## 4 Results and Discussion
### Parametric Sensitivity Analysis of Models  
In order to study the sensitivity of ML model predictions to variations in input feature values, a parametric sensitivity analysis is carried out. For this exercise, the input feature values are systematically varied and are fed into the ML models. As a baseline case, all input feature values are kept at their mean values and the predicted AAR corresponding to this scenario is the most likely value (MLV) of prediction for a particular model. In addition, each input feature is varied to include two other values: mean minus standard deviation and mean plus standard deviation. The predictions of the ML models are obtained by this  
Figure 14: Boxplots of MAE in the predictions of AAR of machine leaning models during final five-fold cross validation tests of models. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Ave. MAPE** & **Ave. MAE** & **Ave. RMSE** \\ \hline Simple linear regression (SLR) * & 0.228 & 0.148 & 0.232 \\ Multivariate linear regression (MLR) & 0.225 & 0.139 & 0.185 \\ Support vector regression (SVR) & 0.162 & 0.103 & 0.145 \\ k-nearest neighbors regression (KNN) & 0.145 & 0.092 & 0.137 \\ Random forest regression (RFR) & 0.144 & 0.090 & 0.124 \\ Adaptive boosting regression (ABR) & 0.144 & 0.090 & 0.125 \\ Gradient boosting regression (GBR) & 0.143 & 0.092 & 0.133 \\ Artificial neural network regression (ANN) & 0.128 & 0.083 & 0.113 \\ \hline \hline \end{tabular}
* Statistics-based (non-ML) model.  
\end{table} 
Table 3: Summary of average MAPE, MAE and RMSE (testing errors) of models in final five-fold cross validation tests. method using these two extreme values for a certain input feature, while all other input feature values are kept at their mean value. As there are five input features, this method results in eleven combinations of input features. The results of this parametric sensitivity analysis are presented in Figure 15 for four models in the form of \"tornado diagrams\". In the tornado diagrams presented in Figure 15, the x-axis represents the predicted AAR values by that particular model when the input feature values are varied (mean \(\pm\) standard deviation). Note that in a tornado diagram, the absolute difference between the prediction values corresponding to the two extreme values of an input feature is called the \"swing\", and the input feature that has greatest swing is plotted at the top of the plot (the input features are plotted on the y-axis in descending order of their swing values). Also included in these figures are the most likely value (MLV) of predicted AAR (vertical dashed lines), when all the input features are kept at their mean values. Table 4 presents summary results of predicted AAR in parametric sensitive analysis (MLV, minimum and maximum) for all seven ML models. As the results presented in Figure 15 and Table 4 indicate, the predicted AAR is more sensitive to peak ground acceleration (\(a_{max}\)) than any other input feature for all the models (i.e., \(a_{max}\) produces the maximum swing in predicted AAR). Only the MLR model shows an almost symmetric response around the MLV in tornado diagrams, mostly because it is a linear ML model. The unsymmetric nature of the tornado diagrams of all nonlinear ML models indicates that the relationship between AAR and input features are highly nonlinear. For all six nonlinear ML models, about 45% to 75% of the variance in the prediction of AAR results from the variation in \(a_{max}\) (variance in this context is defined as half-swing divided by the most likely value of predicted AAR). Next to \(a_{max}\), \(C_{r}\) and \(A/A_{c}\) have more effect on model predictions in general when compared to \(h/B\) and \(I_{a}\). This trend is consistent with the experimental results plotted in Figure 3, where \(a_{max}\) and \(C_{r}\) are identified as the key variables that dictate AAR.