## 4 Results and Discussion
### Artificial Neural Networks Modelling Results
#### 4.1.1 Cross-Validation Results  
Regarding the k-fold CV, the value of k was set to 5 and consequently, by having 129 available experimental data about the physical and mechanical properties of the considered HMA mixtures, each training data set was randomly made up of 103 observations, whereas the test data set was made up of the remaining 26 observations. The four indicators' values for each of the case studies considered in the model selection procedure (Table 1) are shown in Tables 4-6, one for each mechanical parameter considered; the models with the best prediction performance are indicated in bold type. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number of** **Hidden Layers**} & \multirow{2}{*}{**Test Performance**} & \multicolumn{6}{c}{**Positive Linear Transfer Function**} \\ \cline{3-6}  & & \multicolumn{3}{c}{**Number of Neurons in Each Hidden Layer**} \\ \cline{3-6}  & & **6** & **8** & **10** & **12** & **14** \\ \hline \multirow{3}{*}{\(1\)} & \(R_{cp}\) & 0.846 & 0.850 & 0.848 & 0.846 & 0.868 \\  & \(MSE_{cp}\) & 202,780 & 196,190 & 202,814 & 203,437 & 175,946 \\  & \(\sigma_{cp}\) & 46,847 & 36,339 & 57,885 & 51,690 & 53,240 \\  & \(\Delta\) & 59,087 & 48,988 & 56,148 & 57,797 & 72,179 \\ \hline \multirow{3}{*}{\(2\)} & \(R_{cp}\) & 0.854 & 0.882 & 0.865 & 0.891 & 0.869 \\  & \(MSE_{cp}\) & 191,976 & 156,746 & 176,240 & 145,174 & 166,594 \\  & \(\sigma_{cp}\) & 33,192 & 56,422 & 23,829 & 39,205 & 21,230 \\  & \(\Delta\) & 48,569 & 89,536 & 63,076 & 74,872 & 79,697 \\ \hline \multirow{3}{*}{\(3\)} & \(R_{cp}\) & 0.877 & **0.881** & 0.874 & 0.861 & 0.853 \\  & \(MSE_{cp}\) & 160,046 & **158,193** & 162,541 & 183,040 & 185,407 \\  & \(\sigma_{cp}\) & 34,825 & **28,282** & 69,860 & 22,155 & 36,253 \\  & \(\Delta\) & 79,948 & **34,142** & 87,021 & 80,075 & 84,381 \\ \hline \multirow{3}{*}{\(\begin{array}{c}\textbf{Number of}\\ \textbf{hidden layers}\end{array}\)} & \multirow{3}{*}{**Test performance**} & \multicolumn{6}{c}{**Hyperbolic tangent transfer function**} \\ \cline{3-6}  & & \multicolumn{3}{c}{**Number of neurons in each hidden layer**} \\ \cline{3-6}  & & \multicolumn{3}{c}{**Number of neurons in each hidden layer**} \\ \cline{3-6}  & & \multicolumn{1}{c}{**6**} & \multicolumn{1}{c}{**8**} & \multicolumn{1}{c}{**10**} & \multicolumn{1}{c}{**12**} & \multicolumn{1}{c}{**14**} \\ \hline \multirow{3}{*}{\(1\)} & \(R_{cp}\) & 0.887 & 0.882 & 0.878 & 0.874 & 0.886 \\  & \(MSE_{cp}\) & 154,281 & 161,179 & 169,033 & 175835 & 163,735 \\  & \(\sigma_{cp}\) & 69,853 & 51,885 & 72,373 & 77341 & 73,525 \\  & \(\Delta\) & 103,312 & 116,789 & 128,318 & 134397 & 124,963 \\ \hline \multirow{3}{*}{\(2\)} & \(R_{cp}\) & 0.850 & 0.842 & 0.841 & 0.843 & 0.871 \\  & \(MSE_{cp}\) & 224,187 & 216,034 & 217,966 & 214,564 & 180,046 \\  & \(\sigma_{cp}\) & 81,612 & 89,560 & 80,727 & 85,947 & 78,098 \\  & \(\Delta\) & 190,187 & 185,333 & 191,538 & 188,164 & 149,247 \\ \hline \multirow{3}{*}{\(3\)} & \(R_{cp}\) & 0.847 & 0.846 & 0.827 & 0.834 & 0.827 \\  & \(MSE_{cp}\) & 220,608 & 219,592 & 253,499 & 235,365 & 260,324 \\ \cline{1-1}  & \(\sigma_{cp}\) & 97,328 & 100,695 & 118,710 & 101,692 & 137,263 \\ \cline{1-1}  & \(\Delta\) & 194,246 & 192,887 & 229,045 & 213,887 & 235,464 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimates of stiffness modulus models’ predictive performance by five-fold cross validation. Table 4, which reports the ITSM models' results, shows that the hyperbolic tangent transfer function leads to higher values of \(MSE_{cv}\), \(\sigma_{cv}\), and \(\Delta\) as the number of hidden layers increases. Conversely, \(R_{cv}\) decreases for an increasing number of hidden layers (for example, 0.878 to 0.827 for 10 neurons passing from one to three hidden layers), as would be expected. This trend is verified whatever the number of neurons. Accordingly, hyperbolic-tangent ANNs exhibit an overfitting problem when the number of hidden layers is greater than one.