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Enhanced Ali Baba and the forty thieves algorithm for feature selection

Feature Selection (FS) aims to ameliorate the classification rate of dataset models by selecting only a small set of appropriate features from the initial range of features. In consequence, a reliable optimization method is needed to deal with the matters involved in this problem. Often, traditional methods fail to optimally reduce the high dimensionality of the feature space of complex datasets, which lead to the elicitation of weak classification models. Meta-heuristics can offer a favorable classification rate for high-dimensional datasets. Here, a binary version of a new human-based algorithm named Ali Baba and the Forty Thieves (AFT) was applied to tackle a pool of FS problems. Although AFT is an efficient meta-heuristic for optimizing many problems, it sometimes exhibits premature convergence and low search performance. These issues were mitigated by proposing three enhanced versions of AFT, namely: (1) A Binary Multi-layered AFT called BMAFT which uses hierarchical and distributed frameworks, (2) Binary Elitist AFT (BEAFT) which uses an elitist learning strategy, and, (3) Binary Self-adaptive AFT (BSAFT) which uses an adapted tracking distance parameter. These versions along with the basic Binary AFT (BAFT) were expansively assessed on twenty-four problems gathered from different repositories. The results showed that the proposed algorithms substantially enhance the performance of BAFT in terms of convergence speed and solution accuracy. On top of that, the overall results showed that BMAFT is the most competitive, which provided the best results with excellent performance scores compared to other competing algorithms.


Introduction
The world is currently witnessing the spread of the Internet among people, governments and organizations. Due to this, a substantial amount of data is engendered on a daily basis in several forms, with these datasets usually being an extension of knowledge [1]. In this respect, the society can produce data from various areas of life such as agriculture, health, education, finance and industry and many more fields [2,3]. These datasets can be classed and thereupon utilized for knowledge, predictions and insights. Raw datasets are often futile and not always ready for use until a suitable automated method is applied to draw out valuable information from them. Oftentimes, traditional data analysis methods cannot be used effectively to process immense volumes of datasets. In some cases, the unconventional nature of the data means that traditional methods are hard to use ably and adeptly even if the size of the dataset is relatively small. In some other cases, questions that require an answer could not be addressed using existing data analysis procedures, and thereby, new techniques need to be used [4]. On account of the above debates, Data Mining (DM) conceptions and principles could be exploited to locate the perceptive patterns behind the raw datasets [5]. DM is a general process that consists of a series of transformation steps, from a pre-processing step of the data to a post-processing step of the results generated by a pre-built DM method [6]. Thus, data preparation methods are a basic request to fetch the most out of the data, and thus create a classification model with an optimum search performance. Often, aggregated raw datasets may contain redundant, irrelevant or unimportant data, as DM methods mandate the use of explicit and clean data [7]. In this, there is a necessity to apply pre-processing stages to the collected data to refine and prepare it for further stages of machine learning techniques [8]. Data preprocessing is probably the most laborious and time-consuming step in the comprehensive knowledge discovery process [9].
From these prospects, this study chiefly highlights on Feature Selection (FS) to cope with the aforementioned hurdles. FS is a key pre-step that seeks to exclude unnecessary data, whether unrelated or redundant, from the dataset to be processed [10]. In other words, FS techniques mainly intend to improve the efficacy of classification accuracy by identifying the smallest subset of proper and related features from the initial wide range of features of the datasets. On that account, the virtues of FS extend from reducing over-fitting and data dimensionality, enhancing accuracy to improve insights and lessening complexities within a dataset, among many other benefits. The complex and burdensome nature of most real-world problems demands abundant solution space on account of interdependence and non-linear needs between the datasets' features [11]. In the course of creating different subsets, there is a necessity to test each created subset so that one can decide what fits to an evaluation method such as minimizing error or maximizing classification rate [12]. This process is computationally expensive, exhaustively impractical, and requests a great deal of computational burden, especially for high-dimensional datasets. Thereupon, traditional FS methods are incapable of dealing with such convoluted problems. For example, filter-based techniques have crucial issues ranging from their inability to enlarge consumption time, to providing adequate performance, complexity, and much more issues. These and other challenges have prompted researchers to explore various other ways to obtain superior performance options during classification tasks [13]. Hence, the quest for better methods for FS problems with optimal search performance has been directed to the use of meta-heuristics [4].
Meta-heuristic methods have confirmed their broad suitability in various fields of science due to their general efficiency in addressing practical problems in reasonable time with low computational efforts [14]. Their peculiarity of surmounting the curse of dimensionality by improving the performance rate of classification, extenuating the large use of computational resources and the number of features are other virtues of their use [4]. Fitly, a reliable and effectual meta-heuristic is needed to solve FS problems with a high accuracy rate, quick processing and few evaluations. As there is a huge amount of works on the use of meta-heuristics for FS problems, related works are reviewed later in another section.
The pressing need for FS in various domains as well as the breathtaking features of meta-heuristics have enlivened to use a newly developed algorithm to solve a variety of FS problems. This algorithm is called Ali Baba and the Forty Thieves (AFT) [15], which has been adopted here to evolve a binary version, referred to as Binary AFT (BAFT), to solve FS problems. This is the first motive of this work. As FS problems are challenging, a robust meta-heuristic is needed to efficiently solve such problems. This is the second and most important impetus of this paper, in which three enhanced versions of AFT were developed, briefly described as follows: (1) As AFT is a population-based iterative method that uses the information created by interactions among the individuals. However, its interactive strategy is modest so that individual's search capabilities are bounded. As per this, a hierarchical population structure called multi-layered AFT was constructed to direct individuals, which could be suitable for the individuals to boost their search capabilities. (2) Elitist AFT (EAFT) was drafted in this work through the development of an Elite Learning Strategy (ELS). The ELS in AFT is devised to be implemented merely on the globally best search agent and merely in a convergence status. The adaptive ELS could preserve the diversity of the population to jump from the potential local optimum. (3) A Selfadaptive AFT (SAFT) that encodes the tracking distance parameter in AFT and optimizes this parameter along with the position of the individuals during run time is proposed. In this iteration-number-based strategy, the exploration gradually fades out and exploitation gradually fades in. This process may boost both population diversity and search effectiveness.
To assess the performance of the proposed binary versions of AFT, they were applied to 24 FS problems drawn from publicly available datasets. Then, a comparison was made between the best proposed version of AFT as well as the basic one with a set of promising algorithms applied to FS problems. The arrangement of the rest of this paper is laid out as follows: Sect. 2 presents a brief review of some of the existing literature on FS methods. Section 3 introduces the AFT algorithm. The proposed variants of AFT are fully detailed in Sect. 4. The proposed FS approach is presented in Sect. 5. Section 6 then provides a performance comparison between the basic and proposed FS methods. A comparison of the basic, the best proposed method, and other meta-heuristics is also given in Sect. 6. Finally, Sect. 7 reaches a conclusion and gives insightful directions for the future.
The FS method presented in this work is a wrapper-based technique. Therefore, filter-based FS and hybrid filterwrapper-based FS methods are outside the scope of this work. Numerous meta-heuristics have been expansively adapted by many academics to design wrapper-based approaches to tackle various types of FS problems in a variety of domains [17,18]. The wrapper-based methods use machine learning methods like k-Nearest-Neighbor (k-NN) and Support Vector Machine (SVM) to evaluate the fitness of the resulting subset of features. The following is a review of some meta-heuristics classified according to the category of algorithms used, which have been used to design wrapper-based FS methods similar to those presented in this work, and which has given sensible performance in the literature.

Swarm intelligence-based FS
Swarm Intelligence (SI) optimization methods have been widely used to solve FS problems of diverse complexity and dimensions and have shown encouraging results in large-scale problems [19,20]. In a promising study, the most distinct features were selected by Chaotic Dragonfly Algorithm (CDA) in which ten chaotic maps were embedded into the optimization process of DA in order to fine-tune the key parameters of dragonflies' movements [21]. CDA was used to identify features of datasets pulled from the Drug bank database. CDA consisted of a synthetic minority over-sampling technique, as a pre-step, to address the issue of an unbalanced dataset, a features selection stage to select the most distinguishable feature, and an SVM for classification tasks. The obtained results affirmed the ability of CDA compared to DA and other metaheuristics-based FS methods.
Anand et al. [22] used Chaotic Selfish Herd Optimizer (CSHO) with various chaotic maps to solve FS problems. The obtained results demonstrated the efficiency of CSHO in getting the optimum feature subset by achieving a high level of accuracy associated with minimal number of features compared to other meta-heuristics-based FS methods. Xian-Fang et al. [23] developed a FS algorithm, referred to as HFS-C-P, with three stages, described as follows: (1) in the first phase, the non-significant features were excluded using C-relevance, (2) the Mth feature cluster was utilized to aggregate similar features into the same cluster in the second phase, and (3) an amended variant of Particle Swarm Optimization (PSO) was employed to identify the optimal feature combination. Xian-Fang et al. reported that HFS-C-P realized encouraging results on 18 datasets with respect to fitness values, number of selected attributes as well as the computational burden time, in all of the problems considered and better than those realized by other similar methods. Pashaei and Pashaei [20] proposed a FS method on the basis of the Chimp Optimization Algorithm (ChOA) for classifying biomedical data. Two variants of this algorithm were evolved for FS problems. In the first variant, binary ChOA was obtained from the continuous ChOA with the use of Sand V-shaped transfer functions. A crossover operator was utilized in the second variant to ameliorate the exploration conduct of ChOA. The efficacy of the proposed approaches was validated on high-dimensional biomedical datasets and other datasets from diverse fields. The results of the binary ChOA divulged better classification accuracy than other FS algorithms in most test cases.
An enhanced binary version of the Rat Swarm Optimizer (RSO) integrated with PSO's local search model was developed to address 24 FS problems [19]. In this method, three crossover operators were integrated and managed by a switch probability to boost the diversification of their solutions. Whilst these FS methods revealed promising performance in terms of many relevant criteria, their convergence rates are rather modest. Another good work was evolved using binary versions of the Horse herd Optimization Algorithm (HOA) in [4]. In that work, three various types of transfer functions were employed to transform these versions from continuous to binary. These versions were strengthened with three kinds of crossover operators, where 15 versions of binary HOA obtained, where they were tested on 24 FS datasets. The version with a single crossover operator and an S-shape transfer function has provided the best results among all the other BHOA's versions and other FS methods out there. However, implementing these versions requires significant computational efforts.
The above FS methods have fulfilled reasonable performance levels and efficacy results in providing potent solutions to FS problems aimed in the studied datasets. They have also demonstrated that they are able to locate the minimum number of features from a specified subset. However, some FS methods like those presented in [4,24] suffered from premature convergence and local optima. Moreover, they gave reasonable outcomes in some considered problems but not all problems.

Evolutionary algorithms-based FS
Evolutionary Algorithms (EAs) have been extensively adapted to evolve applicable methods for solving FS problems with reasonable performance using low computational effort [25,26]. Awadallah et al. [27] designed a FS algorithm based on a Binary JAYA algorithm named BJAM, with the assistance of an adaptive mutation operator managed using a predetermined mutation rate to diversify the population. The binary variant of the JAYA algorithm was collected using an S-shape transfer function.
To obtain the optimal settings for the adaptive mutation rate used, extensive experiments were performed. The proposed BJAM was examined on 22 FS problems, where promising performance was realized compared to other FS methods. A Multi-Objective Binary DE with self-learning, referred to as MOFS-BDE, was designed by Zhang et al. [28] for FS problems. A mutation, one-bit purifying search and non-dominated sorting operators were combined with MOFS-BDE to improve its search performance. The results showed that the performance of MOFS-BDE is competitive against some hopeful FS methods. In a good aspect of improving EAs to handle high-dimensional FS problems to mitigate the curse of dimensionality, Cheng et al. [29] presented a Steering-Matrix-based MOEA, named SM-MOEA, where a steering matrix was exploited to direct population evolution to get high quality feature subsets. Dimensionality reduction and individual repairing operators were used in this matrix to powerfully guide the population evolution in each generation. Further, an update strategy for the steering matrix is also designed to further improve the performance of SM-MOEA. The results of SM-MOEA on 12 high-dimensional problems showed that it outperformed several FS algorithms with respect to the number and quality of selected features.

Physics-based algorithms -based FS
Physics-based (PB) algorithms have been widely adopted to develop viable methods to handle FS problems with a respectable level of performance that requires little computational burden [30,31]. Yan et al. [32] introduced an improved binary coral reefs optimization algorithm, referred to as BCROSAT, to select the best feature subsets. A tournament selection scheme was adopted to augment the diversity of the population. Experimental results of BCRO-SAT on 13 public medical datasets illustrated that BCRO-SAT outperformed other state-of-the-art FS methods. Gharehchopogh et al. [31] embedded various chaotic maps with the search process of a binary version of the Vortex Search Algorithm (VSA) for solving FS problems. The results of that method on 24 FS datasets showed that chaotic maps are able to enhance the performance of VSA. The results clearly showed that the fitness of the proposed method is very low in achieving the optimal feature subset with utmost accuracy and minimum number of features, and it has a higher percentage of accuracy compared to other FS algorithms. Elgamal et al. [30] proposed Chaotic Harris Hawks Optimization (CHHO) for FS problems. In CHHO, two amendments were conducted, the first one is the use of chaotic maps to enhance population diversity, and the second one is the application of Simulated Annealing (SA) algorithm to improve HHO's exploitation. Based on the results obtained, CHHO affirmed its preeminence compared to the standard HHO algorithm and other algorithms in the majority of FS problems. The performance of CHHO was applied to 14 medical datasets, and this algorithm was compared with the original HHO and some famous and recent meta-heuristics.
In several contexts, many hybridization models of PB algorithms with EAs or SI algorithms have been introduced in the literature to address FS problems. Chantar et al. [33] solved FS using a hybridization of an improved Binary version of DA and SA (BDA-SA). In this hybrid approach, SA was used to enhance the ability of DA in selecting the best subset of features for classification tasks. A set of datasets commonly used in this field was used to assess the performance of BDA-SA. Results of BDA-SA showed that this approach has superior performance compared to the basic BDA and other wrapper-based methods. Last but not least, a hybrid version of the HHO algorithm based on Bit-wise operations and SA (HHOBSA) was presented in [34] to solve FS problems for classification tasks. AND and OR Bit-wise operations were applied to randomly transfer the most useful features from the best solution to others in the population to boost their goodness. HHOBSA was evaluated on 24 benchmark datasets and 19 artificial datasets of different dimension sizes, where it presented superior results in terms of several measures compared to other methodss.

Human-based algorithms for FS
Human-based algorithms, of which the considered AFT algorithm belongs to this class, are slightly used for solving FS problems, although they have been used successfully to solve various types of continuous real-world problems. Some of the new human-based algorithms used effectively to develop wrapper-based methods in binary domains can be found in [35][36][37]. Agrawal et al. [35] evolved an amended version of the Gaining-Sharing Knowledge (GSK) method to identify the best features in addressing FS problems. First, a Binary variant of the GSK algorithm was represented using a probability estimation operator (Bi-GSK) on the two key pillars of this algorithm. Next, ten different kinds of chaotic maps were used to adapt the parameters of this method. The performance of the evolved binary methods was evaluated on 21 datasets. The results in terms of different types of related measures indicated that Chebyshev chaotic map performed the best among all other chaotic maps and outperformed the original algorithm and other meta-heuristics. In another study, Agrawal et al. [36] proposed two methods using GSK to solve FS problems: 1) FS-BGSK, a binary variant of the GSK method that employed knowledge agent 1, and 2) FS-pBGSK, a method that used a scheme for population reduction to strengthen exploration and exploitation of FS-BGSK. These methods were validated on 21 FS datasets, where the obtained results indicated that these methods surpassed other ones in most of the datasets with respect to accuracy, convergence behavior and solution quality.
Although meta-heuristics do not guarantee that optimum solutions will be found in all experimental runs, they can score relatively credible solutions in a timely manner. This implies that finding the optimum feature subset is not assured. This may be ascribed to the fact that metaheuristics may drop into local solutions when solving highdimensional problems. This stimulated this study to ameliorate the AFT algorithm by proposing three enhanced versions of it, and using binary variants of these versions to explore their abilities to solve well-known FS datasets.

Basic AFT algorithm
The mathematical model of the basic AFT algorithm, as presented in [15], comprises of three states that can be modeled and described as follows: Case 1 The position of the thieves when pursuing Ali Baba based on the information they got from someone can be simulated as shown below: þ Td t y t i À m t aðiÞ r 2 sgnðrand À 0:5Þ where x tþ1 i is the position of the ith thief, y t i stands for Ali Baba's position regarding thief i, best t i identifies the best position arrived at by thief i, gbest t is the global best position got up to now by any thief, m t aðiÞ stands for Marjaneh's wit level employed to disguise thief i, Td t is the tracking distance of the thieves as defined by Eq. (2), Pp t is the thieves' potential perceptual ability for Ali Baba as defined in Eq. (3), r 1 , r 2 , rand, p and q are random values created within [0, 1], p ! 0:5 presents either 0 or 1, sgnðrand À 0:5Þ can be -1 or 1 and a can be defined as shown in Eq. (4).
where t and T refer to the present and maximum iteration values, respectively, s 0 (s 0 ¼ 1) denotes an initial estimate of the tracking distance and s 1 (s 1 ¼ 2:0) is a fixed value used to control exploitation and exploration.
where k 0 (k 0 ¼ 0:1) is the final approximation of the probability that the thieves will achieve their goal at the edge of the search process and k 1 (k 1 ¼ 2:0) is a fixed value used to control exploration and exploitation.
where f ðÁÞ represents the fitness function's score. Case 2 Thieves may realize that they have been duped, so they will scout new random areas as defined below: where r is a random figure created in the range [0, 1], u j and l j denote the upper and lower bounds of the search space at dimension j, respectively. Case 3 Thieves could search in positions other than those found using Eq. (1) to further boost the exploration and exploitation sides of AFT; this case can be drafted as follows: The iterative pseudo-code steps of the basic AFT algorithm can be summarized as shown in Algorithm 1.

Issues of the basic AFT algorithm
Even though AFT can promisingly search for optimum solutions for real-world roblems, its search capability is limited due to its native mathematical models, which may result in local optima. This phenomenon is manifested in that AFT is likely to experience early convergence. The key is that the thieves of AFT rely on their memory and their best position to regularly cooperate and search, but their memory cannot ensure that they will not be locked into a local optima or they can flee from local slump. Thus, there must be some mechanisms that help thieves' memory.
In addition to the case of early convergence, AFT faces another problem of low exploration and exploitation capabilities. An adequate exploration process can find an encouraging area where the optimum solution prevails in the whole search space. Then, adequate exploitation can eventually identify this optimum solution. However, the exploration capacity of AFT in the foremost search stage is inadequate, so the weak exploitation ability of this algorithm devises it hard to identify the global optimal solution in the latter search stages. In such a situation, local optimum is in general derived. Thus, a proper balance should be constructed between exploration and exploitation conducts for the purpose of strengthening its search capacity. In AFT, thieves' searches for Ali Baba are carried out through the incorporated memory concept of the best thieves and global best position offered by the best thieves. This strategy can enhance understandings between thieves, and the best thieves act a pivotal role in guiding and assisting them. However, the best thieves can speed up the process of early convergence of the population once they are caught in the local optima, and without any ability to block or remedy this situation. Thus, it is inappropriate to reinforce the best thieves only utilizing their memory. Besides, the parameter Td t progressively reduces with time, which means that the best thief's abilities to direct all the thieves are diminished so that the exploitation of AFT is depressed more in the last iterations of the search process. Simultaneously, it is difficult for them to assist the thieves escape from the local optima. Thus, it is essential to enhance the exploitation capability of AFT in the late stages. This is not only to speed up the convergence rate of AFT, but also to avert suffering from early convergence.
To deal with the above issues, three enhanced versions of AFT were proposed as shown below.

Multi-layered AFT
A hierarchical population topology structure was utilized to promote the underlying AFT algorithm, so that the early convergence process that might be encountered could be mitigated. Moreover, the thieves will possess a great chance to get away from the local optima and approach the global optimum solution. This Multi-layered hierarchical structure of the basic AFT (MAFT) is formed as shown below [15]: 1. Hierarchical framework: A four-layer hierarchical architecture is created in MAFT, which can be described as below [15]: • Population topology layer (Layer 1): All thieves who make up the population are placed at this layer. The movements of the thieves and distribution of their positions are carried out at this layer by the other layers mentioned below. Accordingly, this layer principally provides a complete search space for the population of the MAFT algorithm, and all thieves can constantly change their locations during the search process. • Best iteration layer (Layer 2): This layer intends to identify many of the best thieves in the current population to direct the population topology layer (i.e., Layer 1). The M best thieves are picked at this layer, where this means that the best iteration layer (i.e., Layer 2) entices Layer 1 to realize the exploration and exploitation aspects of MAFT. Thieves constantly change their positions at Layer 1 with the assistance of the parameters (i.e., Pp and Td) provided by Layer 2. Additionally, the number of thieves at this layer dynamically alters as a result of the linear descend of the M best individuals over time. • Local best layer (Layer 3): This layer was latterly created for the purpose of reinforcing exploration capacity and prohibiting the M best thieves from falling into the local optima. Each thief has his personal best historical position while he is on the search process, which was named as the best thief. This thief denotes the best solution got so far. Because the M best thieves who only change their positions in accordance with their tracking distance (Td t ) are only liable to bear from early convergence, there is a need to find other thieves who will assist them avert or flee from the local optima. So, the M personal best thieves who corresponds to the M best thieves are picked to give more details to further direct the thieves at Layer 2, where each personal best thief specifically directs the corresponding thief. In the same way, the number of thieves at Layer 3 alters in compliance with the thieves at Layer 2. • Global best layer (Layer 4): The exploitation feature of MAFT for the aim of helping the population to divest of the local optima and to speed up the convergence process occurs at this layer. The global best search agent (i.e., thief) in this layer is adopted to direct the populations on Layer 1 more. The global best thief points to the best solution in the population has come up with so far. Accordingly, the effect of this layer on Layer 1 is that all thieves are heading in the direction of the best global thief so that a better solution could be located.
As an illustrative example, Fig. 1 axiomatically shows the hierarchy of the proposed MAFT algorithm. At Layer 1 in Fig. 1, there are N thieves (i.e., x 1 ; x 2 ; . . .; x N ), and at Layer 2, there are M best thieves (i.e., I 1 ; I 2 ; . . .; I M ). The population at Layer 1 is pulled by Layer 2 according to the parameters of AFT (i.e., Td t and Pp t ). Concurrently, there are M best thieves at Layer 3 (i.e., best 1 ; best 2 ; . . .; best M ), and at Layer 2 there is the global best thief (i.e., gbest). Layer 3 routes Layer 2 by the best thieves one to one (i.e., best 1 directs I 1 , best 2 directs I 2 , and so on). The global best layer guides the population topology layer more, where this means that gbest provides additional guidance for the population.
2. Hierarchical Interactivity: The number of types of hierarchical activities among the four layers of MAFT is three, as characterized below [15]: (a) The interactivity between Layer 1 and Layer 2: The M best thieves at Layer 2 allure all thieves at Layer 1 by the tracking distance and potential perceptual between them. Appropriately, the interactivity between these two layers adheres to the principle of the original AFT algorithm. This is to provide effective leadership to the search process of the thieves, where the best predefined thieves are rated at this layer. At each iteration loop, Layer 2 instructs Layer 1 to update the positions of the thieves. Properly speaking, every thief demands a large value of Td t and a small value of Pp t . With this, the entire search space can be explored globally in the initial iterations, and these parameters are progressively altered over the iteration loops. This implies that AFT's exploration needs to be boosted by a suitable large Td t , while the exploitation ability needs to be strengthened by a small Td t . Thus, to ameliorate the exploration in AFT, a new formula for the tracking distance, Td t , was used to replace the native formula given in Eq. (2). In MAFT, a log-sigmoid function was utilized to devise a mathematical model for Td t with the formula defined in Eq. (8) [15].
where Td 0 denotes the largest tracking distance, which is equal to 2, s is a threshold value indicating the center point of a curve as per the maximum count of iterations, and L identifies the inclination of the curve. After extensive experimental testing, it was found that with s ¼ 2 and L ¼ 100, reasonable results could be got. (b) Interactivity between Layer 2 and Layer 3: In order to efficiently direct the best thieves, the personal best ones are employed. Each personal best thief at Layer 3 particularly guides his corresponding one at Layer 2. This type of oneto-one control can enhance the search property of all thieves more in compliance with their historical best positions. This strategy can be presented as shown below [15]: where g i specifies the updated position of the ith best thief, randðÀ1; 1Þ represents a uniform random value generated in the range ðÀ1; 1Þ, and P i opt stands for the ith personal best thief corresponding to the ith best thief at Layer 2. (c) Interactivity between Layer 4 and Layer 1: A global best thief manages the population to improve the exploitation capacity of population Fig. 1 Illustrative structure of the proposed MAFT algorithm [15] Neural Computing and Applications and speed up the convergence of MAFT in the late search stage. The concrete control is carried out as shown below [15]: where c i represents the updated position of the directed thief x i , g opt refers to the global best thief and randð0:5; 1Þ stands for a uniform random number that lies in the range (0.5, 1). As mentioned above, the interactivity between Layers 4 and 1 is to principally foster the performance degree of the population in the lagging search process. In consequence, a reverse sigmoid function is applied to characterize the interactivity duration between the other layers.
To be more specific, the interactivities between Layers 1, 2 and 3 pay attention to the aspect of exploration, while the interactivity between Layers 4 and 1 focuses on the exploitation aspect. A random value in the range rand(0.5, 1) assures that the thieves can proceed rapidly around the global best thief. On this account, the population can gain further speeds to further enhance its position through the interaction between these two layers. In conformity with the three types of hierarchical interactions between the four layers, thieves eventually move and search for Ali Baba with their updated position as presented in Eq. (11) [15].
In consequence, new exploration positions efficiently smooth and expedite the movement of thieves so that the population can conduct better during the search process. In this, Eq. (11) was proposed in MAFT to update the position of the thieves on Layer 1 which could efficaciously mitigate the premature convergence rate of AFT and ameliorate its performance score.

Elitist AFT
The proposed Elitist AFT (EAFT) is combined with an Elitist Learning Strategy (ELS) that is applied to the globally best individual to assist leap out of local optimum areas when the search process is determined to be in a convergence state. Opposite to other individuals (i.e., thieves), the global commander does not have exemplars to pursue [38]. It demands new impetus to ameliorate himself. Due to this, a perturbation-based EAFT was evolved to assist gbest propel himself out into a possibly better area. If another better area is located, then the remaining of the individuals will pursue the commander to jump and head to a new area . The EAFT selects one dimension of gbest's historical best position at random denoted by P d for dimension d. One dimension is only chosen due to that the local optimal is more probably to have a good structure of the global optimal, and therefore, this should be protected.
Since each dimension has the same probability of being selected, EAFT can be considered to operate on each dimension in a statistical manner. Accordingly, elitist learning is implemented using a Gaussian perturbation: The search scope L d min ; U d max h i is the same as the lower and upper limits of the problem under study, the Gaussian l; r 2 ð Þ represents a random value of a Gaussian distribution with mean l of zero and a standard deviation (SD) r, which is called the ''elitist learning rate''. In this, it is proposed that r decreases in a linear direction with the iteration value, which is defined by: where r L and r U stand for the lower and upper limits of the parameter r, which implements the learning scale to arrive at a new area in the search space. The experimental study showed that r L ¼ 0:1 and r U ¼ 1:0 perform well in the majority of the feature selection problems considered (Fig. 2). The ELS used in the proposed EAFT is described as shown in Fig. 3. In a statistical way, a decrease in SD offers a larger learning rate in the former phase for gbest to leap out of a potential local optimal, while a smaller learning rate in the latest stage directs the gbest to improve the solutions. In the proposed EAFT, the solution of the new position will be counted on if and only if the fitness value of the updated position is more better than the current gbest. Conversely, the new position is employed to supersede the individual with the worst fitness value in the population of individuals.

Self-adaptation of the parameter Td of AFT
The individuals of the AFT algorithm who fall into local optima need a good exploration aspect to get rid of these local optimums. Considering this, some individuals demand a relatively big tracking distance for the task of exploring the search space and other individuals require a small distance for exploiting the search space. To deal with such issues, a Self-adaptive tracking distance of AFT, or so-called SAFT, was used, where the formula given in Eq. (8) was used for Td t in SAFT. This distance indicates that the exploration feature fades away step by step and the exploitation feature fades in step by step. In the event that the fitness score of an individual is worse or does not alter, this designates that the individual can determine the local optimum region of the search space. Accordingly, a large tracking distance is required for individuals to get away from this region. If an individual's fitness improves, this points out that the individual possesses a high opportunity of approaching towards the optimal solution. As a result, the tracking distance value at iteration t must be increased to enhance the tracking distance of the individuals. A demonstration of the self-adaptive tracking distance in the proposed SAFT is shown in Fig. 3 The spheres, pentagonal, arrowhead and circle shapes in Fig. 3 represent the local optimal region, the global optimal region and the tracking distance and individual, respectively. In this figure, Case A results in that if an individual falls into local optimum region, it requires an adequate tracking distance to promote its ability to break out of that region. In contrast, Case B shows that an individual quickly transfers to the global optimum region with an ascending tracking distance. The counters, cs i and cf i , given in Eqs. 14 and 15, respectively, are used to record the fitness values of individual i in order to judge the status of the individuals in SAFT [15].
Equations 14 and 15 were developed to fine-tune the parameter Td t as per the fitness values of the fitness criterion, where they are used to evaluate the search state of the individuals. If an individual flops to have a better solution quality in several iterations, it falls into the local optimal region with a high probability. If an individual is modified in several iterations, it may go forward to the global optima. A probability q and a threshold h are used to manage the amendment of the tracking distance of individuals over the path of iterations. If cs t i surpasses h, the tracking distance will be extended to accelerate the convergence process of individual i towards the most powerful individual. Similarly, if cf t i exceeds h, the tracking distance is improved to boost SAFT's capability to avoid the local optimum during the search process. The self-adaptive Td t i of individual i is given as defined in Eq. (16) [15]: where the parameter count comprises of the counters cs t i and cf t i , and r z stands for a random number generated in the interval [0, 1].
In Eq. (16), when count passes h and r z transcends q, the individual demands an expanded tracking distance to enhance its exploration ability, as it is multiplied by r t i . In addition, there is another control parameter concerned with  Neural Computing and Applications the perception potential of the thieves for Ali Baba. Thereby, the tracking distance and perception potential constants of individual i were utilized to adapt the parameter Td t i . The ratio of these two parameters of the individuals can be applied as a value of the new tracking distance. Here, r t i represents the adjusted value of Td t i as defined below [15]: where the parameter v denotes the proportion of the tracking distance parameter to the perception potential parameter, which is given in Eq. (18) [15].
Equation 17 demonstrates that r t i is defined by the reciprocal of the parameter v when it is less than 1. This is to ascertain that the individual can possess a sensibly big tracking distance, otherwise, r t i is identified to v.

Proposed binary feature selection methods
In this study, four wrapper-based FS approaches were designed using a binary version of the basic AFT and three other proposed binary versions of AFT to solve a set of FS datasets. These methods intend to find out the optimal subset of features by finding out the most pertinent features and disposing the insignificant ones. AFT was originally introduced in [15] to deal with continuous optimization problems. In light of this, individuals of AFT have continuous solutions that must be converted into binary ones to solve FS problems with discrete search spaces. This is attributed to the familiar nature of these problems, where the search space of such problems can be specified with binary values. As per this, binary versions of AFT, MAFT, EAFT and SAFT were obtained in this work to develop a search strategy for FS methods. In this, some operators of these algorithms need to be amended to generate binary versions of them, which can give output in binary format as either '1' or '0'. To do so without modifying the structure of the algorithms, there is a requirement to use a Transfer Function (TF), as the way most often used in the FS field. The goal is to establish a probability that an element in the feature subset, x i , will be limited to a binary format that can be either '1' or '0'. The procedure for converting continuous AFT, MAFT, EAFT and SAFT into binary versions, referred to as BAFT, BMAFT, BEAFT and BSAFT, respectively, is performed using an S-shape TF as presented in Eq. (19), with the fitness function for these binary versions is explained in subsection 5.1.

S x ðm;jÞ
where S refers to the transfer vector, S x m;j À Á stands for the probability value of the S-shape value, x ðm;jÞ t stands for the jth element in the solution x at the mth dimension and t denotes the current iteration.
The values of the elements in the solution x are transformed to either '1' or '0' using Eq. (20).
where r is a function that creates a random value in the range [0, 1]. Equation (20) presents that the position of the current individuals will be updated based on the probability value S x ðm;jÞ t obtained from Eq. (19).

Fitness function
In wrapper-based FS methods, a learning algorithm is needed to assess the selected features. Furthermore, an objective function is needed in the design of any optimization problem, which implements a key aspect of any wrapper-based FS approach. In such a context, two master points must be taken into account when formulating FS as an optimization problem: (1) how to represent the solution, and (2) how to evaluate the solution. In this work, a feature subset is expressed by a binary vector of length equal to the number of attributes in the dataset. The feature selection problem is a multi-objective optimization problem that imposes two objectives as follows: (1) reducing the number of selected attributes, and (2) enhancing the classification accuracy reached by the classifier used. It is understood that as the number of features selected in a solution decreases, the quality of the solution is improved. With this in mind, the solution with the fewest number of attributes and which has the highest classification rate is the best solution to achieve. These two contradictory goals must be put forwarded in a single fitness function in FS methods. By this, these two opposing goals were represented in one fitness function for each solution in the proposed BAFT, BMAFT, BEAFT and BSAFT. The k-NN classifier [39] was used in this paper to evaluate the accuracy of classifying the solution as defined in the criterion given in Eq. (21).
where g t stands for the rate of classification error reached by k-NN, V j j and F j j represent the number of selected and native attributes in the dataset, respectively, a and b stand for two counteractive parameters in the range of [0, 1], where b is the complement of a, i.e. b ¼ 1 À a ð Þ, these parameters represent, respectively, the weights of the classification quality and selection ratio of the selected features.
The fitness function presented in Eq. 21 that considers the balance between minimizing the selected features in each solution vector and maximizing the classification accuracy of the learning model is used to assess the selected subsets of attributes of the proposed BAFT, BMAFT, BEAFT and BSAFT. In sum, three proposed binary versions of AFT were designed for FS problems in pursuit of arriving at the optimal number of features for each dataset with minimal error rate with the help of the classifier used.

Experimental results and comparisons
This section presents insights into the experiments conducted and the results obtained to demonstrate the effectiveness of the binary AFT algorithm along with other three enhanced binary versions of this algorithm when applied for the purpose of FS in well-known highly-regarded classification problems. The focus here is on demonstrating the intrinsic characteristics of the proposed algorithms in terms of exploration and exploitation. We not only present details of the experimental results, but also provide an indepth discussion and fair comparison with other wellestablished meta-heuristic techniques harnessed in the field of feature selection targeting classification enigmas.

Description of the medical datasets used
The performance of the basic and proposed algorithms of AFT was validated on 23 benchmark medical datasets collected from the UCI repository source [40], KEEL, Kaggle, and Arizona State University [41]. Table 1 shows a detailed description of these datasets, which cover a wide range of applications and have been extensively studied and used by many researchers in the literature [4,19]. For additional details about these datasets and others, interested readers are advised to access the UCI repository source.
From Table 1, the datasets consist of different numbers of features and samples with great diversity in the dimensions of these datasets. This is beneficial for testing the efficiency of the proposed algorithms as well their performance compared to other algorithms when used to address the same FS problems. These datasets are highly regarded by researchers in this field in that they have applications in diverse areas and vary in their complexity from low to medium moving up to high as can be seen from Table 1. This is particularly important in examining the exploration and exploitation properties of optimization algorithms utilized in FS approaches. That is, the performance of proposed algorithms was examined on FS problems with diverse structures, which can prove the effectiveness of these algorithms in several dimensions. The adopted dimensionality in the basic and proposed feature selection algorithms were based upon the number of features related to each individual dataset listed in Table 1.
To illustrate the datasets in Table 1 in more detail, the forth column in Table 1 indicates the available count of samples in the datasets. The third and sixth columns of Table 1 denote, respectively, the number of classes and dimension of each dataset. The fifth column indicates the number of features available in each of the 23 datasets. These features are the basis in any classification or clustering technique. However, not all of these features are necessary because in most cases there will be an abundance of redundancy in these features and using all the features during classification or clustering can only complicate and slow down the whole classification process. This is where feature selection methods come into play for selecting subsets of these features. This is an optimization game, because using a couple of these features on the one hand hurts the accuracy of classification and clustering techniques. On the other hand, using more than necessary of the features defies the very core of the feature selection method used. As is usual in the literature concerned with feature selection approaches, individual datasets are split into training and test sets, where 70% of the instances are randomly selected during the training process. This leaves 30% of the instances in each dataset to test the proposed feature selection method under experiment.

Real-world dataset
To add more challenge to the performance of the proposed algorithms on real-world FS problems, they were evaluated on the real-world Corona-virus disease (that is, SARS-CoV-2 or widely known as COVID-19) dataset. In it, the World Health Organization (WHO) announced that severe respiratory syndrome Corona-virus 2 (SARS-CoV-2) began attacking China in late 2019 and rapidly expanded throughout the world. Currently, artificial intelligence has become a breakthrough in current technology, and it can be applied in medical diagnosis, prevention and detection to fight against COVID-19. Here, the proposed algorithms were used to predict the health of a COVID-19 patient. A detailed description of this patient dataset is broadly available in [42]. This dataset consists of 15 features as in Table 2.
The aim of studying this real case study is to predict the conditions of death and recovery relying on the specified factors. Patient data with missing values for both ''death'' and ''recovery'' cases are eliminated from the master dataset. For the purpose of evaluation, the dataset of this case study was partitioned into two sets, where 30% of the dataset was employed for testing and the remaining 70% was employed for training.

Experimental setup
The binary version of the basic AFT and the proposed binary versions of the enhanced AFT were implemented with the help of the k-NN classifier with an Euclidean distance of 5. For this classifier, the input was the closest training states in the feature space, while the output represents the class membership. The labeling method is based on the majority of votes. Moreover, the parameters a and b of the objective function presented in Eq. 21 are set to 0.99 and 0.01, respectively. These values have often been used in many published works [4,19]. The lower and upper bounds used for each FS problem under study were set to 0 and 1, respectively. On top of that, the adopted dimensions for each problem were set based on the number of features related to each dataset entered in Tables 1 and 2. Table 3 shows the settings of the parameters of the proposed algorithms of AFT.
The performance of the basic and proposed algorithms of AFT was assessed with 30 independent runs with different random seeds for the purpose of substantiating the stability and robustness of these optimization algorithms in solving FS problems. The results I'll be showing were conducted on a PC with an Intel Core i3-7020U CPU, 2.30 GHz clock rate, and 4.0 GB of RAM. These experiments were implemented in MATLAB R2018a and run in a Microsoft Windows 10 Pro environment with 64-bit. To unitize the experimental setup for the basic binary AFT algorithm, the three proposed binary variants of AFT, and the state-of-the art optimization methods found in the literature, the same population size (this corresponds to the number of search agents in this context) and a fixed value of the maximum number of iterations were used for each of the experimented algorithm. Specifically, the adopted population size was 30 and a value of 100 was used for the maximum number of iterations. This ensures a fair comparison between the contestant algorithms.

Evaluation measures
The confusion matrix used to assess the k-NN classifier in the proposed FS algorithms in terms of accuracy is provided in Table 4.
To quantify the performance level of the proposed binary algorithms of AFT, a pool of six assessment evaluation measures was computed, which are defined as shown below: -Accuracy: this evaluation method was employed to correctly represent the classification of the data as defined by Eq. 22.
-Sensitivity: this metric computes the percentage of the prediction of positive patterns in accordance to the selected attributes of the best solution, as defined by Eq. 23.
-Specificity: this metric computes the percentage of the prediction of negative patterns that appear in the selected attributes of the best solution as defined in Eq. 24.
-Fitness: this criterion defined in Eq. 21 was used to compute the solution quality for the purpose guiding the search process for the optimization algorithms used. -Number of the selected features: this value is employed to reveal the capacity of the optimization algorithms in solving FS problems for the studied datasets where a small number of features is preferred. -Computational time: this metric indicates the computational time that the FS method takes during the iterative search process.   Symptoms of patients

Evaluation of the proposed algorithms
This subsection evaluated the three enhanced versions of the AFT algorithm along the basic one when used to solve FS problems. These algorithms are compared and contrasted to each other using various performance metrics such as the average classification accuracy (Acc), the average classification sensitivity (Sen), the average classification specificity (Spe), the average selection attributes (Att), the average fitness values (Fit), and the average computational CPU times. These measures were computed over all of the 30 independent runs of the evaluated algorithms. In addition to these metrics, the standard deviation (Std) values are also provided for each of the mentioned performance metrics related to the presented FS algorithms. The obtained results of the basic and proposed algorithms of AFT can be found in Tables 5, 6 , 7, 8, 9, 10, where Table 5 shows the results related to the Acc and Std for BAFT, BMAFT, BEAFT and BSAFT methods. It should be mentioned here that Acc corresponds to the selected features from the 24 datasets. Table 6 summarizes the statistical results of the proposed algorithms in terms of the average sensitivity and Std values, where Table 7 displays the results related to the average specificity alongside the Std values. Table 8 has the Fit and Std values. Table 9 contains Att and Std values. The average CPU times can be viewed in Table 10. In each of these tables, unsurpassed figures are displayed in bold for clarification and better comparison. It can be obviously observed from Table 5 that BAFT and the proposed binary algorithms of AFT successfully solved the feature selection problems under study. In regards to the standard deviation results of Table 5, they are very small which in many cases amount to zeros, which means that BAFT and the proposed binary algorithms of AFT are stable and this superiority is stable. Finally, due to the small difference between the accuracy results with problems in different number of features and complexity of Table 5, it can be concluded that the performance of BSAFT and BEAFT pales slightly around BMAFT, where the latter algorithm is still more settled and powerful against the curse of dimensionality, that is, it is generally steady with increasing the dimensionality of the problems.
The average sensitivity and standard derivation findings of the three proposed binary variants of AFT as well as the basic one are displayed in Table 6. A better optimization algorithm is indicated by a higher sensitivity value. The proposed BMAFT performs better than the other binary variants of AFT because it produced the best results in the majority of the datasets. The algorithm BEAFT, which was ranked second, produced results that were not better than those of BMAFT but better than those obtained by both of BSAFT and BAFT. Finally, BAFT came in last as it outperformed the proposed algorithms only in the Coimbra dataset. Table 7 displays the specificity results of the four proposed binary variants of AFT in terms of average and standard deviation. Better performance is a result of the higher specificity outcomes. It is evident that the proposed BMAFT achieved the best specificity results in the majority of datasets, while BSAFT and BEAFT ranked second and third, respectively, by achieving the best specificity results in those positions, and BAFT ranked last after BMAFT, BSAFT, and BEAFT. By reading the results in Table 7 once again, one may observe that BMAFT, BSAFT, BEAFT, and BAFT are stable by generating low standard deviation values in these datasets.
The goal of optimization algorithms is to reduce fitness functions. Table 8 shows the mean and standard derivation results of the fitness values got by the four proposed binary variants of AFT. A better algorithm's performance is indicated by a smaller fitness value. As shown in Table 8, the proposed binary algorithms, namely BAFT, BMAFT, BSAFT and BEAFT, showed reasonable low averages of fitness values correlated with reasonable low standard deviation values. In specific details, in 15 out of the 24 datasets, BMAFT performed better than BAFT and the other proposed variants, which put it in first place. Table 9 presents the average number of selected features and the standard derivation of the three proposed binary variants of AFT and the basic BAFT algorithm. Similar to classification accuracy and fitness values, the number of selected features is a significant metric used to assess the effectiveness of FS methods. One could say that choosing as few features as possible is best. Reading that BSAFT got the minimal Std results in 12 datasets, it can be said that it is more stable than other binary variants of AFT and the basic one in terms of the average number of selected features.
The time it takes for the algorithm to finish the calculations is of major importance. From this perspective, the run times for the basic BAFT and the other binary variants of AFT were measured in seconds, and the results were recorded in Table 10. The best of these proposed binary versions is BMAFT, which was able to reduce the execution times in 11 out of 24 datasets. However, in terms of the execution times, the proposed BSAFT algorithm did not perform as well as BMAFT, BEAFT and BAFT.  These results confirm the efficiency and reliability of BMAFT, BSAFT and BEAFT in striking reasonable exploration and exploitation behaviors as well as a good balance between these two aspects. It is noteworthy that higher sensitivity figures imply better performance degree. It can be noticed from Table 6 that BMAFT ranked first by recording the best exclusive sensitivity results in 15 out of the 24 datasets. The BEAFT algorithm was ranked second by presenting the best exclusive sensitivity outcomes in 5 datasets (i.e., Retinopathy, ILPD Lymphography Parkinsons COVID-19) and got the largest sensitivity results in a total of 6 datasets, where the sixth dataset is the Breast dataset. BSAFT came in third with the best exclusive sensitivity results only in 3 datasets, namely, BreastEW, Spectfheart and Heart, and obtained the largest sensitivity results in an overall of 4 datasets, by adding the Breast dataset to the previously mentioned three datasets. Also, it was reported sensitivity results similar to those obtained by BAFT and BEAFT in the Colon dataset with a sensitivity value of 85:33%, while the value reported by BMAFT is 86:00%. For the Coimbra  Table 6 again, one can underline that there is a relatively big difference between the results reported by BAFT and those reported by its proposed companions. Table 7 presents the specificity results of the basic binary and proposed binary variants of AFT in respect of their mean and standard deviation results. It is clear that BMAFT came first with the best exclusive privacy scores in 12 datasets and has the largest specificity scores in a total of 13 datasets. This version is respectively followed by BSAFT and BEAFT, where both had the best exclusive specificity results in 4 and 3 datasets and achieved the greatest results in a total of 5 and 4 datasets, respectively, with only a slight difference between these two versions in the results arrived at for the Leukemia dataset. For the Breast dataset, there is a very little difference between the specificity value reported by BMAFT and that reported by BEAFT, where the former having a specificity of 99:39% and the latter having a specificity of 99:38%. Regarding the COVID-19 dataset, both of BMAFT and BEAFT reported the highest specificity results of 98:29%, with the values reported by BSAFT and BAFT were 97:69% and 97:08%, respectively. Regarding the Std values shown in Table 7, either zero or very close to zero values can be noticed, which means that the robustness of the proposed algorithms is rooted.

Discussion of the results
The aim of the proposed binary algorithms of AFT for FS problems is to reduce the fitness values obtained by the criterion given in Eq. (21). This is embodied in Table 8 as the average fitness values, where this table contains the averages and standard deviations of the basic and proposed algorithms of AFT over 30 runs for all of the 24 FS problems. In general, from Table 8, it can be noticed that BMAFT is qualified to consistently and exclusively find the global optimal solution in 15 datasets and achieve the largest fitness results in a total of 16 datasets. Obviously, it came in the first rank by getting these findings. BSAFT ranked second by having the minimum mean fitness values in 4 problems exclusively, with the minimum fitness scores Although classification accuracy is very important for any classification algorithm, the number of features used in the classification process is no less important than any feature selection algorithm. In view of this, Table 9 shows the average number of attributes used in the classification of the 24 datasets for each of the proposed binary algorithms of AFT. In Table 9, one can notice a larger variation when the four algorithms are contrasted. BMAFT was the most successful algorithm in lessening classification features, where the average number of selected attributes was the minimum among the four proposed algorithms. It has the exclusive minimum number of attributes in 16 out of 24 datasets, with a total of lowest average number of selected features is 17 datasets. BSAFT has the exclusive minimum number of attributes in the Breast, Coimbra and Heart datasets, shared the minimum number of selected features for the PimaDiabetes dataset with BMAFT and BEAFT, and shared the minimum number of selected features for Saheart with BEAFT which exclusively caught the minimum number of attributes in the Lymphography, HeartEW and Hepatitis datasets.
From another point of view, the time it takes for the algorithm to finish the computations is of paramount importance. In this light, feature selection algorithms respect short execution times. To shed light on this crucial parameter, the CPU execution times of the four proposed Neural Computing and Applications algorithms were calculated in seconds and the results were reported in Table 10. The winner out of these algorithms is BMAFT, which was able to reduce the execution times in 11 out of the 24 datasets. The second winning algorithm is BAFT, which was capable of minimizing the execution times in 6 out of a total of the 24 datasets. BEAFT hit the minimum execution time in 4 datasets, which are Diagnostic, Prognostic, Hepatitis and ProstateGE, while BSAFT came in the last place by reporting only minimum execution times in the Coimbra, Dermatology and Parkinsons datasets.
Finally, due to that the absolute performance of the proposed algorithms in relation to each other is not fair enough to rate these algorithms, it is better to find the relative performance of one or all of the algorithms when their performance is compared with that of other algorithms. In this, by revisiting the results shown in Tables 5-10, we can clearly see that BMAFT outperformed all other algorithms in all assessment criteria. Therefore, in the next subsection, the results of both the basic BAFT and the proposed BMAFT are compared with those of broadly known optimization algorithms mentioned in the relevant literature.

Comparative study with other methods
To substantiate the overall performance and thorough assessment of the proposed BMAFT, its findings were compared with the basic BAFT and other seven wellknown algorithms in the relevant literature when evaluated on the same FS problems mentioned above. In specific, these opted binary algorithms are: Binary Success-History based Adaptive Differential Evolution with Linear population size reduction (BLSHADE) [44], Binary Crow Search Algorithm (BCSA) [45], Binary Honey Badger Algorithm (BHBA) [46], Binary Moth-Flame Optimization (BMFO) algorithm [47], Binary Particle Swarm Optimization (BPSO) [48], Binary Covariance Matrix Adaptation with Evolution Strategy (BCMA-ES) [49] and Binary Teaching-Learning-Based Optimization (BTLBO) [50]. In this comparison, not only the details of the experimental results are presented, but also a deep discussion and a fair comparison with those comparative optimization techniques were provided. For a fair comparison of the proposed BMAFT with the comparative optimization techniques, the same experimental conditions were applied to all the compared algorithms. The parameter settings for each of the chosen algorithm can be found in Table 11.
To unitize the experimental setup for the standard AFT algorithm, the proposed binary BMAFT, and all other competing optimization algorithms, the same population size and maximum number of iterations were used for each of the tested algorithms. Specifically, as presented in Tables 3 and 11, the adopted population size was 30 and a value of 100 was chosen for the maximum number of iterations. This, as already mentioned, guarantees a fair comparison between the contestant algorithms.
To reach statistically significant results, the experimental runs were repeated 30 independent times. Then the statistical results were reported based on the general capabilities and findings during these runs. The dimension of each solved FS problem is equal to the number of features in each corresponding dataset. The average (Ave) and standard deviation (Std) measures of classification accuracy (Acc), average fitness values, average selected attributes and CPU execution times were employed as performance indicators for the accuracy and stability of the proposed BMAFT method and other rival methods. The average measure was employed to assess the accuracy of the methods, while the standard deviation values were computed to assure stable performance of the algorithms during the independent runs.
The results of comparisons between the binary AFT algorithm, the binary MAFT algorithm and other competing binary algorithms in terms of average achieved classification accuracy, average selected features, average fitness values, and average CPU execution times associated with their related standard deviations are exhibited in Tables 12, 13,14, 15, respectively. In all tables, best results are discriminated in bold to give them more importance out of the other results.
Looking at the classification accuracy results shown in Table 12, the superiority of the proposed BMAFT is evident from various facets, where it has way outperformed the basic AFT as well as all other rivals. In this table, we can see that BMAFT was able to achieve the highest exclusive accuracy figures with reasonably good margins in 9 out of 24 considered datasets, and achieved the largest accuracy figures in an overall of 17 out of 24 datasets. BMAFT was only able to achieve 100% accuracy in one dataset which is the Leukemia dataset, at it has a standard deviation value of 0.0000 for this dataset. On the other hand, no other method was able to achieve 100% accuracy in any dataset. In other words, BMAFT has beaten BHBA and BCSA in 9 datasets, where these three methods shared the highest accuracy values in 8 datasets; BMAFT was inferior to BHBA in only two datasets, the Heart and Lymphography datasets with not-that-bad accuracy of 84.67% and 64.01% compared to 85.00%, 64.02%, respectively. BCSA outperformed BMAFT and BHBA in the Dermatology dataset with no more than 0.33% in the classification accuracy. Similarly, BHBA defeated BMAFT and BCSA in the ILPD dataset of over than 15% and astonishingly passed BAFT with a value exceeding 3.6% in the same dataset. For the ProstateGE dataset, BMAFT realized an accuracy of 94.00% compared to 90.00% realized by BHBA and BCSA with only about 4% deviation. By comparing the proposed BMAFT with the other competing algorithms, one can see that BTLBO, BPSO and BMFO lag far behind BMAFT as Table 12 illustrates. Furthermore, one can observe several significant improvements in the Std index of BMAFT compared to other algorithms. As a general conclusion and based on these results, one can claim that the proposed BMAFT is better suited to FS problems than the other rival algorithms.
We then move on to compare the BMAFT algorithm with the other competing algorithms in terms of the fitness values that the cost function aim to reduce. Regarding the results contained in Table 13, without any doubt, BMAFT had the smallest fitness values compared to its companion competitors. The standard-deviation values of BMAFT are also the smallest. Precisely, the proposed BMAFT exclusively achieved the smallest fitness values in 14 datasets, with the smallest fitness values in an overall of 15 datasets out of 24. This reveals the superiority of the proposed BMAFT algorithm in comparison with the state-of-the-art search algorithms. Regarding the standard divisions of BMAFT, they are small figures that underline that the preeminence of this algorithm is solid.
Although classification accuracy is so important for evaluating any proposed feature selection algorithm, the number of attributes selected in the classification process is as important for any feature selection algorithm. In Table 14, one can notice a greater variation when the proposed BMAFT, the basic BAFT and the other seven competing algorithms are compared. The division among the nine competing algorithms in respect of the average features was as follows: the BMAFT algorithm was the most effective one, where the average number of selected attributes was the minimum among the nine competing algorithms exclusively in 11 out of 24 datasets. BCMA-ES had the exclusive minimum number of attributes in the Hepatitis dataset and shared the minimum number of attributes for the ILPD dataset with BCSA and BHBA, where BCSA exclusively got the minimum number of attributes in the Coimbra and COVID-19 datasets and shared the minimum number of attributes for the Breast and Heart datasets with BMFO and BHBA, respectively. BMFO got the least number of attributes in 2 problems, and shared the minimum number of attributes in other two problems; BPSO lessened the number of attributes in 2 problems, and shared the minimum number of attributes in one problem, PimaDiabetes. The BAFT, BLSHADE and BTLBO algorithms were not capable of achieving any minimum number of attributes in any considered dataset.
The fourth comparison between the competing algorithms is in terms of the delay the algorithm takes to complete the computations. In the comparison outlined in Table 15, we see that BCMA-ES was the fastest feature selection algorithm among the nine competitors, where it had the minimum execution time in 8 datasets out of the 24. The computation times of BMAFT are the shortest in 5 datasets where this figure is similar to the number of datasets that BPSO has succeeded to reduce the execution Awareness probability (AP) 0.1

BTLBO No especial parameters Population size is 5 as this method has two main stages
Neural Computing and Applications     Fig. 4. In Fig. 4, the convergence behaviors of the nine optimization algorithms is shown over the trajectory of 100 iterations determined on the x-axis versus the fitness values on the y-axis. To obtain the correct and precise value from Fig. 4, we must look at it in conjugation with Table 13. In these convergence behaviors, the favored optimization method is the one that exhibits swift convergence accompanied by a low fitness value. This insinuates that we prompt the optimization method that settles on a very low fitness value in the fewest number of iterations. Examining Fig. 4, one can observe significant differences in the convergence curves of the competing algorithms. This difference is due to how the optimization algorithm behaves in respect of exploitation and exploration aspects, which also differs from one dataset to another for the same algorithm relying on the nature of the problem examined. Generally, one can discern that the convergence curves rest on or beyond 50 iterations in many datasets. Broadly speaking, BMAFT did well in several datasets, where one can perceive that BMAFT is excellent in Diagnostic, Prognostic, BreastEW, Lymphography, Parkinsons, ParkinsonC, Cleveland, Spectfheart, Thyroid, Leukemia, Colon, Pros-tateGE and COVID-19. BCSA performed well in the Breast, Coimbra, Dermatology, SPECT, Hepatitis, Saheart and PimaDiabetes datasets. BHBA has done well in ILPD, HeartEW, Saheart, Heart and PimaDiabetes. BCMA-ES revealed pretty convergence conduct in the ILPD, Saheart and PimaDiabetes datasets, while BLSHADE displayed sensible convergence conduct only in the PimaDiabetes dataset. Lastly, BPSO exhibited reasonable convergence behavior in the Retinopathy and Dermatology datasets.
Inspecting each convergence plot individually, one can realize that BMAFT eclipsed its companion algorithms in the Diagnostic dataset, where it reached a very low fitness value in less than 20 iterations. The curve continued to fall after that but only slightly. In the Breast dataset, BCSA was the best, where 30 iterations were enough for this method to settle at its lowest fitness value. For the Prognostic dataset, no one algorithm was capable of going below 0.1052 fitness, where BMAFT hit its minimum very lately and in more than 75 iterations. For the Coimbra dataset, one can notice a surprising fall in the fitness values experienced by all rival optimization methods, but BMAFT was the best one after 80 iterations. For the BreastEW dataset, BMAFT and BCMA-ES were the best and behaved analogously with small margins in between, where BHBA is the third best algorithm. In Retinopathy, BPSO was the best and the curve continued to come down after the 50th iteration. For Dermatology, BPSO and BMAFT acted similarly in the convergence aspect. For ILPD, BCMA-ES required 13 iterations to remain at its minimum. For Lymphography, BMAFT and BCMA-ES methods behaved approximately the same, with BMAFT better than BCMA-ES by a relatively good margin. The proposed BMAFT only needed 8 iterations to settle on its minimum in the Parkinsons dataset. In ParkinsonC, BMAFT surpassed the other competing algorithms in just 20 iterations, where it converged to the lowest minimum fitness very lately, and it continued to drop up to iteration 78, where it then settled on the minimum fitness. In SPECT, BCSA was the best in 16 iterations, with a slight difference better than BMAFT. In Cleveland, BMAFT, BLSHADE, BCSA, BHBA and BCMA-ES were nearly the same with only modest variations between them. The behavior is different in the HeartEW dataset, where a decline in the convergence curves of BHBA and BCSA took place very early, and before the 10th iteration. In Hepatitis, BCSA outperformed the others after 90 iterations. For Saheart, BCSA, BHBA, BCMA-ES and BTLBO behaved similarly with slightly better convergence behavior than BMAFT. Although BMAFT was the best in Spectfheart, it needed about 25 iterations to converge. For Thyroid, all competing algorithms except BTLBO behaved in like manner. BHBA was the best for the Heart dataset in less than 10 iterations, with both BMAFT and BCSA having nearly the same convergence behaviour. Once again, all the competing algorithms behaved in the same convergence manner in the PimaDiabetes, ProstateGE and Leukemia datasets, with BMFO and BPSO lagging behind the other algorithms in the leukemia dataset. For the Colon dataset, BMAFT showed better convergence process than the other algorithms, where it arrived at its minimum fitness in around 40 iterations. For COVID-19, all the algorithms exhibited the same convergence behavior except for BTLBO which lagged behind, although it showed relatively good convergence but less efficient than the other algorithms.
Again, looking back at the results presented in Tables 12, 13, 14, and 15 and observation of Fig. 4, one can clearly realize that BMAFT outperformed the other competing algorithms in average classification accuracy, average fitness and average number of selected features, while BCMA-ES excelled in reducing the average execution times while still having reasonable performance levels.

Statistical test analysis
In the above subsections, we used several performance indicators to evaluate the accuracy of the proposed BAFT and BMAFT algorithms, and to compare their results on FS problems with other optimization algorithms mentioned in the literature. The average and standard deviation were used as statistical measures to compute the optimal solutions (i.e., classification accuracy and fitness values) over 30 independent runs. These measures come up with an overall understanding of the accuracy of the proposed algorithms in tackling FS problems. The first measure demonstrates how an algorithm performs on average, while the second measure indicates how stable the algorithm is throughout all the independent run times. Even though these measures were able to demonstrate the overall performance and strength of the method evaluated, they could not contrast each of the independent runs separately. In other words, they have revealed that the proposed algorithms have promising exploitation and exploration behaviors, but they are incapable to show how good these algorithms are. To do this, Friedman's and Holm's statistical tests were conducted to emphasize the importance of the outcomes, and to show that these outcomes were not generated by chance. Friedman's test seeks to decide if there is a root difference between the outcomes of the compared methods, in which it is applied on the basis of the null hypothesis that there is no difference in the accuracy of all the compared methods. The method with the best performance always wins the lowest rank, while the method with the worst performance always gets the highest rank. The p-value of the Friedman's test is required to be found for the results of the problems under study, where if this value is equal to or less than the degree of significance, which is 0.05 in this work, the null hypothesis is rejected. By achieving this case means that there are statistically large variations between the performances of the compared methods. This statistical test is then chased by a post-hoc test method, which is the Holm's test procedure in this work, which can be applied to explore pairwise comparison of the compared algorithms. The method with the lowest rank got by Friedman's test is generally used as a control one for post-hoc analysis. In sum, Friedman's and Holm's statistical methods [51] were utilized to divulge that the average fitness and classification accuracy outcomes of the proposed BMAFT algorithm presented in Tables 12 and 13 are statistically worthy of attention and do not swerve from the results of other promising algorithms in a statistical manner. Table 16 displays the ranking outcomes of the Friedman's test based on the findings given in Table 12.
The p-value calculated by Friedman's method as per the average classification accuracy outcomes presented in Table 16 is 1.1637E-10. The null hypothesis of similar performance was rejected to reveal the existence of statistical significant difference between the accuracy of the compared algorithms. In compliance with the statistical results exhibited in Table 16, BMAFT is statistically considerable and the best effective binary method among all other binary ones. Accordingly, BMAFT has the first rank which is 2.2499 with a level of significance of 5%. Particularly, BMAFT was followed in order by BHBA, BCSA, BCMA-ES, BLSHADE, BAFT, BMFO, BPSO, and BTLBO in the last place.
Holm's test procedure was then applied after Friedman's test to determine if there are statistically large differences between the control algorithm that has the lowest rank (i.e., BMAFT) and the other competing methods shown in Table 16. The statistical results got by Holm's procedure on the basis of the obtained results, reported in Tables 12  and 16, are presented in Table 17. In this table, z denotes the statistical difference between two compared methods, R 0 denotes the Friedman's rank allocated to the control method, R i denotes the Friedman's rank allocated to method i and ES stands for the effect size of the control algorithm on algorithm i.
A comparison of BMAFT with other algorithms was made using Holm's test as presented in Table 17, where this test ignores hypotheses with a p-value 0:0166. As can be understood from the outcomes in Table 17, BMAFT is a powerful FS method in getting auspicious results for the datasets under investigation, and it is considerably better than other adept methods used to solve these FS problems. The p-values also improve the degree of performance of BMAFT on FS problems and show how effective this algorithm is. In short, the results in Tables 16  and 17 show that BMAFT has successfully avoided local optimal solutions and is considered statistically significant by achieving high classification accuracy values compared to the obtained classification accuracy rates by the other competing algorithms.
A supplementary statistical comparison was conducted between the proposed BMAFT and the other competing binary algorithms in respect of the average fitness outcomes where the results are presented in Table 13. The ranking outcomes of the statistical evaluation using Friedman's test based on the fitness values are given in Table 18.
As per the fitness scores exhibited in Table 13, the pvalue determined by Friedman's test is 7.6470E-11. Due to the outcomes presented in Table 18, BMAFT is the best efficient FS algorithm among all other competing ones. In this table, BMAFT has got the first rank with a value of 1.7708, with a level of significance of a ¼ 5%. The BHBA method is the second best one method with a rank of 2.4166 with a little difference from the rank obtained by BMAFT. In view of the results displayed in Table 18, BMAFT acted much better than the other algorithms in the FS task, where it ranked first, followed successively by BHBA, BCSA, BCMA-ES, BLSHADE, BAFT, BMFO, BPSO, and BTLBO.
These results are consistent with those formerly found in Table 16. Holm's procedure is then applied to see if there are statistically noticeable differences between the control algorithm (i.e., BMAFT) and the other comparative algorithms. The results got as a result of applying Holm's test with a ¼ 0:05 to the outcomes displayed in Table 18 are given in Table 19.
Holm's procedure discarded the hypotheses with a pvalue 0:0166 for the results in Table 19. It is evident from these results that BMAFT achieved a high performance score when conducting FS tasks over those got by other algorithms. These findings show that BMAFT has brilliantly succeeded in evading local solutions during exploration and exploitation of the search space. The results presented in Tables 16 to 19 indicate that the results of BMAFT are statistically significant, where this method is superior to other methods with splendid FS results over other well-known developed optimization algorithms. This consequence confirms that the proposed binary MAFT as a binary extension of the AFT algorithm is highly effective in tackling a wide variety of well-known FS problems.

Conclusion and future work
In this paper, four Feature Selection (FS) methods with S-shaped transfer function using binary variants of Ali Baba and the Forty Thieves (AFT) algorithm are proposed. These four binary FS algorithms are: BAFT, BMAFT, BSAFT and BEAFT, where they are respectively, referred to as Binary AFT, Binary Multi-layered AFT that utilizes hierarchy and distributed frameworks, Binary Self-adaptive AFT that uses an adapted tracking distance of AFT, and Binary Elitist AFT that utilizes an elitist learning strategy. These algorithms were experimented on a benchmark consisting of 24 well-regarded medical datasets. A detailed evaluation of the proposed algorithms using various performance metrics is provided. First, the proposed binary variants of AFT are assessed and compared among themselves to find the best one in reliably dealing with FS problems, among which BMAFT got the best results in all evaluation criteria. Second, considering that the strategies evolved or used in AFT have distinct merits, the proposed BAFT and BMAFT are placed in comparison with other algorithms in addressing FS problems. The obtained results showed the efficacy of the proposed BMAFT in obtaining a high level of performance in many evaluation methods. The strength of this algorithm is attributed to its success in balancing between exploration and exploitation. Obviously, the experimental results and analysis evidenced the excellency of the proposed FS algorithms, as promising potential tools, for solving tough FS problems. In the future, there is a plan to apply the proposed algorithms in feature selection challenges in other domains. The proposed FS tools can be wrapped for classification tasks with other common learning methods such as support vector machines and convolutional neural networks. Also, a multi-objective AFT algorithm can be evolved to deal with other combinatorial optimization problems or other highdimensional FS problems.