## 3 Background
### Binary Harris Hawks Optimizer (BHHO)
Binarize the population using transfer functions, including S-shaped and V-shaped func-tions, to convert continuous position values into binary values (0 or 1) for feature selection
12. Return Xrabbit as the best solution, representing the optimal feature subset

### Binary Grey Wolf Optimization (bGWO)

bGWO mimics the hierarchical and cooperative hunting behavior of gray wolves to optimize FS. This metaheuristic algorithm adapts the original Grey Wolf Optimization to binary search spaces using a sigmoidal transfer function, enabling effective navigation of feature subsets. The sigmoidal function, shown in Equation (8), converts continuous positions into probabilities. Allowing the updated positions to be expressed as binary values using Equation (9), where _rand_ is a random number between 0 and 1, and \(x_{d}^{t+1}\) is the updated binary position in dimension d at iteration t. \(X_{\alpha}\), \(X_{\beta}\), and \(X_{\delta}\) are the positions of the top three wolves [27]. \[sigmoid(a)=\frac{1}{1+e^{-10(x-0.5)}}, \tag{8}\]

\[x_{d}^{t+1}=\left\{ \begin{matrix}1,\ if\ sigmoid\left(\frac{x_{\alpha}+x_{ \beta}+x_{\delta}}{3}\right)\geq rand\\ 0,\ otherwise\end{matrix} \right.. \tag{9}\]

The process starts with initializing a population of n gray wolves, each assigned a random binary position within the range [0, 1], ensuring diverse starting points for an effective exploration of the solution space. After initialization, the algorithm evaluates the fitness of each wolf (\(\mathbf{wolf_{i}}\)) to identify the best solutions, labeled as \(\alpha\), \(\beta\), and \(\delta\), representing the top three solutions. This identification is crucial for guiding the search towards optimal feature subsets. During each iteration, the position of every wolf in the pack is updated using the sigmoidal function and stochastic thresholding to ensure binary representation. Parameters \(a\), A, and C are updated iteratively to balance exploration and exploitation using Equations (10), (11), and (12), respectively. Specifically, parameter \(a\) is linearly updated in each iteration to range from 2 to 0, controlling the trade-off between exploration and exploitation. After updating positions, the fitness of all wolves is re-evaluated, and the best solutions, \(\alpha\), \(\beta\), and \(\delta\), are updated accordingly. \[a=2-t\frac{2}{T}\, \tag{10}\]

\[A=2a\left(r_{1}\right)-a, \tag{11}\]

\[C=2(r_{2})\, \tag{12}\]

where t is the current iteration, T is the maximum number of iterations, and \(r_{1}\) and \(r_{2}\) are random numbers in the range [0, 1]. These parameters collectively refine the wolves' positions and encourage variability, enhancing search efficiency. The algorithm iterates until the stopping criteria are met, which may include reaching the maximum number of iterations or achieving a predefined fitness threshold. Once the stopping condition is satisfied, the procedure returns the optimal binary position \(x_{a}\) and the best fitness value \(f\left(x_{a}\right)\). This structured approach enables the bGWO algorithm to effectively navigate the feature selection space, using the social dynamics of gray wolves to find optimal or near-optimal solutions. The pseudocode in Algorithm 3 details this process, emphasizing the role of the sigmoidal transfer function in balancing exploration and exploitation. By effectively reducing feature dimensions, bGWO excels in high-dimensional datasets, making it a powerful tool for FS. ### Genetic Algorithms (GAs)

GAs are optimization techniques inspired by natural selection, where potential solutions, represented as binary chromosomes, evolve through iterative selection, crossover, and mutation processes to explore and exploit the search space efficiently. Guided by a fitness function, typically based on classification accuracy in FS tasks, GAs effectively explore high-dimensional search spaces [38; 39]. Fitter individuals, determined by a fitness function fi, are more likely to be selected for reproduction.