# Machine Learning with Evolutionary Parameter Tuning for Singing Registers Classification
## 2 Materials and Methods
### Computational Framework  
The proposed approach basically consists of three steps: (i) data collection, (ii) model development, and (iii) performance evaluation and further analysis. Figure 5 illustrates these steps. The data collection stage begins by reading the audio files. These are then submitted to the feature extraction process, which, in this case, was carried out by the TSFEL library. At the end of these steps, the database containing the attributes of all the signals is built. After that, the model development stage begins. This part was set up to run 50 times independently to obtain more statistically robust results. Initially, the previously built database is randomly divided into a training set (70%) and a test set (30%). From the training set, the DE algorithm was used in conjunction with the cross-validation method to optimize the parameters of the classification model used (SVC, MLP, XGB). The best set of hyperparameters is then selected to adjust the model, which is then evaluated from the perspective of the test set. Finally, once the classes have been predicted, it is possible to evaluate the classification performance and carry out both parametric and feature importance analysis. In the present study, DE was employed to optimize the hyperparameter of the selected ML models. In particular, the DE configuration involved four key parameters, as detailed in Table 3. Such parameters guided the search process to identify the optimal configurations for the chosen ML models. The stopping criteria for DE were twofold:  
1. Maximum generations: The search terminated after reaching a user-specified maximum number of generations, i.e., \(G_{max}\). In the present study, \(G_{max}\) was set to 30 generations. 2. Convergence threshold: The search halted when the standard deviation of the Euclidean distance between individuals within the population fell below a predefined threshold, i.e., \(10^{-4}\). Such criterion ensured convergence towards a stable solution. In the present research, each candidate solution within the DE population, denoted by \(\mathbf{\theta}\), encodes the hyperparameters of a specific employed ML model (SVM, MLP, or XGB). The specific encoding scheme for each ML model is presented in Table 4.  
Figure 5: Flowchart of the methodological process used. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(NP\) & Population Size & 20 \\ \(G_{max}\) & Maximum Generations & 30 \\ CR & Amplification Factor & 0.9 \\ F & Mutation Rate & 0.8 \\ \(f(\mathbf{\theta})\) & Objective Function & F1-Score \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The employed DE algorithm configuration. Each reported parameter is followed by its respective description and selected parameter value. Moreover, the classification performance of the ML models with hyperparameters optimized by DE was assessed relying on the following performance metrics: Accuracy (Acc), Precision Score (PS), Recall Score (RS), F1-Score (F1), and Jaccard Score (JS), reported as follows:  
* Accuracy: it is determined relying on Equation (10), and it represents the percentage of accurately classified samples compared to the total amount of samples, as follows \[Acc=\frac{1}{N}\sum_{k=1}^{N}I(f(\mathbf{x}_{k})=c_{k})\] (10) where \(f(\mathbf{x}_{k})\) denotes the predicted class label of a test sample, \(c_{k}\) represents the true class label of this sample, and \(I(true)=1\) and \(I(false)=0\) are indicator functions. * Precision Score: It is the proportion of accurately predicted positive instances to all positive expected cases. Equation (11) shows its mathematical formulation as follows: \[PS=\frac{TP_{k}}{PP_{k}}\] (11) where \(TP_{k}\) and \(PP_{k}\) are the number of true positives and predicted positives for the class \(c_{k}\).