## 4 Results and Discussion
### Artificial Neural Networks Modelling Results
#### 4.1.2 Standard Approach Results  
Concerning the standard ANNs, 103 data (80% of the data set) were used for the training of networks and the remaining 26 (20%) for the test. Tables 7-9 show the \"best\" and the \"worst\" (over the 10 identified random partitions of the data set) prediction performance of the standard ANNs with a different number of neurons in the hidden layer on the three mechanical parameters considered, namely the stiffness modulus, Marshall stability, and quotient. The standard deviation of the test \(MSE\), included in Tables 7-9, represents a measure of the variance of the performance scores over the 10 groups. Figure 10: Training result for the Marshall quotient ANN model. As can be observed, on the basis of the standard approach, the network performance can vary significantly depending on the test set considered, whatever the number of neurons in the hidden layer; for example, in terms of the stiffness modulus, with regards to 10 neurons, test MSE values are within the range \(88,046\)-\(550,323\), with a standard deviation equal to \(138,381\) (Table 7). On the basis of the 10 different data samplings carried out, for each considered number of neurons, models characterized by the \"worst\" performance showed large differences between training and test MSE values; for example, with regards to the stiffness modulus, the MSE values for testing are an order of magnitude higher than the ones for training (\(550,323\) vs. \(47,722\) for 10 neurons, Table 7). These results demonstrate that standard ANN models, on the basis of the \"worst\" random data partition, do not have the capacity to adequately generalize independently by the number of neurons. Therefore, in such cases, it is absolutely necessary to repeat the standard procedure, namely, random data sampling along with training and test phases, following a trial-and-error philosophy. In this way, it is possible to obtain good predictions (i.e., \"best\" cases), but these results depend on the data sets used for training and testing the model. Such data sets are randomly determined on the basis of a trial-and-error procedure that requires repetitions of the random partition; this means that a favorable data partition has to be looked for on the basis of a random repetition of the procedure, without any rational line guide. Hence, the standard approach, based on a few data sampling repetitions (up to 10 in this study), cannot be considered as the best suited method for ANN model optimization because the best results can be obtained only for comparable training and test sets variability. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Model Performance**} & \multirow{3}{*}{**Statistical Indicators**} & \multicolumn{8}{c}{**Numbers of Neurons in Each Hidden Layer**} \\ \cline{3-11}  & & \multicolumn{2}{c}{**6**} & \multicolumn{2}{c}{**8**} & \multicolumn{2}{c}{**10**} & \multicolumn{2}{c}{**12**} & \multicolumn{2}{c}{**14**} \\ \cline{3-11}  & & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** \\ \hline \multirow{3}{*}{Best} & MSE & 60,863 & 77,337 & 88,399 & 68,540 & 88,046 & 68,607 & 62,276 & 64,221 & 49,390 & 63,133 \\  & R & 0.970 & 0.942 & 0.950 & 0.949 & 0.943 & 0.951 & 0.914 & 0.960 & 0.985 & 0.949 \\ \hline \multirow{3}{*}{Worst} & MSE & 476,723 & 46,517 & 387,522 & 33,425 & 550,323 & 47,722 & 503,815 & 53,480 & 333,328 & 12,046 \\  & R & 0.716 & 0.970 & 0.728 & 0.977 & 0.640 & 0.968 & 0.560 & 0.964 & 0.738 & 0.992 \\ \hline \multirow{3}{*}{Test MSE standard deviation} & \multirow{3}{*}{130,512} & \multirow{3}{*}{86,718} & \multirow{3}{*}{138,381} & \multirow{3}{*}{151,325} & \multirow{3}{*}{97,811} \\  & & & & & & & & & & & \\ \end{tabular}
\end{table} 
Table 7: Effect of data sample variability on the stiffness modulus model’s performance.