## 5 Construction and Evaluation of Prediction Model  
Figure 1 illustrates the prediction model's creation process. In this case, 80% and 20% of the dataset were chosen as training and test sets, respectively, based on statistical integrity. Second, the predictive model was constructed using the trial-and-error approach \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Dataset**} & \multirow{2}{*}{**Statistical Parameters**} & \multicolumn{3}{c}{**Seismic Parameter**} & \multicolumn{3}{c}{**Geotechnical Parameter**} & \multicolumn{3}{c}{**Topographic Parameter**} & \multicolumn{1}{c}{**Output**} \\ \cline{3-10}  & & \(M\) & \(a_{max}\) & \(R\) & \(D50_{15}\) & \(F_{15}\) & \(T_{15}\) & \(W\) & \(D_{H}\) \\ \cline{3-10}  & & - & \(\mathbf{g}\) & km & mm & \% & m & \% & \(\mathbf{m}\) \\ \hline \multirow{4}{*}{Training} & Minimum & 6.4 & 0.15 & 0.5 & 0.04 & 1 & 0.2 & 1.64 & 0 \\  & Average & 7.26 & 0.41 & 15.10 & 0.36 & 18.83 & 7.80 & 11.69 & 2.45 \\  & Maximum & 9.2 & 0.68 & 100 & 7.7 & 70 & 16.7 & 57.7 & 10.16 \\  & Standard deviation & 0.51 & 0.15 & 11.61 & 0.65 & 13.71 & 5.16 & 9.96 & 2.26 \\ \hline \multirow{4}{*}{Testing} & Minimum & 6.4 & 0.15 & 0.5 & 0.07 & 2 & 0.5 & 2.11 & 0 \\  & Average & 7.3 & 0.38 & 16.10 & 0.39 & 13.96 & 7.98 & 10.04 & 2.17 \\ \cline{1-1}  & Maximum & 9.2 & 0.68 & 60 & 1.98 & 66 & 16 & 48.98 & 8.39 \\ \cline{1-1}  & Standard deviation & 0.49 & 0.13 & 10.61 & 0.42 & 12.14 & 5.20 & 9.78 & 2.21 \\ \hline \end{tabular}
\end{table} 
Table 2: Statistical parameters for free-face condition. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline _Parameters_ & \(M\) & \(a_{max}\) & \(R\) & \(D50_{15}\) & \(F_{15}\) & \(T_{15}\) & \(W\) & \(D_{H}\) \\ \hline \(M\) & 1.000 & & & & & & & \\ \(a_{\text{max}}\) & \(-0.341\) & 1.000 & & & & & & \\ \(R\) & 0.761 based on training set utilizing the optimum hyperparameters configurations. Iterative method was utilized to find optimum values for the hyperparameters after setting them to random values (within a reasonable range). The values of the key kernel parameters, omega (\(\omega\)) and sigma (\(\sigma\)) are 0.4 while noise is 0. 3 in the GPR model after multiple trials. Finally, the testing data was used to evaluate the proposed GPR model's performance using four common evaluation metrics: coefficient of determination (\(R^{2}\)), coefficient of correlation (\(r\)), mean absolute error (\(MAE\)), root mean square error (\(RMSE\)), ratio of the root mean square error (\(RSR\)) to the standard deviation of measured values, and Nash-Sutcliffe coefficient (\(NSE\)). The \(R^{2}\) and \(NSE\) values that are higher, and \(RSR\) values that are lower, imply that proposed model's prediction accuracy is better. Waikato Environment for Knowledge Analysis software was used throughout the whole calculation process The Pearson VII function-based kernel [41] was employed in this study for the GPR model. The generated model's performance was assessed using \(R^{2}\), \(r\), \(MAE\), \(RMSE\), \(RSR\), and \(NSE\). \[R^{2}=1-\frac{\sum_{i=1}^{n}(x_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(x_{i}-\overline{x })^{2}} \tag{8}\]  
\[r=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sqrt{\sum_{i =1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}} \tag{9}\]  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-y_{i}) \tag{10}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-y_{i})^{2}} \tag{11}\]  
\[RSR=\frac{\sqrt{\sum_{i=1}^{n}(x_{i}-y_{i})^{2}}}{\sqrt{\sum_{i=1}^{n}(x_{i}- \overline{x})^{2}}} \tag{12}\]  
\[NSE=1-\frac{\sum_{i=1}^{n}(x_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(x_{i}-\overline{x })^{2}} \tag{13}\]  
where \(n\) denotes the set of data points, \(x_{i}\) and \(y_{i}\) denotes the actual and estimated output of data's \(i\)th sample, respectively; \(\overline{x}\) and \(\overline{y}\) represents the mean actual and estimated output of the dataset, respectively. The \(r\) value varies from \(-1\) to \(1\).