## 3 Results
### Hybrid (GP-XGBoost) Predictions  
Figure 7 shows the scatter plot that compares actual and predicted \(C_{c}\) values for the GP-XGBoost model. This figure highlights the strong predictive performance of this hybrid method. Most points from both the training and testing datasets align closely with the 1:1 reference line. This performance shows a high level of agreement between predictions and actual values. The points are tightly distributed within \(\pm\)10% deviation lines and confirm the model's accuracy and consistency. This demonstrates the hybrid model's ability to effectively handle nonlinear relationships in the data and make it a reliable choice for predicting \(C_{c}\). A few slight deviations suggest areas for potential improvement in feature representation or model assumptions. The histogram, boxplot, and violin plot in Figure 7 provide additional insights into the model's performance. The histogram shows that predicted values closely follow the actual data distribution and peak around the mean \(C_{c}\). The boxplot reveals minimal differences between median values and tightly grouped interquartile ranges and it further confirms the model's accuracy. The violin plots give a more detailed view of data distribution. They show consistent density and effectively address outliers. Together, these visualizations emphasize the GP-XGBoost model's robustness, precision, and reliability and make it a strong tool for geotechnical predictions that require high confidence. Table 11 shows how well the Hybrid GP-XGBoost model predicts \(C_{c}\) values for both training and testing datasets. The model delivers excellent results, with an \(R^{2}\) of 0.966 for the training set and 0.927 for the testing set. The results represent high accuracy and strong generalization. The low mean absolute error (MAE) values of 0.030 for the training set and 0.028 for the testing set, along with a minimal mean squared error (MSE) of 0.001 for both datasets, demonstrate its precision in handling complex relationships. This performance highlights the model's ability to outperform individual methods, proving its reliability and robustness. By effectively capturing the complexities of the data, the Hybrid GP-XGBoost model stands out as a highly dependable approach for accurate predictions. The following equation shows the GP-XGBoost proposed equation. Also, Table 12 represents parameters and constants:  
\[ \begin{split}\text{Y}=((\text{R}_{1}+\text{R}_{2})\times X_{3} \times\text{R}_{1}+(((4\times X_{3}-\text{R}_{3})\times(X_{3}-X_{2}+X_{1}-X_{ 4}))^{2}))+(((\text{R}_{1}+\text{R}_{2}\times X_{4})\times(\text{R}_{1}+\text {R}_{2})\times(X_{4}\times\text{R}_{1})\times X_{2}\times\text{R}_{1})-((X_{3} \\ -X_{4})\times((X_{3}-X_{2}+\text{R}_{3}-X_{4})-((X_{2}-X_{3}) \times(2\times X_{3}-X_{1})))))\\ \end{split}  \tag{24}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Optimized Value** \\ \hline Learning Rate (\(\eta\)) & 0.05 \\ Max Depth (d) & 7 \\ Number of Trees (Nestimators) & 400 \\ Subsample Ratio & 0.85 \\ Colsamplebytree & 0.8 \\ Regulazation (\(\lambda\)) & 1.2 \\ Min Child Weight & 5 \\ Gamma & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Optimized hyperparameters for XGBoost. For the GP-XGBoost hybrid model, the best approach involved iterative refinement:  
Feature Selection: Only the top four most important GP-derived features were used in XGBoost. A stacking ensemble approach was used, where a meta-model determined the optimal contribution of GP and XGBoost outputs. A grid search was performed to find the best weight combination. Table 13 shows optimized hyperparameters for GP-XGBoost. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance Metrics** & **Training Database** & **Testing Database** \\ \hline MAE & 0.030 & 0.028 \\ MSE & 0.001 & 0.001 \\ \(R^{2}\) & 0.966 & 0.927 \\ _RMSE_ & 0.037 & 0.034 \\ _MAPE_ & 12.901 & 11.184 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The performance of the hybrid GP-XGBoost method to predict \(C_{c}\) for training and testing databases.