## 3 Results and Discussion
### Bias-Variance Analysis  
Bias and variance are two error sources causing overfitting/underfitting associated with ML algorithms. Hence, they must be extensively analyzed to identify the performance trend. In this study, the bias and variance scores of the optimized models were estimated using (bias_variance_decomp) function available on the mlxtend library, considering the mean squared error (_MSE_) as a loss function. Table 4 provides the estimated bias, variance, and _MSE_ scores for each model. Figure 9 also provides a rich visualization of the conducted bias-variance analysis. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Dataset**} & \multicolumn{4}{c}{**Metric**} \\ \cline{3-6}  & & \(R^{2}\) & _RMSE_ (kN) & _MAE_ (kN) & _MAPE_ (\%) \\ \hline \multirow{4}{*}{RF} & Training & 0.9782 & 50.5509 & 29.5371 & 8.1684 \\  & Validation & 0.8501 & 129.3699 & 76.1972 & 21.2943 \\  & Testing & 0.8955 & 93.0718 & 62.6384 & 19.0369 \\  & Total & 0.9658 & 61.4556 & 36.1573 & 10.3421 \\ \hline \multirow{4}{*}{ET} & Training & 0.9977 & 16.4073 & 2.4859 & 0.7452 \\  & Validation & 0.8831 & 114.6653 & 66.3175 & 18.2174 \\  & Testing & 0.9034 & 89.4670 & 57.7167 & 16.8640 \\  & Total & 0.9836 & 42.6172 & 13.5320 & 3.9690 \\ \hline \multirow{4}{*}{AB} & Training & 0.9931 & 28.3637 & 13.9013 & 9.0994 \\  & Validation & 0.8605 & 124.2699 & 78.8827 & 24.9258 \\ \cline{1-1}  & Testing & 0.9058 & 88.3623 & 58.9152 & 20.5315 \\ \cline{1-1}  & Total & 0.9800 & 46.9594 & 22.9041 & 11.3858 \\ \hline \multirow{4}{*}{CB} & Training & 0.9971 & 18.3366 & 8.4077 & 2.9568 \\  & Validation & 0.9170 & 96.4863 & 56.8127 & 15.0108 \\ \cline{1-1}  & Testing & 0.9422 & 69.2136 & 47.1074 & 14.4932 \\ \cline{1-1}  & Total & 0.9889 & 35.0298 & 16.1476 & 5.2641 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Overall performance evaluation of optimized ML models. Figure 7: CV performance for optimized ML models. Figure 8: Bar plots of evaluation indices for optimized ML models. (**a**) \(R^{2}\) score. (**b**) \(RMSE\) score. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Bias** & **Variance** & \(MSE\) \\ \hline RF & 10,147.7445 & 2497.0147 & 12,644.7592 \\ ET & 8881.1901 & 2532.4058 & 11,413.5955 \\ AB & 8308.8276 & 5013.4415 & 13,322.2690 \\ CB & 6134.7950 & 2407.6251 & 8542.4200 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Bias-variance evaluation for optimized ML models. Figure 9: Plot of bias-variance analysis for optimized ML models. The CB model exhibited the lowest bias and the lowest variance among all evaluated models. The low bias indicates that CB is very effective at accurately capturing the underlying patterns within the training dataset, leading to less systematic error in the prediction. The low variance suggests that its prediction is stable and not highly sensitive to the specific dataset used for training. This combination resulted in the lowest \(MSE\) score. The ET model exhibited moderate bias and low variance. Its bias is higher than CB and AB but lower than RF. Its variance is very close to that of CB and RF but substantially lower than AB. This balance between bias and variance resulted in the second-lowest \(MSE\) score. ET offers a good compromise between capturing the true patterns and maintaining prediction stability, although its \(MSE\) score is high. Hence, it can still generalize well. The RF model exhibited the highest bias and the second-lowest variance among the models. The high bias suggests it might not capture the underlying patterns as effectively as other models, potentially leading to underfitting. While its variance is not as high as AB, it resulted in a higher \(MSE\) score compared to CB and ET. This indicates that RF, in this context, is not achieving as optimal a bias-variance balance as the top two models. The AB model exhibited a moderate bias, which is lower than RF and ET but higher than CB. However, it showed the highest variance by a considerable margin.