## 2 Theory and Calculation
### Model Selection Procedure and Error Estimation  
Quantities and functions illustrated in Section 2.3 represent the networks' hyperparameters that are set in the model selection procedure in order to ensure better efficiency of the training process. As mentioned, the number of hidden layers (N) and the number of neurons in each hidden layer (S) were modified to obtain 15 case studies for each selected transfer function (positive linear or hyperbolic tangent): N was changed from 1 to 3, S from 6 to 14, by increments of 2 neurons; such research design is reported in Table 1. In the literature, the use of a lower number of neurons and a greater number of hidden layers is recommended in order to increase the capacity of an ANN and avoid overfitting problems [72]. The hyperparameters' selection is based on the evaluation of the models' prediction capacity [83; 89], which is obtained by means of statistical indicators. In fact, the aim of the model evaluation is to estimate the generalization error of the selected model, calculated on the basis of a defined test set. The term \"generalization\" means the ability of a model to return a good prediction on unseen data, i.e., that have not been used for training. The most used statistical indicators, especially in the \"conventional\" procedure, are the mean squared error (\(MSE\)) and the correlation coefficient (\(R\)). The first one (whose analytical expression is reported in the last side of Equation (4)) calculates the average squared difference between the experimental targets (\(t\)) and ANN-computed outputs (\(y\)); it is also employed as performance function (also called the \"loss\" function) to optimize the ANN training process [78]. In general, the lower the mean square error, the closer the ANN-calculated data are to the experimental ones (i.e., calculated values are dispersed closely to experimental ones). Instead, the correlation coefficient, \(R\), measures the linear relationship between \(t\) and \(y\), and it is expressed by the ratio between the covariance of the two considered variables (\(t\) and \(y\)) and the product of the respective standard deviations:  
\[R=\frac{Cov(t,y)}{\sigma_{t}\sigma_{y}}=\frac{\sum_{l}\bigl{(}t_{l}-\overline{ t}\bigr{)}(y_{l}-\overline{y})}{\sqrt{\sum_{l}\bigl{(}t_{l}-\overline{t} \bigr{)}^{2}}\sqrt{\sum_{l}\bigl{(}y_{l}-\overline{y}\bigr{)}^{2}}}\quad\quad l =1,\ldots,\ M, \tag{10}\]  
where \(\overline{t}\) and \(\overline{y}\) are the mean value of the targets and of network outputs, respectively. In general, the closer the value of \(R\) is to the unity, the stronger the results of the linear relation between \(t\) and \(y\), thus confirming that the training has been completed successfully (if \(R\cong 1\) for the training data set) and that the degree of generalization achieved can be considered optimal (if \(R\cong 1\) for the testing data set). The mean squared error and the correlation coefficient have already been used in previous performance analysis of some ANNs designed to predict the mechanical parameters of HMA mixtures [65; 66; 72; 73; 74; 75; 76; 77; 78; 790]. On the contrary, when a k-fold CV is performed, the ANN performance evaluation takes place through three statistical indicators [82; 83; 84; 85; 86], namely the CV mean squared error (\(MSE_{cv}\)), the MSEs' standard deviation (\(\sigma_{cv}\)), and the CV correlation coefficient (\(R_{cv}\)), which is the average of the test correlation coefficient (\(R_{i}\)) over k rounds. A fourth indicator is given by the mean difference (\(\Delta\)) between the training and test MSEs over k-folds, which allows the occurrence of an overfitting problem to be assessed. In fact, when a model over-adapts to the training data set, the difference between training and test errors increases [82; 85]. Therefore, the hyperparameters that allow the lowest values