## 3 Materials and Methods
### Classification  
For classification, three different classifiers are used to confirm that any trends emerging among feature families and their performances do not depend on a specific classifier. The classifiers are an SVM with a linear kernel, an Artificial Neural Network, a Random Forest, and an Adaboost classifier.  
SVMs represent a standard in ML-based SR and remain a very viable choice especially when dealing with feature-based analysis where features and classifiers need to be somehow interpretable. Interpretability and speed are the main advantages of SVMs, making them suitable for such a study with respect to deep learning architectures. Since the aim is to build SR models that rely on specific, interpretable features, the \"black-box\" nature of DL networks was considered less suitable. Besides, linear SVM has often been demonstrated to be a viable solution for generalized feature-based problems in voice analysis [43; 44].  
The linear kernel SVM is based on the identification of the optimal linear separation hyperplane within the feature space without higher-dimensional projection, as suggested by [PERSON]'s theorem [11]. Outliers, i.e., data points that fall out of the separation hyperplane, are accepted by introducing a penalty or \"complexity\" value, C, that worsens the Lagrangian hinge loss function according to the proximity of the outlier with respect to the maximum margin hyperplane:  
\[L=C||w||^{2}+\frac{1}{n}\sum_{i=1}^{n}\max(0,1-y_{i}H), \tag{4}\]  
where \(w\) is a vector normal to the maximum margin hyperplane \(H\), \(n\) is the number of samples, and \(y\) is the label coded as \(-1\) or \(1\).  
A linear SVM with a complexity value \(C\) of \(1\) was trained on MFCC and RASTA feature combinations using [PERSON]'s Sequential Minimal Optimization [45], and performances were evaluated in terms of balanced accuracy, obtained through 5-fold cross-validation.  
A Random Forest classifier was also employed due to its proven performance in problems involving complex feature maps, reducing overfitting [46]. It is an ensemble of decision trees, each trained on a slightly different version of the dataset generated by bootstrap aggregation (bagging); the end result is obtained by majority voting. Each \"bag\" is obtained via sampling with repetition of the original dataset, and the operation is also performed on the features. The bags used in the proposed classifier were 80% of the original dataset and the RF models were trained for 1000 iterations.  
Finally, a more computationally complex classifier was introduced with Adaptive Boosting (Adaboost), as defined by [PERSON] and [PERSON] [47]. It employs a \"boosting\" technique that aims to better generalize an ensemble of weak learners by having it run on the training data using various weight distributions, finally combining the classifiers thus obtained into one. Different versions of the learners are trained on distributions of data where samples are given a weight, which is iteratively updated to reflect samples that generate more errors. The final prediction is the result of the following formula:  
\[Y=argmax_{Y_{k}(i)}\sum_{k:Y_{k}(i)=g_{i}}\ln\frac{1-err_{k}}{err_{k}}, \tag{5}\]  
This is the method, among the possible labels \(g_{i}\), that maximizes the logarithm of the \"beta ratio\", which decreases with the error for the \(k\)-th iteration. The error here is defined as the sum of the weights of the samples used to generate prediction errors.