# Machine Learning Algorithms for the Prediction of the Seismic Response of Rigid Rocking Blocks
## 2 Materials and Methods
### Dataset Description
Thus, ultimately, two datasets were obtained, both with 20 features, i.e., the 18 IMs shown in Table 1, as well as the frequency parameter \(p\) and slenderness parameter \(\tan(\alpha)\). The first dataset employed in the classification task had 5,782,400 instances, and the target variable was whether the block was overturned or not (safe). The second dataset, which was employed in the regression task, had 3,256,553 instances with the same features, corresponding to those cases in the first dataset where overturning did not occur. In that case, the performance of the examined structural systems is evaluated in terms of the absolute maximum rotation angle \(|\theta_{\text{max}}|\) normalized to the critical overturning rotation\(a\). As such, this engineering demand parameter (EDP) is considered as target variable in the regression models. Figure 2 summarizes the distribution of these target variables. We can readily observe that most instances had a small maximum rotation, approximately following a gamma distribution. \begin{table}
\begin{tabular}{c c c} \hline \hline
**IM\({}_{i}\)** & **Description** & **References** \\ \hline PGA & \(\max|a_{g}(t)|\) & [55] \\ PGV & \(\text{PGV}=\max|v_{g}(t)|\) & [55] \\ PGD & \(\text{PGD}=\max|d_{g}(t)|\) & [56] \\ \(T_{m}\) & \(T_{m}=\frac{\sum_{i}C_{i}^{2}/f_{i}}{\sum_{i}C_{i}^{2}}\) & [56] \\ \(\frac{\text{PGV}}{\text{PGA}}\) & - & [57] \\ \(\frac{\text{PGD}}{\text{PGV}}\) & - & [25] \\ \(A_{\text{RMS}}\) & \(A_{\text{RMS}}=\sqrt{\left(t_{\text{tot}}\int_{0}^{t_{\text{tot}}}a_{g}(t)^{2 }dt\right)^{-1}}\) & [58] \\ \(V_{\text{RMS}}\) & \(V_{\text{RMS}}=\sqrt{\left(t_{\text{tot}}\int_{0}^{t_{\text{tot}}}v_{g}(t)^{ 2}dt\right)^{-1}}\) & [58] \\ \(D_{\text{RMS}}\) & \(D_{\text{RMS}}=\sqrt{\left(t_{\text{tot}}\int_{0}^{t_{\text{tot}}}d_{g}(t)^{ 2}dt\right)^{-1}}\) & [58] \\ \(L_{c_{a}}\) & \(L_{c_{a}}=\text{PGA}\cdot T_{m}^{2}\) & [59] \\ \(L_{c_{y}}\) & \(L_{c_{y}}=\text{PGV}\cdot T_{m}\) & [28] \\ \(I_{A}\) & \(I_{A}=\frac{\pi}{2\kappa}\int_{0}^{t_{\text{tot}}}a_{g}(t)^{2}dt\) & [60] \\ \(I_{\text{C}}\) & \(I_{\text{C}}=\sqrt{A_{\text{RMS}}^{3}t_{\text{tot}}}\) & [61] \\ \(I_{\text{F}}\) & \(I_{F}=\text{PGV}\sqrt{t_{\text{sig}}}\) & [62] \\ SED & \(\text{SED}=\int_{0}^{t_{\text{tot}}}v_{g}(t)^{2}dt\) & [55] \\ \(t_{\text{sig}}\) & - & [52] \\ CAV & CAV & \(\text{CAV}=\int_{0}^{t_{\text{tot}}}|a_{g}(t)|dt\) & [63] \\ CAD & CAD & \(\text{CAD}=\int_{0}^{t_{\text{tot}}}|d_{g}(t)|dt\) & [63] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ground motion description input features. Figure 2: Visualization of the target variables for the regression and classification sub-problems. (**a**) Histogram with kernel density estimation for the normalized maximum rotation. (**b**) Distribution of the “Overturn” variable in our dataset. The elevated number of response data enables the training of a reliable model within the considered range of rocking block features. Due to the fact that different rocking structures can be modeled by a rigid block of equivalent frequency parameters and the same slenderness, the trained model can also be used to estimate their maximum response. However, these predictions may be conservative since the database is generated assuming Housner's coefficient of restitution.