kahya et al. 2020). The binary version of the binary whale optimization algorithm (BWOA) algorithm is also provided based on a time-varying transfer function to solve feature selection problems (Chu and Tsai 2007). In this paper, a new time-varying V-shaped transfer function is employed to convert the continuous space to the binary space in the CSO algorithm. The proposed transfer function has an adaptive parameter striking a balance between two exploration and exploitation processes in the proposed algorithm.
In the feature selection problem, the F1 and F2, which are inversely related functions, naturally convert this problem into a multi-objective one. The F1 and F2 functions refer to the number of features selected and classification error rate, respectively. In this research, singleobjective and multi-objective approaches are proposed to solve the feature selection problem. The proposed algorithm is applied to 15 benchmark datasets to optimize the feature selection. The experimental results indicate that the proposed binary algorithm has reported better performance in comparison to other well-known binary optimization algorithms.
The contributions of this investigation are summarized as follows: • The improved CSO binary algorithm is presented.
• An initial combined population with a suitable diversity is presented to solve binary problems based on OBL and uniform distribution. The OBL mechanism is utilized in the evolution of population members of the middle generation.
• A new time-varying V-shaped transfer functions are employed to balance two exploration and exploitation phases. • Single-objective and multi-objective approaches are provided to solve the gene selection problem. • The MR and k parameters in the proposed algorithm are adapted over time. • The proposed algorithm results are investigated on 15 datasets pertinent to microarray data of various cancers, and the outcomes are reported.
The structure of this article is as follows: A summary of the CSO algorithm and a review of OBL and TF are presented in Sect. 2. In Sect. 3, the proposed SBCSO algorithm is presented. In Sect. 4, the proposed SBCSO algorithm's test results compared to other well-known binary optimization algorithms are reported. Finally, a conclusion is presented.

Background
In this section, the fundamental concepts of cat swarm optimization algorithm, OBL, and a review of the transfer functions are presented.

The standard cat swarm optimization algorithm
The cat swarm optimization algorithm is an algorithm inspired by the group behavior of cats in nature. Cats usually conduct two behaviors in nature. First, they rest most of the time, watching their surroundings intelligently. Second, they are aware of everything happening around them, and as soon as they see a target, they move quickly toward it. These two behaviors are considered to be the principle of the design and implementation of the CSO algorithm (Kumar and Singh 2018).
In the CSO algorithm, these two behaviors are stimulated in two phases of seeking and tracing. In the seeking phase, cats observe their surroundings, and in the tracing phase, they move toward a suitable target. In the CSO algorithm, the number of cats falling under the tracing phase category is determined based on the mixture rate (MR) parameter, and the rest of the cats fall into the seeking phase category. Figure 3 demonstrates the general process of the CSO algorithm.
As depicted in Fig. 3, cats search the entire problem space in the seeking phase (i.e., the exploration phase). In this phase, four parameters of counts of dimension to change (CDC), self-position consideration (SPC), seeking range of the selected dimension (SRD), and seeking memory pool (SMP) are defined and reported in Table 1.
Several copies of every cat in the seeking phase will be created (the copy number equal to SMP parameter), and each copy will change separately based on SRD and CDC parameters (similar to the mutation operator in a genetic algorithm). The value of the SPC parameter is either TRUE or FALSE. In case it is TRUE, the present cat competes with other modified copies, as well. Finally, the best solution in terms of fitness is transferred to the next algorithm generation. In the seeking phase, the position of each copy is updated based on Eq. (1).
x d m ðt þ 1Þ ¼ x d i ðtÞ AE SRD Â x d i ðtÞ; m ¼ 1; 2; :::; SMP ð1Þ After producing all the copies, the probability of choosing each solution is calculated based on Eq. (2).
FS m is the fitness value of the mth solution. The values of FS min and FS max are, respectively, the minimum and maximum fitness values among all copies. In a single-objective optimization problem, the FS m value in the general case is calculated as follows: gðxÞ ¼ ðg 1 ðxÞ; g 2 ðxÞ; :::; g k ðxÞÞ ! 0 hðxÞ ¼ ðh 1 ðxÞ; h 2 ðxÞ; :::; h p ðxÞÞ ¼ 0 where x ! ¼ ðx 1 ; x 2 ; . . .; x n Þ and n is the number of decision variables, k is the number of unequal constraints, and p is the number of equality constraints. Note that every singleobjective optimization problem can include one or more constraints. The penalty function is the most common and primary approach in evolutionary algorithms to handle the constraints. Any cat in the tracing phase will move to the  position of the cat in the best position (Kumar and Singh 2018).
where x is the inertia weight; v d i t ð Þ is the velocity value of the ith cat in the dth dimension; x d Global t ð Þ is the position of the best cat among all cats in the tth generation of the algorithm; x d i t ð Þ is the current position of the ith cat in the dth dimension; c is the coefficient of group experience; and r is the randomization rate for the velocity equation with its value being between 0 and 1. Algorithm 1 shows the pseudo-code of the cat optimization algorithm.
Researchers have introduced several improved versions of the CSO algorithm, so far. Particularly in the article (Abd Elaziz and Mirjalili 2019), an improved version of this algorithm is presented based on the normal mutation to achieve a faster convergence. The article (Pappula and Ghosh 2018) has also improved the global and local search of this algorithm by modifying some of the equations.

The binary cat swarm optimization algorithm
In the research literature, some binary versions of the cat optimization algorithm are presented. For the first time in 2013, a binary version of the cat algorithm addressing the zero-one knapsack problem was introduced (Siqueira et al. 2018). In this algorithm, a simple sigmoid function was introduced to convert the continuous space into binary space. In the article (Siqueira et al. 2020), one binary version of the cat algorithm based on the V-shaped transfer function is presented to solve the one-zero knapsack problem. In the article (Haznedar et al. 2017), the updating equations of a cat position in the tracing phase changed, and the zero-one knapsack problem is solved based on it.
The pseudo-code of the original binary cat swarm algorithm (BCSO) is indicated in Algorithm 2. Figure 4 shows both the tracing and seeking phases in the binary version of the CSO algorithm. In the seeking phase, the probability of mutation operation (PMO) parameter is employed for the probability of changes in the genes of a solution (Fig. 4a). Also, in the tracing phase, the parameter t k is utilized to update the position of a solution (Fig. 4b). It is noteworthy that t k is calculated in each algorithm step using the following equation (sig is the sigmoid function). Fig. 4 Graphical representation of two phases of the BCSO algorithm A self-adaptive binary cat swarm optimization using new time-varying transfer function for gene… 7963

Opposition-based learning (OBL)
In the evolutionary algorithms, the population members' diversity plays a pivotal part in preventing the early premature convergence and also getting stuck in a local optimum. Accordingly, various methods, including OBL and chaotic maps, are presented to increase the population members' diversity (Mahdavi et al. 2018;Zhao et al. 2020;Rojas-Morales et al. 2017). The concept of OBL is demonstrated in Fig. 5 (also see Eq. 10). x The upper and lower values are the ceiling and the floor of the search space, respectively. Given that the search space is binary in the feature selection problems, the opposite of a solution is calculated employing the following equation.
In recent decades, researchers have taken advantage of various OBL mechanisms in some evolutionary optimization algorithms. In this regard, we can name quasi-opposition, super-opposition, quasi-reflection OBL, and generalized OBL methods (Dhargupta et al. 2020;Kennedy and Eberhart 1997). In this paper, the OBL mechanism is utilized to improve the proposed algorithm's population members' diversity.

A brief overview of existing transfer functions
Various transfer functions have been employed to convert the continuous space to the binary space in evolutionary optimization algorithms to solve feature selection problems. Among the well-known transfer functions capturing researchers' interests in recent years are V-shaped and S-shaped transfer functions. It is worth mentioning that the shape of these transfer functions can vary by the time during the optimization process. Accordingly, the shape of these transfer functions can be either static or time-varying. In Table 2, a review of some transfer functions is presented.

The proposed self-adaptive binary cat swarm optimization algorithm
This section presents the binary version of the self-adaptive cat swarm optimization algorithm for the gene selection in DNA microarray expression cancer data. In the proposed binary algorithm named SBCSO, the OBL mechanism is utilized to increase the population members' diversity.
Enhancing the population members' diversity contributes highly to preventing the evolutionary optimization algorithms from getting stuck in a local minimum. In sub-Sect. 3.1, the method of using the OBL mechanism in the proposed algorithm is presented. The feature (gene) selection is a binary optimization problem. Therefore, the transfer functions are required to convert an optimization algorithm in continuous space to binary space. In sub-Sect. 3.2, a new time-varying V-shaped transfer function containing a time-varying parameter is presented. The role of this parameter is to balance two phases of exploration and exploitation. The cat swarm optimization algorithm comprises two phases of seeking and tracing, in which the number of cats in each phase is determined based on the mixture rate (MR) parameter. In subSect. 3.3, an automatic mechanism for adapting the proposed algorithm's parameters, including the MR and k parameters, is presented (k is a controlling parameter balancing the exploration and the exploitation phases). In subSect. 3.4, the cost function value for solving the feature selection problem is defined in the form of single and multi-objective functions. In sub-Sect. 3.5, the final conditions of the proposed algorithm are presented.
In order to obtain the best subset of features, the K-nearest neighbor (KNN) algorithm with K = 5 is employed to determine the performance of each feature's subset for classification. The block diagram of the SBCSO algorithm for selecting the optimal feature subset of a dataset is demonstrated in Fig. 6. The details of the proposed algorithms are presented below. Moreover, the pseudo-code of Algorithm 3 illustrates the general procedure of the proposed algorithm.

Initial population strategy
The OBL mechanism is employed in two parts of the proposed algorithm to improve the population members' diversity. First, a population titled the subset {p} with a uniform distribution is initially created in the first part, as indicated in Fig. 7. In the next step, a population titled {OP} is created using the OBL mechanism. N better solutions for the initial population are selected from {P, OP} set to calculate their cost functions. Most evolutionary algorithms come across stagnancy over time as population diversity decreases. In this research, as demonstrated in Fig. 8, if the proposed algorithm comes across the stagnancy condition, the OBL mechanism will become active, improving the population members' diversity. In addition, the exact details of the steps of the OBL mechanism in the proposed algorithm are provided in the pseudo-code of Algorithm 4.
In the SBCSO algorithm, the OBL mechanism can become active under particular circumstances. As indicated in Fig. 9, two scenarios are introduced. In the first scenario, if the algorithm comes across a stagnancy in terms of a decline in the cost function (fitness) value, the OBL mechanism will become active. In the second scenario, the OBL mechanism is active after several defined steps. In the proposed algorithm, the first scenario is employed. In fact, if the value of the cost function remains unchanged after ''step'' successive steps, the OBL mechanism will be employed (the value of step in this article is considered 10).

Position updating based on new timevarying V-shaped transfer function
The evolutionary algorithms usually employ two types of transfer functions, S-shape and V-shape, to convert continuous space to binary space. In the S-shaped transfer functions, the solution is converted to the binary space on the basis of Eq. 12.
The expression x d i (t + 1) is the value of dth dimension of the ith solution at the tth generation. The phrase rand is a random number with a uniform distribution in [0, 1] range.
In the V-shaped transfer functions, the solution is converted to the binary space on the basis of Eq. 13.
In this paper, a new time-varying V-shaped transfer function is presented as follows: In this equation, k is a controlling parameter balancing the exploration and the exploitation phases in the optimization process. In Fig. 10, the influence of the parameter on the proposed transfer function is demonstrated.
As shown in Fig. 10, for the small values of k, the transfer function has values approximate to one. Also, according to Fig. 11, the variations on the solution are high, and the algorithm is in the exploration phase. For the larger values of k, on the other hand, the transfer function has values approximate to zero. According to Fig. 11, the variations on the solution are low, and the algorithm is in the exploitation phase. Figure 12 demonstrates an overview of the different shapes of the proposed transfer function. The evolutionary algorithms usually start to operate with the exploration phase and enter the exploitation phase over time. If such a scenario is supposed to occur in the proposed algorithm, it ought to start with small value of the k parameter, and k must be increased over time. This matter is easily performed as the k parameter linearly increases according to Eq. 15. This method's major drawback is that it compels the algorithm to move in a predefined path from the Fig. 9 The scenarios of employing the OBL mechanism Fig. 8 The use of the OBL mechanism for avoiding the stagnancy conditions exploration phase to the exploitation phase. And sometimes, it makes the algorithm get stuck in the local minimum and be unable to get out it. In subSect. 3.3, a new method for determining the k parameter's value in each step of the proposed algorithm is presented. Fig. 10 The effect of the k parameter values on the shape of the proposed transfer function Fig. 11 The effect of k value on the intensity of the changes on a solution A self-adaptive binary cat swarm optimization using new time-varying transfer function for gene… 7969

The self-adaptive parameters in the proposed method
The k and MR parameters in the proposed SBCSO algorithm play a pivotal role in placing the population members in two exploration and exploitation phases. In each step of the proposed algorithm, the MR parameter value indicates how many members of the population are in the exploitation phase. Also, the value of the k parameter in the transfer function indicates the intensity of changes on one solution. Figure 13 demonstrates the self-adaptive approach of the k and MR parameters in the proposed SBCSO algorithm. As can be seen in this figure, the binary cat optimization algorithm operates in two phases of seeking and tracing. In the seeking phase, the concentration is on global search (increase in exploration), and the tracing phase concentrates on local search (increase in exploitation). The evolutionary algorithms cannot have the best performance until they can divide their population members well between these two phases and determine the phase in which they should operate more during the optimization process. This investigation employs k and MR parameters  The self-adaptive approach of k and MR parameters in the proposed algorithm to control the balance between these two phases at any moment. The update procedure of these parameters will be discussed later.
As shown in Fig. 14, the cost function curve can be considered as feedback of the algorithm optimization process's performance. The variation of fitness value is calculated in two successive steps according to the following equations.
BestFitness t is the best cost function value in the tth step. a t is the difference in the value of the cost function changes in two consecutive steps of t and t-1. a tÀ1 is the difference in the value of the cost function in two successive steps of t-1 and t-2.
In the proposed algorithm, k and MR parameters are adapted based on a t and a tÀ1 following pseudo-code (As shown in Algorithm 5). When the algorithm gets stuck in the local optimum, the cost function curve remains unchanged (a t ¼ a tÀ1 ¼ 0). In this case, to increase the diversity of the population members, the search mechanism should be in the exploration phase (by reducing the values of k and MR). When the slope of the cost function curve's variations is low (a t \a tÀ1 ), it indicates that either the population members' diversity decreases or we are close to a local or global optimum. Accordingly, in order to increase the population members' diversity, the search mechanism ought to be in the exploration phase (by reducing the k and MR values). When the slope of the cost function value's variations is high (a tÀ1 \a t ), it means that the search route is suitable, and we can increase the convergence speed and place the search mechanism in the exploitation phase (by increasing the values of and MR).
As can be seen, values of both MR and TR parameters are updated in each generation using the Algorithm 5. Also, it is of note that the values of both MR and TR depend on the values of a tÀ1 and a t in each generation of the algorithm, which are updated using Eqs. 16 and 17, respectively. Furthermore, the values of both a tÀ1 and a t are calculated based on the value of the fitness function of the best member of the population in each generation. Indeed, the parameters values depend on the severity of changes in the fitness value of two successive steps of the algorithm. Figure 15 depicts the details of the steps of updating the values of both MR and TR parameters in each generation of the algorithm.

Fitness function
The definition of a cost function for an optimization problem is of paramount importance. A gene selection problem as a minimization problem is expressed as follows: Minimum The f 1 ðxÞ is classification accuracy and f 2 ðxÞ is the number of selected genes. The objective functions of f 1 ðxÞ and f 2 ðxÞ for a specific dataset are defined as follows:

Single-objective approach
In the single-objective approach, the fitness function is defined as follows: The values of w 1 and w 2 are the weights to control the effect of objective functions on the final fitness function's final value. In the test results, the values 0.95 and 0.6 are considered for parameter w 1 .

Multi-objective approach
In the single-objective approach to the gene selection problem, the objective functions of f 1 ðxÞ and f 2 ðxÞ are minimized simultaneously, and ultimately, an optimal solution is obtained. In the multi-objective approach, according to Fig. 16, a set of answers titled Pareto front (PF). Based on the non-dominated sorting algorithm, the best answers are ranked first. As demonstrated in Fig. 16, solution 'A' comprises the maximum selected genes and the minimum classification errors. On the contrary, solution 'B' contains the minimum selected genes and the maximum classification errors.
Note that the proposed SBCSO algorithm is a singleobjective version. In a multi-objective approach, the cost function is defined as follows: A definition of the concept of the crowding distance (CD) is: The multi-objective approach of the proposed SBCSO algorithm runs exactly as its single-objective version does. The only difference is that in the single-objective approach, the SBCSO algorithm aims to minimize the cost function of Eq. 21. Nevertheless, in the multi-objective approach, the SBCSO algorithm aims to minimize the cost function of Eq. 22.

Termination criteria
One of the evolutionary algorithms' challenges is that it is not clear when the algorithm will reach the optimal solution. Also, it might not reach the optimal solution at all and get stuck in a local optimum. Therefore, in evolutionary algorithms, usually, the end of the search process is considered based on either the number of evaluations of the objective function (NFE) or the number of steps. In this According to Algorithm 6, the initial population is generated based on Algorithm 4. Afterward, based on the value of the MR parameter, the population members are classified into two subgroups: seeking and tracing. In this process, the population members are divided in each step of the algorithm. Then, each member of the population is updated based on its position. Next, based on the proposed transfer function (Eq. 14), the position of all members will be converted from continuous space to binary space. It is noteworthy that the values of k and MR parameters are updated based on Algorithm 5 in each step of the algorithm. In the proposed algorithm, if stagnation occurs, the diversity of population members will be enhanced using the OBL mechanism.

Complexity analysis
The computational complexity of the SBCSO algorithm depends on three main processes: generating the population, calculating the fitness function of the population members, and updating the position of the cats. The initial population is generated based on the hybrid mechanism described in subSect. 3.1. If the number of population members is equal to N, their OBL should be calculated; hence, the number of 2 Â N members should be assessed by their fitness value. Therefore, the complexity will be equal to Oð2 Â NÞ. It is noteworthy that since the fitness calculations depend on the type of optimization problem, we do not consider its effect on the complexity of calculations. In updating the cats' position, if M members of the population update their positions on average in each generation of the SBCSO algorithm, the algorithm's time complexity is OðT Â MÞ for T successive generations. Also, the value of M is in the interval of ½N; a Â SMP þ b, where a and b are equal to N À N Â MR and N Â MR, respectively. Thus, the final computational complexity of the SBCSO algorithm is equal to OðT Â M þ 2NÞ, which is highly dependent on both MR and SMP parameters.

Experimental results and discussion
In this section, in order to investigate the proposed algorithm performance, the results of the 15 datasets simulation related to the microarray data of different types of cancers, compared to the other common methods, are reported in several subsections as follows. The assessment results of the SBCSO algorithm, compared to the binary single-objective optimization algorithms, such as the BACO ( (Soyel et al. 2011), and BMOBBA ( Carrasco et al. 2020), are reported in subSect. 4.6. All of the simulations were executed in MATLAB on a computer Fig. 15 The updating process of the parameters of the proposed algorithm in each generation Fig. 16 The Pareto front for solving the gene selection problem equipped with a Core i7 CPU and 16G RAM. In this study, the results of the proposed SBCSO algorithm are compared based on the statistical tests together with different algorithms. Friedman, Quade, and Wilcoxon are among the statistical tests used in the assessment of the results (Tubishat et al. 2020;Derrac et al. 2011). In the experimental results section, bold values in all tables refer to the best algorithm.

Dataset description
Some details of 15 datasets of different types of cancer microarray data, including the numbers of samples, genes, and data class of each dataset, are indicated in Table 3. Each dataset is categorized into two training and testing parts based on the k-fold cross-validation in the results' assessment. In this research, to ensure the experiments' results on different datasets, the k = 10 value is used for the k-fold cross-validation mechanism. The optimization process of all feature selection algorithms is executed on train data, and after selecting the optimal features, the assessment results of classification on test data are reported.
In the suggested wrapper-based method, a K-nearest neighbor (KNN) algorithm is used to select an optimal subset of features. The KNN algorithm obtains the optimal features based on the train data with the lowest classification error. According to the gained features, the classification accuracy of the test data is achieved. In the KNN algorithm, the K value and also the distance method are obtained based on the trial-and-error method. In this study, the KNN algorithm, K = 5, and Euclidean distance method are taken into consideration.

Parameter settings
All of the values related to the parameters of the different algorithms are provided in Table 4 Table 4. The results obtained from 20-times independent execution of the proposed algorithm, compared to the other binary algorithms, are reported in Tables 5, 6 , 7, 8, 9, 10, 11, and 12. In order to assess the SBCSO algorithm performance, the two values of 0.6 and 0.95 have been used for the W1 parameter in formula fitness (see Eq. 21). The results of Tables 5, 6, 7, and 8 are based on W1 = 0.95. Accordingly, the results of Tables 6, 7, 8, and 9 are on the basis of W1 = 0.60. The fitness values of all benchmark datasets are indicated in Tables 5 and 6. As demonstrated in Table 5, the SBCSO algorithm has the lowest fitness value in 12 out of 15 benchmark datasets. The BGA algorithm has obtained better results in datasets In the ranking of the algorithms, the white box indicates the first rank and the black box indicates the last rank among all.
Tables 8 and 9 demonstrate the results of the best classification error along with several features selected from each dataset for all of the algorithms ( Table 8 for the values of W1 = 0.95 and Table 9 for the values of   Figures 19,20,21,22,23,24,25,26,27, and 28 indicate the fitness decline curve for different algorithms during the optimization process. As can be seen, the convergence velocity and the slope of fitness decline in the SBCSO are more than the other algorithms.
The Friedman, Sign, and Quade statistical tests are employed to analyze more of the proposed algorithm's results than other algorithms. The two procedures of pairwise comparisons and multiple comparisons are employed in the results of the reports. The Sign and Wilcoxon tests fall into the pairwise comparisons classification, and the Friedman and Quade tests fall into the multiple comparison's classification.
As indicated in Table 10, the SBCSO algorithm has the best performance in comparison with other algorithms in Quade and Friedman tests and attained the 1.45 and 1.31 ranks for the Friedman and Quade tests, respectively. The BGA and BBA algorithms attained the second and third ranks in the Friedman and Quade tests, respectively. The BACO algorithm is ranked the last among all of the algorithms. Table 11 reports the results of the Sign test. Table 11 indicates the number of times each algorithm wins another algorithm at both 0.05 and 0.1 levels. Table 11 clearly demonstrates that the SBCSO algorithm with the certainty level of a = 0.05 has won all of the other algorithms. The BGA and BBA algorithms are ranked second and third, respectively, and the BACO algorithm is ranked the latest among all other algorithms. Table 12 reports the results of the Wilcoxon test. In Table 12, the values of R, -R ? , and p-value are calculated for all binary comparisons pertinent to SBCSO. -Table 12 clearly demonstrates that the proposed algorithm wins over all of the other algorithms with the certainty level of a = 0.01.

Classification performance
The influence of employing the SBCSO algorithm in the feature selection process for each dataset is calculated to examine the performance and evaluate the SBCSO algorithm. In Table 13, the FSL and FSA values refer to the number of main features and the number of the selected feature of each dataset. The Accuracy and Time values for each dataset are also calculated before and after the feature selection process. The KNN algorithm with the value of K = 5 and the k-fold validation mechanisms with the value of k = 10 are employed to calculate the accuracy. As indicated in Table 13, the SBCSO positively influences the accuracy and time, and in all of the datasets, better performance is reported (with a much lesser number of features).

2019) algorithms
To further evaluate the performance of the SBSCO algorithm, the SBCSO algorithm was compared with other  Table 8 The best Table 9 The best classification error and the number of genes selected for each dataset (W1 = 0.60) common algorithms such as the binary whale optimization algorithm (BWOA) (Too et al. 2019), binary Harris hawk optimization (BHHO) (Abd Elaziz et al. 2022), and binary marine predators algorithm (BMPA) (Blum et al. 2011). Some details of seven datasets of different types of cancer microarray data, including the numbers of samples, genes, and data class of each dataset, are indicated in Table 14.

#DS
The value of k = 10 was used in the k-fold cross-validation mechanism to validate the results of tests on different datasets. Table 15 shows the comparison of the results obtained from 20 independent runs of the proposed algorithm with those of other algorithms.   The performance of the SBCSO algorithm was assessed by assigning the values of 0.6 and 0.95 to W1 in the fitness function. According to Table 15, the SBCSO algorithm obtains the lowest classification error value in five out of seven benchmark datasets for W1 = 0.95. Meanwhile, in the DS04 dataset, the BGA algorithm obtains a better result than the SBCSO algorithm in the classification error criterion for W1 = 0.95. Besides, it gets similar results using the SBCSO algorithm in two datasets of DS01 and DS05. BCSO and BGWO algorithms obtain the same result as the SBCSO algorithm in terms of the classification error criterion for W1 = 0.95 in the DS01 dataset. Also, in terms of fitness, the BMPA algorithm obtains first place compared to other algorithms in the value of W1 = 0.95. Ultimately, as shown in Table 15 and Fig. 29, SBCSO, BWOA, BHHO, BMPA, BGA, BCSO, BPSO, and BGWO Fig. 20 The curve of average fitness decline for the DS07 dataset for different algorithms Fig. 21 The curve of average fitness decline for the DS02 dataset for different algorithms A self-adaptive binary cat swarm optimization using new time-varying transfer function for gene… 7983 algorithms are ranked first to eighth in the overall ranking of criteria, respectively. In order to have a quick examination of the algorithms' results, the comparison between all of the algorithms for all of the datasets is conducted in Fig. 29. In the ranking of the algorithms, the white box indicates the first rank and the black box indicates the last rank among all.
In this regard, Figs. 30, 31, 32, and 33 exhibit different algorithms' fitness and classification error reduction curves. It can be observed that the speed of convergence and the slope of the reducing curve in the SBCSO algorithm are higher than in other algorithms (Fig. 34). In this section, the results of the multi-objective SBCSO algorithm procedure are examined. In subSect. 3.4, a discussion has taken place regarding the objective functions and the multi-objective SBCSO algorithm procedure. The proposed algorithm's results have been compared with BNSGA-II, BMODE, BMOCSO, BMOPSO, and BMOBBA algorithms. Given that the Pareto-optimal front related to the datasets is not available, the two D and HV criteria were used for the results' assessment. These criteria are briefly explained as follows. Fig. 24 The curve of average fitness decline for the DS10 dataset for different algorithms Fig. 25 The curve of average fitness decline for the DS13 dataset for different algorithms A self-adaptive binary cat swarm optimization using new time-varying transfer function for gene… 7985 The hypervolume (HV) criterion is used to assess the convergence velocity of the proposed algorithm toward the optimal Pareto front. The HV indicates the dominated space value. A large amount of the HV demonstrates that the Pareto front level is closer to the optimal Pareto front.
The spread metric (D) criterion is used for diversity assessment in the Pareto front level. A small amount of the D indicates that the Pareto front level is more ordered. Fig. 26 The curve of average fitness decline for the DS12 dataset for different algorithms Fig. 27 The curve of average fitness decline for the DS04 dataset for different algorithms Fig. 28 The curve of average fitness decline for the DS05 dataset for different algorithms  The comparison results between two D and HV criteria for 20-times independent execution of each algorithm are reported in Tables 16, 17, 18, and 19. As indicated in Tables 16, 17, 18, and 19, the SBCSO algorithm has obtained better results based on the HV criteria.
Accordingly, it demonstrates that the SBCSO algorithm can obtain more non-dominated solutions rather than the other algorithms, not to mention that it has a higher convergence velocity. Regarding the D criterion, the SBCSO algorithm has obtained better answers in DS05, DS06, DS08, DS09, DS10, DS11, DS12, DS13, DS14, and DS15.  The BNSGA-II algorithm in the DS03 and DS04 datasets, the BMODE algorithm in the DS01 and DS07 datasets, and also the BMOPSO algorithm in the DS02 dataset have gained better answers based on the D criterion.
To further examine the results of Table 16, the results of statistical tests comparing different algorithms are presented in Table 18 and Table 19. Table 18 shows that SBCSO is the best performing algorithm in the comparison, with a rank of 1.33 and 1.12 for the Friedman and Quade tests, respectively. Figures 35, 36, 37, 38, 39, and 40 demonstrate the Pareto front diagram obtained by different algorithms. In these figures, the vertical axis represents the classification error and the horizontal axis represents the number of the selected features for each dataset. As indicated in the figures, the SBCSO algorithm has a faster convergence to the POF non-dominated solutions (at the same time compared to other algorithms). The multi-objective SBCSO algorithm simultaneously minimizes the number of selected features and the classification error.

Conclusions and future work
This article proposes a new version of cat binary optimization algorithms, titled SBCSO, in order to select genes at DNA microarray expression cancer data. The high number of genes against the low number of samples has always been a challenge to microarray technology. The proposed algorithm consists of four main sections. In the first section, an opposition-based learning (OBL) mechanism is employed for population members' diversity. In the second section, a new time-varying V-shaped transfer function that varies by time is employed. In the third section, the MR and k of the proposed algorithm are adapted over time. And in the fourth section, a single-objective and multi-objective approach is proposed in order to solve the gene selection problem. The 15 datasets pertinent to the microarray data of different types of cancer have been employed in order to compare the proposed method with other methods. The results of the experiments indicate that the proposed algorithm is highly capable of selecting optimal gene sets required for the faster diagnosis of a vast majority of diseases. One of the main advantages of the proposed method is the automatic adjustment of the balance between the two phases of exploration and exploitation in the optimization process, and automatically escaping from the local optimum trap (i.e., stagnation). This automatic adjustment is performed based on the information taken from the optimization environment. On the other hand, one of the disadvantages of the proposed method is the large number of adjustable parameters in the SBCSO algorithm. This algorithm contains four parameters (i.e., PMO, CDC, SPC, and SMP) in the seeking phase and      two parameters (i.e., c and x) in the tracing phase. However, future works could be conducted to automatically update seeking phase parameters, tracing phase, MR, and k parameters based on the fuzzy system. In addition, the impact of chaos mapping on the initial population and the adjustment of some parameters based on chaos mapping can be investigated as a future research.
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