## 4 Framework  
### HPA-C Framework  
The process begins by initializing a random population of wolves according to Equation (10):  
\[\text{wolves}_{i,j}\sim\text{Bernoulli}(0.5),\quad\forall i\in\{1,2,\dots, \text{num\_wolves}\},\forall j\in\{1,2,\dots,\text{num\_features}\} \tag{10}\]  
The initial population goes through a fitness evaluation test. Equation (11) presents the fitness test in terms of classification accuracy:  
\[\text{Accuracy}(\text{wolves}_{i})=\frac{\text{Correct Predictions}}{\text{ Total Predictions}}\times 100 \tag{11}\]  
Following the fitness evaluation, the solutions are sorted based on \(\text{Accuracy}(\text{wolves}_{i})\) in descending order, where \(alpha_{\text{wolf}}\) is termed the best, \(beta_{\text{wolf}}\) the second most appropriate, and \(delta_{\text{wolf}}\) is the least appropriate. They are then subjected to an iterative position update and fitness evaluation process. The technique uses the main optimization loop and computes three new position variables: \(X_{1}\), \(X_{2}\), and \(X_{3}\). \(X_{1}\) uses the main GWO position calculation mechanism, \(X_{2}\) uses the ABC mechanism, and \(X_{3}\) follows the BAT procedure, as expressed by Equation (12). \[ \begin{split} X_{1}&=alpha_{\text{wolf},j}-A1\cdot|C1 \cdot alpha_{\text{wolf},j}-\text{wolves}_{i,j}|\\ X_{2}&=\text{wolves}_{i,j}+\phi_{1}\cdot(\text{ wolves}_{i,j}-beta_{\text{wolf},j})\\ X_{3}&=\text{wolves}_{i,j}+[\phi_{3}\cdot(delta_{ \text{wolf},j}-\text{wolves}_{i,j})+\text{velocities}_{i,j}]\end{split}  \tag{12}\]  
where  
* \(X_{1}\): variable updated using the Gray Wolf Optimization (GWO) update equation
* \(X_{2}\): variable updated using the Artificial Bee Colony (ABC) update equation
* \(X_{3}\): variable updated using the bat update equation (BAT)
* \(alpha_{\text{wolf},j}\): position of the alpha wolf in dimension \(j\)
* \(beta_{\text{wolf},j}\): position of the beta wolf in dimension \(j\)
* \(delta_{\text{wolf},j}\): position of the delta wolf in dimension \(j\)
* \(A1\), C1: random coefficients for GWO
* \(\phi_{1}\): Random coefficient specific to ABC
* \(\phi_{3}\): Random coefficient specific to BAT. For exploitation, a single position variable (\(P\)) is computed by taking the maximum of the three update mechanisms:  
\[P=\text{max}(X_{1},X_{2},X_{3}) \tag{13}\]  
The motivation behind choosing the maximum function is that it retains the most significant feature importance score among three incorporated mechanisms in HPA-C (Algorithms 1-3). This ensures that highly discriminative features emphasized by at least one technique are not suppressed. Unlike averaging, which can dilute strong feature contributions, taking the minimum may underestimate feature significance. The maximum function ensures that the best-ranked features from any algorithm are preserved. This enhances the robustness of the selected feature subset and improves classification accuracy. The wolves' positions are bounded using Equation (14), and binary quantization (thresholding at 0.5) is applied as expressed by Equation (15):  
\[\text{wolves}_{i,j}=\text{max}(\text{min}(P,\text{ub}_{j}),\text{lb}_{j}) \tag{14}\]  
\[\text{wolves}_{i,j}= \begin{cases}1&\text{if wolves}_{i,j}>0.5\\ 0&\text{otherwise}\end{cases}  \tag{15}\]  
The choice of 0.5 as a threshold ensures a balanced decision boundary. The feature selection scores are normalized between 0 and 1. Our choice makes retention and discarding equally probable. This approach aligns with widely adopted binary discretization techniques in metaheuristic-based feature selection to ensure methodological consistency. Other threshold choices (0.4 and 0.6) showed no significant performance gains, with 0.5 providing stable results across datasets.