# Machine Learning with Evolutionary Parameter Tuning for Singing Registers Classification
## 2 Materials and Methods
### Support Vector Classifier  
Support Vector Classifiers (SVC) are powerful classification models usually leveraged in the context of ML, initially introduced by [PERSON] [29]. SVC can effectively learn non-linear decision boundaries in high-dimensional feature spaces by finding a hyperplane that maximizes the margin between the data points of different classes [30]. The latter capability is achieved by identifying a subset of training data points, called support vectors, that define the optimal hyperplane.  
Formally, given a training set \(T=\{(\mathbf{x}_{i},y_{i})\}_{i=1}^{N}\), where \(\mathbf{x}_{i}\) is a data point and \(y_{i}\) its corresponding class label, an SVC solves the optimization problem showed by Equation (1):  
\[\min_{\mathbf{w},\mathbf{b},\xi}\frac{1}{2}\mathbf{w}^{T}\mathbf{ w}+C\sum_{i=1}^{N}\xi,\quad C>0\] (1) subject to \[y_{i}(\mathbf{w}^{T}\phi(\mathbf{x}_{i})+b)\geq 1-\xi_{i},\quad \xi_{i}>0 \tag{2}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Feature Descriptions** & \multicolumn{2}{c}{**Feature Descriptions**} \\ \hline \(X_{1}\) & Area under the signal waveform. & \(X_{8}\) & No. negative signal turning points. \\ \(X_{2}\) & Signal autocorrelation. & \(X_{9}\) & No. peaks from a defined signal neighborhood. \\ \(X_{3}\) & Time axis centroid. & \(X_{10}\) & No. positive signal turning points. \\ \(X_{4}\) & Mean absolute signal differences. & \(X_{11}\) & Signal traveled distance. \\ \(X_{5}\) & Mean of the signal differences. & \(X_{12}\) & Signal Slope. \\ \(X_{6}\) & Median absolute differences of the signal. & \(X_{13}\) & Sum of absolute signal differences. \\ \(X_{7}\) & Median of differences of the signal. & \(X_{14}\) & Signal Zero-crossing rate. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Temporal features description computed with the Python TSFEL library, employed in the present study to retrieve information from the collected audio signals. The theoretical description of each reported feature may be retrieved from [PERSON] et al. [28].  
where \(\phi(\mathbf{x}_{i})\) maps the input vector \(\mathbf{x}_{i}\) to a higher-dimensional space, \(C\) is a regularization parameter controlling the trade-off between maximizing the margin and minimizing training errors, and \(\xi_{i}\) represents the slack variables allowing for misclassified points. Due to the potentially high dimensionality of \(\mathbf{w}\), the problem is typically solved in its dual form according to the following Equation (3):  
\[\min_{\mathbf{u}}\frac{1}{2}\mathbf{u}^{T}\mathbf{Q}\mathbf{u}- \mathbf{1}^{T}\mathbf{u}\] (3) subject to \[\mathbf{y}^{T}\mathbf{u}=0,\quad 0\leq u_{i}\leq C \tag{4}\]  
where \(\mathbf{1}\) is a vector of ones, \(\mathbf{Q}\) is a positive semi-definite kernel matrix defined as \(Q_{ij}=y_{i}y_{j}K(\mathbf{x}_{i},\mathbf{x}_{j})\), and \(K(\mathbf{x}_{i},\mathbf{x}_{j})\) represents a kernel function that computes the similarity between data points. Standard kernel functions include linear, RBF, and sigmoid (Table 2). The relationship between the weight vector \(\mathbf{w}\) and the dual solution \(\mathbf{u}\) is given by Equation (5):  
\[\mathbf{w}=\sum_{i=1}^{N}y_{i}u_{i}\phi(\mathbf{x}_{i}) \tag{5}\]