## 4 Discussion
### Evaluation of the Models  
Table 3 presents the optimized critical hyperparameters of the XGBoost model. These hyperparameters were fine-tuned through Optuna with a 10-fold cross-validation technique to enhance the model's predictive performance. The Optuna framework iteratively trained and evaluated models using the mean \(R^{2}\) score across the cross-validation folds as the objective metric. By focusing on maximizing this metric, Optuna identified the optimal set of hyperparameters (see Table 3) that balanced accuracy and generalization. The results of the hyperparameter importance analysis are illustrated in Figure 4. As can be observed, the relative importance of each parameter is ranked based on its contribution to model performance. Among the hyperparameters, the learning rate (eta) emerged as the most \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Metrics** & **Formula** & **Ideal Value** & **Equation. No** \\ \hline Correlation coefficient & \(r=\frac{\sum\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left(y_{i}-\overline{ y}\right)}{\sqrt{\sum_{i}\left(y_{i}-\overline{y}\right)^{2}}}\) & 1 & (13) \\ Coefficient of & \(R^{2}=1-\sum\limits_{i=1}^{N}\frac{\left(x_{i}-y_{i}\right)^{2}}{\left(x_{i}- \overline{x}\right)^{2}}\) & 1 & (14) \\ Index of Agreement & \(IA=1-\frac{\sum_{i=1}^{n}\left(y_{i}-y_{i}\right)^{2}}{\sum\left(y_{i}-\overline {y}\right)\left(y_{i}+\left|y_{i}-\overline{y}\right|\right)^{2}}\) & 1 & (15) \\ Kling-Gupta efficiency & \(\sqrt{\left(r-1\right)^{2}+\left(\kappa-n\right)^{2}+\left(\beta-1\right)^{2} }\) & 1 & (16) \\ Mean squared error & \(MSE=\frac{1}{n}\sum\limits_{i=1}^{n}(x_{i}-y_{i})^{2}\) & 0 & (17) \\ Root mean squared & \(RMSRE=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}\left(\frac{x_{i}-y_{i}}{x_{i}} \right)^{2}}\) & 0 & (18) \\ Mean absolute error & \(MAE=\frac{1}{n}\sum\limits_{i=1}^{n}\left|x_{i}-y_{i}\right|\) & 0 & (19) \\ Mean absolute & \(MAE=\frac{1}{n}\sum\limits_{i=1}^{n}\left|\frac{x_{i}-y_{i}}{x_{i}}\right|\) & 0 & (20) \\ Mean square relative error & \(MSRE=\frac{1}{n}\sum\limits_{i=1}^{n}\left|\frac{x_{i}-y_{i}}{x_{i}}\right|^{2}\) & 0 & (21) \\ Mean bias error & \(MBE=\frac{1}{n}\sum\limits_{i=1}^{n}\left(x_{i}-y_{i}\right)\) & 0 & (22) \\ Maximum absolute & \(MaxARE=Max\left(\left|\frac{x_{i}-y_{i}}{x_{i}}\right|\right)\) & 0 & (23) \\ \hline \hline \end{tabular}  
* Let \(X\) represent the measured Vs values and \(Y\) represent the predicted Vs values. Then, \(x_{i}\) represents the \(i^{th}\) value of variable \(X\), \(\overline{x}\) is mean of \(X\), \(y_{i}\) is the \(i^{th}\) value of variable \(Y\), \(\overline{y}\) is the mean of \(Y\), \(r\) represents the linear correlation coefficients, and \(n\) is the number of data points.  
\end{table} 
Table 2: Performance and error metrics used to evaluate Vs predictive capabilities of XGBoost and Bayesian GLM. influential factor. This underscores its pivotal role in determining the pace of model updates during training. Following the learning rate, the subsampling ratio and maximum tree depth (max_depth) were found to significantly affect model performance. The max_depth governs the complexity of the decision trees, allowing the model to capture complicated patterns in the dataset. Notably, the regularization term (lambda) exhibited relatively lower importance indicating that the other hyperparameters already provided sufficient control over model complexity. The partial dependence analysis was conducted to explore the relationships between key input features and the predicted Vs. The results (see Figure 5) indicate that Vs exhibits positive correlations with \(\sigma\),\(\
u\), qc_mean, fs_mean, depth and qc_cv. The relationship between Ic_mean and Vs, however, is non-linear, indicating a complex influence on Vs predictions. This non-linear interaction can be effectively captured by the XGBoost model but could not be accommodated by the linear framework of the Bayesian GLM. Consequently, Ic_mean was excluded from the Bayesian GLM.