## 5 Results and Discussion  
In this section, the performances of the three selected ML tools are compared. As metrics for the said performances, the following statistical indices have been adopted to measure the discrepancy between ground-truth and the predicted values of the seismic load bearing capacity:  
\[\mathrm{R}^{2}=\left[\sum_{i=1}^{n}(Q_{\mathrm{OG2}\ i}-\overline{Q_{\mathrm{ OG2}}})\left(Q_{Model\ i}-\overline{Q_{\mathrm{Model}}}\right)/\sqrt{\sum_{i=1}^{n} \left(Q_{\mathrm{OG2}\ i}-\overline{Q_{\mathrm{OG2}}}\right)^{2}\sum_{i=1}^{n} \left(Q_{Model\ i}-\overline{Q_{\mathrm{Model}}}\right)^{2}}\right]^{2} \tag{4}\]  
\[\mathrm{MAPE}=\frac{1}{n}\sum_{i=1}^{n}\left|Q_{\mathrm{OG2}\ i}-Q_{Model\ i} \right|/Q_{\mathrm{OG2}\ i} \tag{5}\]  
\[\mathrm{SI}=\mathrm{RMSE}/\overline{Q_{\mathrm{OG2}}} \tag{6}\]  
\[\mathrm{BIAS}=\frac{1}{n}\sum_{i=1}^{n}\left(Q_{\mathrm{OG2}\ i}-Q_{Model\ i}\right) \tag{7}\]  
\(\mathrm{R}^{2}\), MAPE, SI, and BIAS being, respectively, the coefficient of determination, the mean absolute percentage error, the scatter index, and the standard bias. Besides these indices, the RMSE introduced in Equation (3) has been adopted too. In these equations \(i=1,\ldots,n\) is an index running over the instances in the dataset. It is also worth recalling that \(Q_{\mathrm{OG2}\ i}\) is the numerical value of the dimensionless load bearing capacity \(Q=q_{u}/\gamma B\) generated by the Optum G2 software, while \(Q_{Model\ i}\) is the corresponding value provided by the trained ML tool (the overbar stands for the average value of the corresponding variable).  
Table 5 gathers the values of all the statistical indices introduced above to assess the performance of each ML approach. It can be seen that MLP is more accurate than GMDH and ANFIS-PSO in catching the structural response, as shown by the \(\mathrm{R}^{2}\), RMSE, SI, and BIAS values. The same trend is also shown in the parity plots given in Figure 15, where the estimations of the seismic bearing capacity \(Q=q_{u}/\gamma B\) of the shallow strip footing are  
Figure 14: ANFIS-PSO: evolution of the RMSE during the optimization by PSO, as a function of data proportioning.  
compared with the ground-truth values. Output provided by MLP is shown to be much less scattered than the others, and the linear interpolation of all the pairs of results is well aligned with the (perfect fit) bisector of the quadrant. In spite of the relatively small value of the BIAS, due to the fact that each single term in the sum is purposely taken with its own sign, the GMDH model performs worse, as reported by the large scatter and also by the deviation from the perfect fit line of the linear interpolant.  
The goodness of the fit of the results has been also studied: Figure 16 provides a comparison between the FELA and the foreseen seismic bearing capacity of the shallow strip footing. For this comparison, only 1% of the instances in the dataset have been randomly selected, so as to provide a clear view of the quality of data fitting. It results that the seismic bearing capacity computed by the ML techniques matches relatively well with their FELA counterparts, and the minor scattering around them is obviously in accordance with the results reported in Figure 15.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & \(\mathbf{R}^{2}\) & **RMSE** & **MAPE** & **SI** & **BIAS** \\ \hline MLP & 0.9955 & 0.0158 & \(-\)0.0101 & \(-\)0.0189 & \(-\)0.0001 \\ GMDH & 0.8417 & 0.0938 & \(-\)0.0750 & \(-\)0.1123 & \(-\)0.0009 \\ ANFIS-PSO & 0.9542 & 0.0505 & \(-\)0.0017 & \(-\)0.0605 & 0.8997 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Performance of the ML algorithms, in term of the adopted statistical indices.  
Figure 15: Performances of the ML algorithms: parity plots showing the ML output against the corresponding ground-truth numerical values for: (**a**) all the ML methods and regarding all the data; (**b**) MLP regarding test and validation data; (**c**) GMDH regarding test data; (**d**) ANFIS-PSO regarding test data.