## 2 Materials and Methods
### Performance Metrics  
After training and optimizing the predictive models, it is crucial to evaluate their performance using several quantitative key metrics, in addition to the robust 10-fold CV, to obtain a comprehensive understanding of the models' predictive capabilities. In the current research, four statistical metrics in regression analysis including \"coefficient of determination (\(R^{2}\))\" (Equation (6)), \"root mean square error (\(RMSE\))\" (Equation (7)), \"mean absolute percentage error (\(MAPE\))\" (Equation (8)), and \"mean absolute error (\(MAE\))\" (Equation (9))  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Hyperparameter** & **Search Space** & **Optimal Value** \\ \hline \multirow{3}{*}{RF} & n\_estimators & Integer (100, 1000) & 783 \\  & max\_depth & Integer (10, 50) & 14 \\  & max\_features & Integer (1, 16) & 8 \\ \hline \multirow{3}{*}{ET} & n\_estimators & Integer (100, 1000) & 798 \\  & max\_depth & Integer (10, 50) & 25 \\  & max\_features & Integer (1, 16) & 15 \\ \hline \multirow{3}{*}{AB} & n\_estimators & Integer (10, 60) & 53 \\  & learning\_rate & Real (0.1, 1.0) & 0.87037279320499 \\ \hline \multirow{3}{*}{CB} & iterations & Integer (100, 1300) & 1300 \\  & learning\_rate & Real (0.01, 1.0) & 0.06344784452431354 \\ \cline{1-1}  & depth & Integer (6, 7) & 6 \\ \cline{1-1}  & l2\_leaf\_reg & Real (0.0, 1.0) & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Hyperparameter spaces and values of optimized ML models.  
Figure 6: Convergence curve of optimized CB model.  
were adopted to evaluate the CB and other developed ML models in predicting the SS of RCDBs [9]. The equations of these metrics are presented below.  
\[R^{2}\ =\ 1-\frac{\sum_{i\ =\ 1}^{N}(P_{i}-T_{i})^{2}}{\sum_{i\ =\ 1}^{N}(P_{i}- \overline{T})^{2}} \tag{6}\]  
\[RMSE\ =\ \sqrt{\frac{\sum_{i\ =\ 1}^{N}(P_{i}-T_{i})^{2}}{N}} \tag{7}\]  
\[MAPE\ =\ \frac{100^{\%}}{N}{\sum_{i\ =\ 1}^{N}}\bigg{|}\frac{P_{i}-T_{i}}{T_{i}} \bigg{|} \tag{8}\]  
\[MAE\ =\ \frac{\sum_{i\ =\ 1}^{N}|P_{i}-T_{i}|}{N} \tag{9}\]  
where \(P_{i}\ =\ \)predicted SS, \(T_{i}\ =\ \)experimental SS, \(\overline{T}\ =\ \)mean value of experimental SS, and \(i\ =\ 1,\ldots,N\) indicates the number of specimens in the database.  
These metrics are commonly used in regression problems, particularly in structural engineering applications. The \(R^{2}\) measures how well the overall established model fits the data and makes a prediction, based on the proportion of total variation of outputs. \(RMSE\) is the measure of the \(SD\) of prediction errors (residuals). It penalizes large errors more heavily than small errors and is sensitive to outliers. \(MAE\) measures error magnitude directly but is less sensitive to outliers than \(RMSE\). Both \(RMSE\) and \(MAE\) express prediction errors in terms of units of the output variable. \(MAPE\) is a scale-independent metric that expresses prediction error as a percentage of the actual value. Statistically, a higher value of \(R^{2}\) and corresponding lower values of \(RMSE\), \(MAE\), and \(MAPE\) indicate the performance level of the model. However, the \(R^{2}\) value is observed as the best metric for evaluation [93].