## 5 Conclusions  
Multiple machine learning (ML) models are developed to predict the maximum acceleration transferred to the center of gravity of structures founded on rocking shallow foundations during earthquake loading. Based on this study, the following major conclusions are drawn.  
* Given the five input features representing the key properties of the rocking foundation and earthquake loading (\(A/A_{c}\), \(h/B\), \(C_{r}\), \(a_{max}\) and \(I_{a}\)), the ML models presented in this paper can be used to predict the maximum acceleration transmitted to structures supported by rocking foundations with reasonable accuracy.
* Based on k-fold cross validation tests, the overall average MAPE in predictions of the KNN, RFR, ABR, GBR, and ANN models are all smaller than 0.145, with ANN being the most accurate and most consistent (MAPE = 0.128). For comparison, the MAPE of the MLR model and statistics based SLR model are around 0.23. This corresponds to an improvement in prediction accuracy of about 43%. Next to the ANN model, the second most accurate model is RFR, and it is followed by ABR, GBR, and KNN. This finding is also supported by another error measure criterion, namely, root mean squared error (RMSE) of model predictions.
* The overall average MAE in predictions of all six nonlinear ML models vary between 0.08 and 0.1, indicating that the maximum acceleration transferred to structures supported by rocking foundations can be predicted within an average error limit of 8% to 10% of the peak ground acceleration of the earthquake.
* Hyperparameter tuning is carried out to obtain the optimum values for hyperparameters and to ensure that the ML models presented in this paper do not overfit or underfit the training data. In terms of the architecture of the ANN model, a relatively simple network (only four hidden layers with 40 neurons in each layer) is found to be  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **MLV \({}^{*}\)** & **Minimum** & **Maximum** \\ \hline Multivariate linear regression (MLR) & 0.530 & 0.389 & 0.723 \\ Support vector regression (SVR) & 0.586 & 0.332 & 1.046 \\ k-nearest neighbors regression (KNN) & 0.435 & 0.349 & 0.634 \\ Random forest regression (RFR) & 0.480 & 0.357 & 0.804 \\ Adaptive boosting regression (ABR) & 0.541 & 0.341 & 1.150 \\ Gradient boosting regression (GBR) & 0.523 & 0.316 & 1.065 \\ Artificial neural network regression (ANN) & 0.469 & 0.295 & 1.018 \\ \hline \hline \end{tabular} \({}^{*}\) Most likely value.  
\end{table}
Table 4: Summary results of predicted AAR in parametric sensitivity analysis of ML models when the input feature values are varied one at a time.  
the optimum and most efficient for the problem considered in terms of accuracy of predictions without overfitting the training data.
* Feature importance analysis using the RFR, ABR and GBR models reveals that the chosen five input features capture the maximum acceleration of structures (through AAR) supported by rocking foundations satisfactorily. Parametric sensitivity analysis of all ML models reveals that AAR is more sensitive to peak ground acceleration of the earthquake motion than to other input features.
* The ML models presented in this paper can be used with numerical simulation results as complementary measures in modeling of rocking foundations or can be combined with mechanics-based models using the emerging framework of theory-guided machine learning. This forms the basis for future research on this topic.  
This research was funded by the US National Science Foundation (NSF) through award number CMMI-2138631.  
Not applicable.  
The data presented in this study and more details about the machine learning models developed in this study are available on request from the corresponding author. The data are not publicly available due to ethical.  
The author declares no conflict of interest.