An artificial bee bare-bone hunger games search for global optimization and high-dimensional feature selection

Summary The domains of contemporary medicine and biology have generated substantial high-dimensional genetic data. Identifying representative genes and decreasing the dimensionality of the data can be challenging. The goal of gene selection is to minimize computing costs and enhance classification precision. Therefore, this article designs a new wrapper gene selection algorithm named artificial bee bare-bone hunger games search (ABHGS), which is the hunger games search (HGS) integrated with an artificial bee strategy and a Gaussian bare-bone structure to address this issue. To evaluate and validate the performance of our proposed method, ABHGS is compared to HGS and a single strategy embedded in HGS, six classic algorithms, and ten advanced algorithms on the CEC 2017 functions. The experimental results demonstrate that the bABHGS outperforms the original HGS. Compared to peers, it increases classification accuracy and decreases the number of selected features, indicating its actual engineering utility in spatial search and feature selection.


INTRODUCTION
High-dimensional gene data classification analysis has received much attention in bioinformatics and computational biology, and it is possible to group genes according to specific structures, such as biological pathways. The expression of these genes in microarray technology 1,2 is simultaneously analyzed and measured to help scholars understand disease at the genetic level. Selecting a set of related genes is preferable to choosing a single gene because selecting a gene ignores the information in the grouping structure is less efficient. Grouping effects indicate that strongly related genes tend to be chosen or not chosen together. For example, tens to hundreds of thousands of genes are measured from individuals in several experimental groups. 3 Meanwhile, genetic expression data is highly dimensional and has extensive features. 4,5 The unrelated and complex gene expression features reduce computational performance and waste resources. [6][7][8] Gene selection is a feature selection (FS) in genes, which reduces unrelated genes and gene dimensions. [9][10][11] Thus, the feature size of high-dimensional gene data becomes small, and classification performance can be improved effectively. [12][13][14] Feature Selection (FS) is a process of determining the most relevant characteristics from a given dataset. By eliminating redundant and ineffective features, the number of features can be reduced to accelerate model training and enhance accuracy, especially in datasets with high dimensions. 15,16 The model's training grows increasingly challenging as the dataset's features increase. The unnecessary features can increase the training time with worse performance in the model. 17 Removing these superfluous features can add to the model's success in feature selection before training. FS inevitably becomes a critical stage, especially in high-dimensional gene datasets, which is a helpful way to reduce redundant features.
Filter, 18 wrapper, 19 embedded, 20 ensemble, 21 and hybrid 22,23 are the five main groups of feature selection approaches. Among them, wrapper methods rely on specific learning algorithms, such as classifiers, to explore a minimal number of feature subsets and can often achieve higher accuracy than filters. 24,25 The search technique and the assessment criterion are the two main components of a wrapper design. 26,27 In the former, a classifier, such as a support vector machine (SVM) or k-nearest neighbors (KNN), 28 is used to evaluate the quality of the feature subset acquired during the search strategy module. 29 In the latter, heuristic search is more efficient computationally than exhaustive and random search, and metaheuristic algorithms (MAs) can swiftly route to the ideal or nearly ideal solution. 30 To find a high-quality solution in a limited time and build on it a mathematical model which maximizes or minimizes the objective function, 31,32 in addition to traditional methods such as particle swarm optimization (PSO), 33 some new algorithms have been proposed recently, including henry gas solubility optimization (HGSO), 34 Archimedes optimization algorithm (AOA), 35 honey badger algorithm (HBA), 36 slime mold algorithm (SMA), 37,38 Runge Kutta optimizer (RUN), 39 colony predation algorithm (CPA), 40 weighted mean of vectors (INFO), 41 rime optimization algorithm (RIME), 42 Harris hawks optimization (HHO) 43 and so on. Recently, many hybrid algorithms have been devised and extensively implemented. For example, Celik 44 proposed improved symbiotic organisms search (ISOS) algorithm for global optimization. Celik et al. 45 propounded a modified salp swarm algorithm that outperformed the original (SSA) algorithm and many recent algorithms. Houssein et al. 46 advocated an improved sooty tern optimization algorithm to solve the feature selection problem. Celik 47 proposed an information-exchanged Gaussian arithmetic optimization algorithm with quasi-opposition learning to solve optimization problems. These algorithms have shown some superiority in different fields such as bankruptcy prediction, 48 scheduling optimization, 49,50 economic emission dispatch, 51 multi-objective optimization, 52 feedforward neural networks, 53 dynamic multi-objective optimization, 54 large-scale complex optimization, 55 constrained multi-objective optimization, 56 global optimization, 57 and feature selection. [58][59][60] In addition, several new hybrid metaheuristic algorithms (MAs) have been developed to address the feature selection issue. Hammouri et al. 61 devised a binary dragonfly algorithm (BDA) by utilizing several ways to update the values of five essential coefficients for feature selection. In summary, the proposed updating technique significantly impacts the algorithm's ability to solve FS problems. Adding chaotic maps to the original population led Tahir et al. 62 to develop a binary chaotic genetic algorithm (BCGA). Affective database AMIGOS and two healthcare datasets with huge feature spaces evaluate the novel BCGA with standard GA and two other state-of-the-art approaches. A new method named OBSSO for social spider optimization (SSO) using an OBL (opposition-based learning) technique was suggested by Ibrahim et al. 63 The accuracy of OBSSO was then compared to that of the standard SSO, the artificial bee colony (ABC), the firefly algorithm (FA), and the sine cosine algorithm (SCA) across ten datasets using a combination of KNN and RF classifiers. An improved salp swarm algorithm (ISSA) was suggested by Tubishat et al. 64 to locate the ideal feature subset using the opposition-based learning approach (OBL)and a new local search algorithm (LSA). On 18 datasets, ISSA's performance was examined and contrasted with four traditional methods. Too et al. proposed a robust hyper-learning binary dragonfly algorithm (HLBDA). 27 The hyper-learning technique was devised to assist DA in breaking out of the local optimum and enhancing the search behavior. To assess the success of their alterations, twenty-one datasets were employed, one of which relates to the coronavirus disease .
In the research above, metaheuristic strategies have outperformed feature selection approaches. 65 Despite its advantages, MAs have certain drawbacks in practice, such as the risk of getting stuck in local optimum, sub-optimal solutions, and slow convergence rate. 43,66,67 From this point of view, a more efficient optimizer is needed to identify the ideal set of dataset features. To reach this goal, a high-performance metaheuristic called Hunger Games Search (HGS) is chosen and used for feature selection in this research for at least the following reasons. Firstly, compared to other optimizers, especially those based on animal behavior, 68 HGS not only takes inspiration from an animal with a particular behavior but also creates a universal metaphor that implies the survival rules of nature. Yang et al., 69 conducted simulations which showed that HGS outperformed six established and nine modern MAs on 23 benchmark functions. Furthermore, HGS was also more successful than nine enhanced algorithms and seven DE variants algorithms on the IEEE CEC 2014 test suite. HGS has been employed to adjust the parameters of a hybrid microgrid system 70 as well as to create a new soft computing model for predicting the intensity of ground vibrations caused by mine blasting. 71 HGS has proven its excellence in AI through its impressive performance in terms of solution quality and computing cost, thereby highlighting its superiority.
Thus, HGS has been adopted by researchers to solve optimization issues. For example, an advanced orthogonal learning and Gaussian barebone hunger games search method were proposed by Zhou et al. for engineering optimization problems. 72 A hunger search-based whale optimization algorithm (HSWOA), as an ensemble of HGS and WOA, 73

RESULTS AND DISCUSSION
To validate the effectiveness of ABHGS, sets of experiments are designed. To prove the cooperation of artificial bee colony strategy and Gaussian bare-bone structure, we created HGS with artificial bee colony strategy, which is named AHGS, and HGS with Gaussian bare-bone structure BHGS. The overall experiments are conducted in the same hardware and MATLAB R2018b software environment. The hardware is a computer with the CPU of 12th Gen Intel (R) Core (TM) I7-12700H (2.30 GHz) Windows 11 data edge of 16.0 GB RAM.
A historical search trajectory test of ABHGS is conducted, as described in the history trajectory section. Balance analysis and diversity analysis section shows the balance analysis and diversity analysis among ABHGS, AHGS, BHGS, and HGS, showing the new balance between global exploration and local exploitation. Then ABHGS compares with AHGS, BHGS, and HGS on CEC2017 test functions to prove its global optimization capacity in the verification of the mechanisms section. ABHGS outperforms the other six conventional algorithms on the CEC2017 functions in the comparative test with conventional metaheuristic algorithms section. Furthermore, the enhanced algorithms are compared with ABHGS on thirty CEC2017 test functions to illustrate its excellent performance in comparative test with several modified algorithms

History trajectory
This section contains several historical trajectories of functions for ABHGS to assess the impact of local exploitation and global exploration on optimized performance. 92,93 Figure 1 shows the historical trajectory of four types of CEC2017 of test functions in 500 iterations, including a unimodal function F2, four basic multimodal functions F5, F8, F9, and F10, and two composite functions, F21 and F24. Figure 1A shows the three-dimensional location distribution. In Figure 1B, the red dots represent the global optimum solution, and the black dots represent positions in every iteration. In Figures 1C, 1D, and 1(c), the red line is ABHGS, and the other line represents HGS. Figure 1A describes the graphical plots of the selected mathematical functions based on ABHGS. Figure 1B shows the individual's historical search of the ABHGS method on the mentioned functions in 500 iterations, distributing around the best solution in the search space. Figure 1C presents the trajectory obtained by ABHGS with iterations in the first dimension. It shows the fluctuating state of the individual to gain the best value. Besides, ABHGS is volatile in the early period but becomes stable later. This phenomenon indicates a high probability of the population spreading around the optimal point. However, for HGS, individuals whose value fluctuates up and down are likelier to stay in the global explorative search stage than in the local exploitative search stage. Figure 1D illustrates the average fitness of agents with iterations. As can be seen from the curve, the mean fitness of ABHGS with fast convergence obtains the minimal value of final convergence. Figure 1E is the curve of functional fitness calculated by ABHGS and HGS. Obviously, ABHGS gains the final optimized minimal value with fast convergence.
In brief, there are the history trajectory graphs of seven representative functions: F2, F5, F8, F9, F10, F21, and F24. It shows the graphical plot and analyzes the search history of ABHGS on the functions. Besides, the comparisons about the trajectories of agents, the average fitness of all agents, and the convergence curves in ABHGS and HGS are shown in Figure 1. In conclusion, ABHGS is superior to HGS in unimodal, multimodal, and composition functions. ABHGS, with a relatively large proportion of global exploration searches in the early stage, always seeks the optimal solution. The capacity of ABHGS to find the best solution is stronger than that of HGS. Therefore, ABHGS gets a smaller convergence value and faster convergence speed than HGS.

Balance analysis and diversity analysis
In this section, further balance analysis of the exploration and exploitation of ABHGS, AHGS, BHGS, and HGS can help us better understand the reason for excellent performance in global optimization cases. The parameter setting in the main function is the same. For example, the maximal evaluation number is set to 300,000. The population size is 30, and each experiment runs 30 times independently. To calculate the increase and decrease in the distance among search agents, a diversity measurement known as the dimension-wise diversity measurement is calculated by Equations 1 and 2 in each iteration.
where medianðX j Þ represents the median of dimension j in the whole population. X j i is the dimension j of search agent i. N corresponds to the number of search agents in the population while D symbolizes the dimension of the search agents. t represents the current iteration. The highest diversity value identified during the entire optimization process is DIV max . The detailed pseudo-code of diversity calculation is presented in Algorithm 1.
The percentage of exploration and exploitation is used to describe the total balance response. Equation (4) and Equation (5) are used to calculate these values in each iteration.
In this model, The Exploration% denotes the degree of exploration, which is the ratio between the diversity in each iteration and the maximum attainable diversity. On the other hand, Exploitation% is a complementary percentage to Exploration%, as it reflects the difference between the maximum diversity and the current diversity of an iteration, which clustering search agents cause.
To assess the artificial bee colony strategy and Gaussian bare-bone structure independently, we design AHGS and BHGS. Through adding the mechanism, the explore and exploit capacities are affected. There is synergy between the different changes. Theoretical and experimental justifications about the effect of modifications on the proposed algorithm can be illustrated and discussed.    iScience Article capability. Inspired by this idea, we combine these two mechanisms to HGS and develop ABHGS, which may lead to a new equilibrium. The experimental results confirm that the concept and the proportion of exploration and exploitation of ABHGS fall between AHGS and BHGS, which is different from HGS. Then, if the exploration search is larger than the exploitation search, the green curve shows an upward trend. A downward trend is presented if the exploration search is less than or equal to the exploitation  iScience Article search. The duration with low or high values with iteration in the figure reflects persistent outcomes of local exploitation or global exploration capabilities in the search strategy. The ten functions evaluated by ABHGS have a higher exploration proportion than HGS, exhibiting its extensive global exploration effects. HGS has a relatively low exploration percentage and strong local exploitation search ability. The decline curve is the largest with the iterations when diversity and intensification are at the same level. To avoid HGS falling into local optima with the stagnation of convergence, ABHGS continues the global exploration phase.
The diversity analysis of mentioned F2, F5, F8, F9, F10, F17, F20, F21, F23, and F24 functions are shown in the last line of Figures 2 and 3. The x axis is the iteration number, while the y axis specifies the diversity measure. Algorithms always start with great diversity because the initialization is random. As the number of iterations increases, the population diversity gradually decreases. The results show that the variety of HGS does not reduce after reaching a value. The diversity values of ABHGS decline faster and are smaller than HGS. The diversity curve indicates that ABHGS converges faster and is earlier in global exploration than HGS.

Verification of the mechanisms
After the above analysis and validation, ABHGS was compared to AHGS, BHGS, and HGS on the IEEE CEC2017 test functions. Besides, the influence of artificial bee colony strategy and Gaussian bare-bone structure are investigated on the optimization issues. Analyzing the mathematical model's elements and their verification and numerical values is crucial to showing its working logic. The entire method was tested in the same software and hardware environment for a fair comparison. The parameter setting in the primary function is the same; for example, the maximal evaluation number is set to 300,000. The population size is 30, and each experiment is run 30 times independently. These methods evaluated their performance using the statistical average value of the optimal function (Avg) and SD(Std). The best result obtained by algorithms for each function in the table is highlighted in bold. Of course, the smaller the value, the better the performance. If the modification is considered significant statistically, the Wilcoxon signed-rank test was less than 0.05; that is, the p value is less than 0.05. The Wilcoxon signed-rank test is a non-parametric statistical test at a significance level of 0.05. The Friedman test is a statistical conformance test, too. The symbols ''+/ = /-'' illustrate that the proposed algorithm performs better, equal, or worse than the other comparative method.
Comparative tests of ABHGS, AHGS, BHGS, and HGS on IEEE CEC2017 are executed to validate the excellent performance of ABHGS. Table 1 displays the statistical average (Avg) and SD(Std) of the involved iScience Article methods for each function independently. The minimal values in every row of Table 1 are marked in bold. Table 2 shows that the p value computed by the Wilcoxon signed-rank test and the values less than 0.05 are observed in bold. It also displays the average ranking result (AVR) value and rank results by the Wilcoxon signed-rank test. Figure 4 shows the Friedman ranking test of ABHGS, AHGS, BHGS, and HGS. From Tables 2 and 3 and Figure 4, the experimental results indicate that ABHGS performs better than AHGS, BHGS, and HGS because an artificial bee colony strategy enhances the global exploration of HGS significantly, and a Gaussian bare-bone improves the local exploitation of HGS in arriving at the optimal solution. As shown in Figure 4 ABHGS has an average 1.5 ranking value by the Friedman test, which is superior to AHGS, BHGS, and HGS. The results show that ABHGS with an artificial bee colony strategy and a Gaussian bare-bone structure can perform well. Therefore, the artificial bee colony strategy and Gaussian bare-bone design significantly affect the performance of HGS.
Convergence curves of the comparison of ABHGS, AHGS, BHGS, and HGS on the CEC2017 test function are presented in Figure 5. There are F1, F9, F10, F13, F15, F16, F20, F22 and F30. As is shown in Figure 5, iScience Article ABHGS has a satisfactory effect after using two strategies compared with the basic HGS. Combining artificial bee colony strategy and Gaussian bare-bone structure assists HGS in arriving at global optima and avoiding the optimal local solution. The advantage of ABHGS is significant, and the effect of the two mechanisms on HGS is positive. Therefore, ABHGS has a better-optimized ability than AHGS, BHGS, and HGS.

Comparative test with conventional metaheuristic algorithms
To validate the global exploration search and local exploitation search abilities of ABHGS, it is compared with some typical conventional algorithms, such as AOA, RSA, NIFO, SMA, CPA, HHO, and RUN. Table 3 lists the specified parameters of the involved MAs. In this section, we conduct IEEE CEC2017 benchmark functions to show ABHGS's global exploration search ability. The simulation results of the involved methods show that ABHGS converges quickly and reaches the minimum, which demonstrates the excellent global exploration and local exploitation performance of ABHGS. Tables 4 and 5 record the mean value, std value, p value, and rank test ARV of the simulation experiment. Figure 6 describes the rank test ARV iScience Article of the Friedman test. Figure 7 presents convergence curves of the mentioned algorithms on the CEC2017 function test set.
As can be seen from Table 4, the minimum value in each function is marked in bold. Though ABHGS has not gained the smallest mean value on F1, F2, F3, F4, F8, F11, F12, F13, F14, F15, F18, F19, F21, F22, F24, F28, and F29, it outperforms the other swarm intelligent algorithms in handling the rest functions. Nevertheless, ABHGS ranks first in the Wilcoxon test in the whole functions in Table 5. The most p value of the Wilcoxon test is less than 0.05, which shows statistical significance. Figure 6 shows the results of the Friedman ranking test and ABHGS with the value of 2.1, ranking first. The results validate the excellent performance of ABHGS when compared with the conventional algorithms involved. ABHGS shows improved global exploration ability compared to the conventional algorithms involved.  Figure 7 presents the convergence curve of the above algorithms on the CEC2017 test functions. There are F6, F7, F10, F17, F20, F23, F25, F26, and F27. We can intuitively discover that ABHGS converges fast and finds the optima accurately. The combination of artificial bee colony strategy and Gaussian bare-bone structure enables HGS to arrive at a higher quality solution in the optimization process, achieving a better equilibrium between global exploration and local exploitation.

Comparative test with several modified algorithms
From the previous analysis, the advantages of the ABHGS algorithm should be further verified on CEC 2017 benchmark functions, compared with several state-of-the-art advanced algorithms. These modified algorithms are adaptive weights levy-assisted SSA (WLSSA), 94 chaotic mutative MFO (CLSGMFO), 95 efficient boosted GWO (OBLGWO), 96 weighted DE (WDE), 97 improved GWO (IGWO), 98 the hybrid algorithm of SCA and DE (SCADE), 99 cloud bat algorithm (CBA). 100 Tables 6 and 7 show the experimental results between ABHGS and eight advanced algorithms on CEC 2017 functions. The minimum of each test function was bold. Table 6 illustrates the comparison's mean value and SD value for each algorithm. As can be seen  Figure 8 describes the Friedman ranking test of advanced algorithms and shows the superiority of ABHGS. The Freidman test value gained by ABHGS is 1.9, also displayed in Table 7, ranking it first. WDE also performs well, ranking second. ABHGS outperforms involved advanced algorithms on CEC 2017 functions. Therefore, ABHGS can be superior to some advanced competitors.
The convergence curves of advanced methods on 30 CEC 2017 functions are presented in Figure 9. The curves of F2, F5, F8, F9, F10, F11, F12, F16, and F27 of ABHGS converge faster than other advanced algorithms except for CBA. However, CBA converges prematurely on these test functions and falls into local optimal solutions. Meanwhile, the proposed method has the best fitness, proving that ABHGS has a strong global search ability and gets rid of local optima. Therefore, ABHGS can produce optimal solutions accurately.  iScience Article In a word, ABHGS optimizes different types of functions on the CEC 2017 functions, outperforming some newly reported advanced techniques. The combination of artificial bee colony strategy and Gaussian barebone structure to the HGS algorithm enables the method to achieve a higher quality solution in the optimization process, making the local exploitation and global exploration in a better equilibrium state.

Feature selection
This section transfers the above continuous ABHGS to a discrete binary version for feature selection. We found the most suitable transfer function for ABHGS and compared it with other optimizers. Fourteen high-dimensional gene datasets are used in this study. The detailed characteristics of these iScience Article high-dimensional gene datasets are displayed in Table 8. Besides, leave-one-out cross-validation is an effective method for feature selection in the datasets; that is, one sample in the dataset is taken as the test set to prove the classification accuracy of the classifier, and the others are used as the training set. Each dataset's validation number equals the number of test datasets. This feature selection task is conducted on the KNN classifier, whose field size k is equal to 1.
Ensuring that the same dataset, settings, and metrics are used when comparing AI techniques is critical to ensure an accurate comparison. 101,102 Different datasets can have varying levels of complexity, which could lead to inaccurate results. 103,104 Additionally, different metrics can measure performance in different ways, making it difficult to compare results between AI techniques. 105,106 Using the same dataset and metrics for comparison ensures that the results are reliable and valid. 107 For fairness, all the algorithms are run on the same primary function with the same random initialization. The parameters of the main function and involved algorithms are set in Table 9. The number of search agents is set to 20. The fold is set to 10, and the maximum number of evaluations is set to 50. The entire algorithms are implemented in the same software and hardware environment.
To prove the effectiveness of the bABHGS_KNN model for feature selection, it is compared with the discrete versions of WOA, SMA, HGS, HHO, RUN, AOA, INFO, and ABC. The parameter settings for the eight comparison algorithms are displayed in Table 9. Compared with the binary MAs, the value of the nearest neighbor in KNN is set to 1 in this study. The 14 dataset's primary data is normalized to À1 and 1 at the beginning. The entire experiment is conducted in the same environmental conditions. Based iScience Article on the machine learning literature, 108-111 10-fold cross-validation (CV) analysis is adopted to classify fair and objective results.
Tables 10, 11, 12, and 13 describe the statistical outcomes of the 14 high-dimensional gene datasets simulated by intelligent swarm algorithms. The minimum value for each dataset is bolded. Table 10 shows the MAs' average error rate, and ABHGS wins the smallest error rate value in each dataset. Therefore, the bABHGS method ranks number one and shows superior performance in terms of error rate, followed by   Table 11, and bABHGS has the smallest number of selected features on 14 high-dimensional gene datasets. In a word, bABHGS is far more competitive than the other optimizers in reducing the features. The best fitness results from the significant measurements are presented in Table 12. The fitness combines classification accuracy and the number of features the objective function accumulates to assess the selected features. Most of the data in bold are from the bABHGS method, which shows its excellent performance on the high-dimensional gene dataset. Tables 10, 11, and 12, the bABHGS method has excellent performance, satisfactory average fitness, and minimal SD in all 14 high-dimensional gene datasets, which performs more stable than other involved optimizers. It can be seen from Table 13 that the average calculation time results of bABHGS are low-ranking, showing the high complexity of the method. It takes more time cost due to the enhancement of the performance. The artificial bee colony strategy and Gaussian bare-bone structure impact the increased time cost. Figure 10 presents the fitness convergence curve of    Compared with other optimizers, bABHGS is the best regarding error rate, number of features, and fitness on 14 high-dimensional gene datasets. Through the calculation time, the cost of the proposed method is not ideal; bABHGS can choose the optimal subset on most microarray datasets in terms of the optimal fitness and the minimal classification error rate without losing the meaningful features. Simulation results prove that the combination of the artificial bee colony strategy and Gaussian bare-bone structure to    iScience Article HGS guarantees its global exploration. Therefore, the proposed method achieves a more effective equilibrium between local exploitation search and global exploration search. In future work, the proposed method can also be applied to more cases, such as the optimization of machine learning models, 112 MRI reconstruction, 113 fine-grained alignment, 114 location-based services, 115,116 Alzheimer's disease identification, 117 renewable energy generation, 118 information retrieval services, 119-122 power distribution network, 123 medical signals, [124][125][126] and iris or retinal vessel segmentation. 127,128 Limitations of the study

As the average values and std values shown in
The present study has several limitations. First, the impact of each method on HGS is not evaluated in feature selection task trials. A preliminary test compares ABHGS, AHGS, BHGS, and HGS on CEC 2017

Conclusions and future works
This article introduces the artificial bee colony strategy and Gaussian bare-bone structure to HGS, namely ABHGS. To validate the global search ability of ABHGS, the simulation is conducted on CEC 2017 functions. Besides, we design experiments to show the excellent performance of ABHGS, such  This model will be further developed for accuracy and stability in future work. We intend to apply the bABHGS method to other practical high-dimensional datasets. Meanwhile, the proposed method can be used in more fields, such as the parameter identification of photovoltaics, engineering optimization, financial prediction, and disease diagnosis. Finally, we can develop an Artificial Intelligence framework for most feature selection problems.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following:

DECLARATION OF AI AND AI-ASSISTED TECHNOLOGIES IN THE WRITING PROCESS
During the preparation of this work the author(s) used chatGPT in order to enhance the english grammar and paraphrase some sentences. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.
where t represents the current iteration, and Maxiter means the maximum number of iterations; rand stands for a random vector that number is in the range of ½0; 1. A variation control E is defined as follows Equation (9).
where FðiÞ denotes the fitness value of the i-th agent and BF stands for the best fitness value of agents in the process. The hyperbolic function Sech is expressed in Equation (10).
As Equation (6) shows, three instructions are divided into two aspects for the overall situation, namely X ! -based and X b ! -based. The early game focuses on independent hunting behavior, with few agents working together as a team. The late two games imitate cooperative foraging behavior with W 1 ! , R ! , and W 2 ! . These different locations explore the optimal solution in the search space.

Hunger role
The hunger-driven behaviors of agents are expressed in mathematical formulas in this part.
The expression for W 1 ! ðiÞ is shown as Equation (11) and W 2 ! ðiÞ can be computed as Equation (12). where hungryðiÞ denotes the hunger level of each individual; N represents the number of agents; SHungry is sumðhungryÞ, which calculated by the hungry amount of the whole individuals; r 3 indicates a random value in the range of ½0; 1; both r 4 and r 5 are random numbers between 0 and 1. The hungryðiÞ is shown as Equation (13) where r 6 is a random number ranging in ½0; 1; WF is the fitness of the worst individual; UB indicates the upper bound in the search space, LB denotes the lower boundary, and hunger H has a lower bound LH.
A brief description of the HGS optimizer is given above, which provides a simple and efficient mathematical model that can be applied to continuous optimization problems. The pseudo-code of the continuous HGS is shown as Algorithm 2. In the HGS algorithm, when food shortage events occur, it forces some agents to a new region to forage. A Gaussian bare-bone mechanism was first proposed by Kennedy, inspired by the distribution histogram of PSO after 1,000,000 iterations. 81 The particle's velocity was removed through Gaussian distribution, and its position in the next iteration was updated. In the barebones PSO algorithm (BBPSO), the following formula is applied to replace the location of the i-th individual. is the arithmetic mean value; pbest t i;j À X b;j is the absolute value function of variance. Gaussian barebone is shown as Equation (17).
Equation (17) where k is a number randomly selected between 0 and 1, the indices k 1 and k 2 represent two different indices derived from the population set 1; 2; .; N, which differ from i. rand is a number ranging in ½0; 1, and the scale factor CR is to control the difference vector. The pseudo-code of the Gaussian bare-bone structure is presented as Algorithm 3.

While ðt % MaxiterÞ
Calculate the fitness of all Individuals; Sort the fitness of all Individuals; Update BF; WF; X b ; Set Hungry by Equation (13) Set the W 1 by Equation (11); Set the W 2 by Equation (12) For each individual Compute E by Equation (9); Calculate R by Equation (7) Update the position by Equation (6); End For t = t + 1;

End While
Return BF; X b (1) A hired bee modifies the position of food sources (solutions) in memory based on local information (visual information) and finds nearby food sources, then assesses the quality of the food. In ABC, the search for adjacent food sources is expressed by Equation (18) where food t + 1 i;j is the candidate food source stands for the position of the ith individual in the jth dimension in the ðt + 1Þ iteration and food t i;j stands for the position in the t iteration; r ! j i represents a random vector which value ranging from ½ À 1; 1, and food t k;j denotes a selected solution k that is different from i.
(2) Once the foraging bees have completed their search, they communicate to the onlooker bees in the dance area the quantity of nectar and its source. This is a trait of ABC artificial bee colonies. The onlookers assess the nectar data from all foragers and select a food source based on probability, which is determined by Equation 19 according to the fitness values of each solution. (3) A random number between 0 and 1 is generated for each source in the ABC. Suppose the probability value prob(i) associated with that source, as stated in Equation (14), is higher than the random number. In that case, the onlooker bee modifies the position of this food source site, similar to what happens with hired bees. After evaluating the source, a greedy selection mechanism is used; the onlooker bee either remembers the new position or retains the old one.
(4) After the entire hired bees and onlooker bees finish their searches in a cycle, a new food source position is created. For every location of a new food source, a random population X c is selected from the population gained by HGS. After being evaluated, the random population selected either memorizes the food source's new position or keeps the old one by greedy selection. Randomly selected population X c are generated by the following formula. iScience Article a completion rate and high classification accuracy. [86][87][88] In recent research, KNN also shows excellent performance in training speed and classification accuracy, 89,90 so this paper adopts K nearest neighbor (KNN) as a classifier for experimental evaluation. It is an instance-based learning model which predicts the class of a new instance according to the majority vote of the k-nearest neighbor class. The minimal distance between the new instance and the training points is used to decide the new instance's class based on the similarity measurement. The similarity is used in the literature most and is measured by the Euclidean distance. The Euclidean distance calculation procedure for two D-dimensional points Z 1 and Z 2 is shown as follows: Due to fast training speed, easy implementation, and excellent efficiency, this paper uses a KNN classifier to evaluate the classification accuracy.

Proposed methodology ABHGS algorithm for global optimization
There are various variants of HGS because the original HGS has some drawbacks that may miss some promising regions and cause population stagnation. To overcome this problem, a new HGS with an artificial bee strategy and Gaussian bare-bone structure is proposed, namely ABHGS. It introduces the artificial bee colony strategy and Gaussian bare-bone structure to the original HGS. In the ABHGS, two cooperative mechanisms provide diversity to the population and improve the objective function's feasibility and convergence capacity. The artificial bee colony strategy contributes to global exploration, and the Gaussian bare-bone structure enhances local exploitation. Therefore, ABHGS can maintain diverse search abilities and meet the population's needs in a certain evolutionary stage under limited computing resources. The detailed pseudo-code of ABHGS is presented in Algorithm 5. In a word, it is easier to understand the dynamic, fitness-wise optimizer. The flowchart of ABHGS is shown in Figure S1. Set Hungry by Equation (13); Set the W 1 by Equation (11); Set the W 2 by Equation (12);

For each individual
Compute E by Equation (9); Calculate R by Equation (7); Update the position by Equation (6);

End For
Update the positions of Individuals X i and food source position food i by Algorithm 3; t = t + 1;

End While
Return BF; X b ll OPEN ACCESS iScience Article comparative test with several modified algorithms; feature selection. The overall experiments are conducted in the same hardware and MATLAB R2018b software environment. In terms of global optimization problem, these methods including ABHGS algorithm evaluated their performance using the statistical average value of the optimal function (Avg) and standard deviation (Std). The smaller the value, the better the performance. If the modification is considered significant statistically, the Wilcoxon signed-rank test is less than 0.05; that is, the p-value is less than 0.05. The Wilcoxon signed-rank test is a non-parametric statistical test at a significance level of 0.05. The Friedman test is a statistical conformance test, too. The symbols ''+/=/-'' illustrate that the proposed algorithm performs better, equal, or worse than the other comparative method. All statistical details of global optimization are provided in Tables 1, 3, 4, 5, 6, and 7, Figures 4, 6 and 8. In terms of feature selection, datasets size (n) information for all analyses is provided in Table 8. The results are evaluated in terms of classification accuracy, number of selected features, and mean and standard deviation of fitness and run time. Tables 10, 11, 12, and 13 describe the statistical outcomes of the 14 high-dimensional gene datasets simulated by intelligent swarm algorithms. All statistical details are provided and explained in the text.

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