## 3 Results
### XGBoost Predictions  
Figure 6 illustrates the strong performance of the XGBoost model in predicting \(C_{c}\) values. Most data points for both the training and testing sets are closely aligned with the 1:1 line, showing a high correlation between the predicted and actual values. The minimal scatter of points indicates excellent accuracy and consistency in the model's predictions. The addition of \(\pm\)10% deviation lines further highlights that the majority of predictions  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variables in Equation (18)** & **Parameters/Values** \\ \hline \(Y\) & \(C_{c}\) \\ \(X_{1}\) & LL \\ \(X_{2}\) & PI \\ \(X_{3}\) & \(e_{0}\) \\ \(X_{4}\) & \(w\) \\ \(R_{1}\) & Constant = 0.782 \\ \(R_{2}\) & Constant = 0.221 \\ \(R_{3}\) & Constant = 0.320 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Variables and parameters in GP proposed equation.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Optimized Value** \\ \hline Population Size & 1000 \\ Generations & 60 \\ Mutation Rate & 0.08 \\ Crossover Rate & 0.75 \\ Selection Method & Tournament (Size = 7) \\ Fitness Function & MSE \\ \hline \hline \end{tabular}
\end{table}
Table 8: Optimized hyperparameters for GP.  
fall within an acceptable error range, demonstrating XGBoost's ability to capture complex and nonlinear patterns in the data. A few points outside these lines suggest areas where fine-tuning hyperparameters or adding features could enhance performance.  
The histogram, boxplot, and violin plot in Figure 6 add more evidence of XGBoost's reliability. The histogram shows a close match between the distributions of predicted and actual \(C_{c}\) values for both the training and testing datasets, suggesting strong generalization to unseen data. The boxplots reveal similar median values and interquartile ranges for the predicted and actual results, with only a few outliers. The violin plots provide a deeper look at data density, showing consistent shapes and patterns between actual and predicted values. These results confirm that XGBoost not only delivers accurate predictions but also maintains robust and consistent performance, making it a dependable choice for geotechnical applications.  
Table 9 highlights how well the XGBoost model predicts \(C_{c}\) values for both training and testing datasets. The model performs great, with an \(R^{2}\) of 0.939 for the training set and an impressive 0.916 for the testing set. This shows that it not only makes accurate predictions but also generalizes well to new data.  
The error values add to this confidence. The MAE is 0.038 for the training set and 0.028 for the testing set, while the MSE is 0.002 and 0.001, respectively. These low error  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance Metrics** & **Training Database** & **Testing Database** \\ \hline MAE & 0.038 & 0.028 \\ MSE & 0.002 & 0.001 \\ \(R^{2}\) & 0.939 & 0.916 \\ \(RMSE\) & 0.049 & 0.037 \\ \(MAE\) & 16.012 & 14.541 \\ \hline \hline \end{tabular}
\end{table}
Table 9: The performance of XGBoost method to predict \(C_{c}\) for training and testing databases.  
Figure 6: Evaluation of actual vs. predicted \(C_{c}\) using XGBoost through various visualizations.  
rates demonstrate the model's precision and reliability. Overall, XGBoost stands out as a highly effective and consistent tool for geotechnical predictions.  
To achieve the best predictive performance, Bayesian Optimization (TPE) was used for hyperparameter tuning. A 5-fold cross-validation approach was applied, and the best parameters were selected based on the lowest MSE. The optimized parameters for the XGBoost model are presented in Table 10.