## 5 Results and Discussion  
### Prediction Accuracy of the Developed Regression Models  
The values of the evaluation metrics obtained during the training and testing of the optimized regression models are presented in Table 7. The values of the training metrics represent the mean validation scores derived from the k-fold cross validation conducted during the training phase, whereas the test metrics were acquired through predictions made on the separate testing dataset reference earlier in this article.  
The tabulated results revealed that the black box models (SVR-BO, DTR-BO, RFR-BO, ERTR-BO, ABR-BO, and XGBR-BO) exhibited superior performance across both training and testing phases compared to the white box models (RR-BO, LR-BO, ENR-BO, and KNN-BO). Among the four white box models, KNN-BO consistently outperformed its counterparts throughout training and testing. However, like the other three white box models, there was a significant difference between the training and testing R\({}^{2}\) scores indicating a model overfit condition. Conversely, among the six black box models, the ensemble models (specifically, RFR-BO, ERTR-BO, ABR-BO, and XGBR-BO) demonstrated  
Figure 13: Convergence graphs of BO plotted during the hyperparameter tuning of the six best performing machine learning regression models.  
better performance than the single model SVR-BO. Notably, the performance of DTR-BO closely resembled that of SVR-BO. In the comprehensive model assessment, the most outstanding performance was observed for the ERTR-BO model, which achieved an R\({}^{2}\) value of 98.8%, IA of 99.7%, and KGE value of 98.4% during the testing phase. This was followed by the testing R\({}^{2}\) values of XGBR-BO (98.0%), RFR-BO (97.8%), and ABR-BO (97.0%). Furthermore, in terms of \(v_{f}\) predictions on the testing dataset, ERTR-BO exhibited the lowest RMSE of 0.866 m/s, the lowest MAE of 0.517 m/s, and the minimal MAPE of 2.126%. This was visually evident in Figure 14, which portrays the actual versus predicted scatter plots for all the optimized regression models, highlighting the remarkable alignment of the ERTR-BO model's predictions with the perfect predictions line.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**R2**} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**MAPE**} & \multicolumn{2}{c}{**IA**} & \multicolumn{2}{c}{**KGE**} \\ \hline
**S No.** & **Model** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** \\ \hline
1 & RR-BO & 0.808 & 0.842 & 3.188 & 3.125 & 2.379 & 2.273 & 11.864 & 11.078 & 0.874 & 0.955 & 0.862 & 0.870 \\
2 & LR-BO & 0.809 & 0.840 & 3.185 & 3.145 & 2.383 & 2.294 & 11.877 & 11.212 & 0.872 & 0.954 & 0.860 & 0.867 \\
3 & ENR-BO & 0.809 & 0.840 & 3.185 & 3.145 & 2.383 & 2.294 & 11.877 & 11.212 & 0.872 & 0.954 & 0.860 & 0.867 \\
4 & KNN-BO & 0.874 & 0.901 & 2.543 & 2.476 & 1.528 & 1.536 & 6.947 & 7.024 & 1.000 & 0.974 & 0.929 & 0.944 \\
5 & SVR-BO & 0.952 & 0.911 & 1.568 & 2.345 & 1.006 & 1.344 & 4.326 & 5.565 & 0.978 & 0.977 & 0.965 & 0.944 \\
6 & DTR-BO & 0.962 & 0.963 & 1.426 & 1.518 & 0.894 & 0.845 & 3.881 & 3.434 & 0.998 & 0.991 & 0.977 & 0.975 \\
7 & RFR-BO & 0.975 & 0.978 & 1.135 & 1.177 & 0.755 & 0.732 & 3.308 & 3.127 & 0.989 & 0.994 & 0.971 & 0.975 \\
8 & ERTR-BO & 0.988 & 0.988 & 0.780 & 0.866 & 0.518 & 0.517 & 2.214 & 2.126 & 0.999 & 0.997 & 0.982 & 0.984 \\
9 & ABR-BO & 0.973 & 0.970 & 1.198 & 1.357 & 0.963 & 1.091 & 4.707 & 5.419 & 0.960 & 0.992 & 0.953 & 0.949 \\
10 & XGBR-BO & 0.978 & 0.980 & 1.066 & 1.113 & 0.779 & 0.761 & 3.543 & 3.311 & 0.997 & 0.995 & 0.978 & 0.982 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Performance metrics for the optimized regression models.  
Figure 14: Actual versus predicted scatterplots for all ten optimized regression models.