## 3 Proposed Method
The objective function in ElasticNet is defined as shown in (5):  
\[\alpha\cdot\lambda{\sum_{i=1}^{k}}\mid\beta_{i}\mid+(1-\alpha){\sum_{i=1}^{k} }\beta_{i}^{2} \tag{5}\]  
where \(\lambda\) controls the overall strength of the penalty, \(\alpha\) determines the balance between L1 and L2 regularization (with \(\alpha=1\) reducing ElasticNet to Lasso and \(\alpha=0\) to Ridge), and \(\beta_{j}\) represents the coefficients of the predictors. By adjusting \(\lambda\) and \(\alpha\), ElasticNet flexibly tunes the regularization to optimize feature selection for each specific dataset. In this study, ElasticNet is applied as a feature selection tool, filtering out less informative features and retaining those that are most predictive. By setting the \(\alpha\) parameter appropriately, ElasticNet performs an effective trade-off between sparsity via L1 regularization and stability via L2 regularization. This feature selection process enables the model to focus on the most relevant features while reducing the dimensionality of the input space, thus enhancing both computational efficiency and interpretability. The ability of ElasticNet to handle collinear variables is important in complex datasets. This property of the model prevents redundant features from obscuring meaningful patterns. This also ensures that only the most powerful predictors contribute to the final model. **Bayesian-Optimized DNN:** In this study, a Bayesian optimization method was used to determine the optimal architecture for a deep neural network. Bayesian optimization is an efficient approach for searching the global optimum of black-box functions that are costly to evaluate. It is particularly advantageous when the objective function lacks a closed-form expression but can be observed at sampled input values. Its effectiveness and versatility make it particularly well-suited for machine learning and deep learning tasks, includingtuning hyperparameters of deep neural networks [43]. Besides, employing Bayesian optimization increases the performance of traditional deep learning models [44; 45; 43]. We used this method in order to select the number of layers, the L2 regularization values, the number of units in each layer, learning rates, and dropout rates. Besides, it allowed a systematic search of the hyperparameter space in order to find a model that provides a balance between complexity and generalization. By using the Bayesian optimization method, we designed the final DNN architecture, which consists of specific configurations, regularization values, and hyperparameters to obtain the best classification performance. In the Bayesian optimization process, prior probabilities determine the initial assumptions about the hyperparameters before any data are observed. In this study, these probabilities were modeled as uniform distributions for the hyperparameters. According to the prior probabilities, the number of units in each layer was sampled from a discrete uniform distribution ranging between 32 and 512, with steps of 32, and the dropout rates were sampled from a discrete uniform distribution between 0.0 and 0.5, with steps of 0.1. On the other hand, the L2 regularization coefficient was sampled from a continuous uniform distribution between 0 and 0.3. Moreover, the number of hidden layers and the learning rate were sampled from a uniform distribution. The number of hidden layers ranged from 1 to 5, and the learning rate was selected from a set of predefined values \([1\times 10^{-2},1\times 10^{-3},1\times 10^{-4},1\times 10^{-5}]\). This uniform prior setup provides a balanced exploration of the hyperparameter space, allowing the model to systematically refine the search as the optimization progresses and posterior distributions are updated. This posterior updating is what enables Bayesian optimization to efficiently identify regions of the search space where the objective function, such as validation accuracy is most likely to improve. The optimization criterion for this process is to maximize validation accuracy.