## 4 Results
### Data-Driven Modeling Results
#### 4.2.3 Classification and Regression Trees (CART)  
After utilizing GP as a grey box AI model for predicting the bearing capacity of shallow foundations, this discussion was further extended to include another grey box AI model, namely Classification and Regression Trees (CART), to complete this group of grey box models. Following a comprehensive evaluation of multiple CART models, the most optimal model was identified. Figure 7 illustrates the results obtained from the best CART model, showcasing a remarkable agreement between the actual and predicted values. These findings highlight the CART model's exceptional accuracy and precision in forecasting bearing capacity of shallow foundations (q\({}_{\text{u}}\)) values. The incorporation of both GP and CART models provides valuable insights into the predictive capabilities of grey box AI techniques for estimating bearing capacity, enabling more reliable and informed decision making in foundation engineering applications. \begin{table}
\begin{tabular}{c c c} \hline
**Metric** & **Training Database** & **Testing Database** \\ \hline MAE & 19.677 & 16.164 \\ MSE & 700.751 & 313.386 \\ RMSE & 26.472 & 17.703 \\ MSLE & 0.018 & 0.012 \\ RMSLE & 0.136 & 0.111 \\ \(R^{2}\) & 0.939 & 0.975 \\ \hline \end{tabular}
\end{table} 
Table 6: Overall performance of the best GP model to predict the ultimate bearing capacity of shallow foundations for both training and testing databases. Table 7 provides a comprehensive assessment of the overall performance of the best Classification and Regression Trees (CART) model in predicting the ultimate bearing capacity of shallow foundations. The evaluation was conducted on both the training and testing datasets, utilizing various metrics to gauge the model's predictive accuracy. In the training dataset, the CART model displayed a Mean Absolute Error (MAE) of 18.388 kPa, indicating that, on average, its predictions deviated by approximately 18.388 units from the actual values. The model achieved a high R-squared (\(R^{2}\)) value of 0.951, signifying its ability to explain a significant portion of the variance in the training data, thereby affirming its robust fit to the dataset. Additionally, the CART model demonstrated precise handling of errors, evidenced from the low Mean Squared Logarithmic Error (MSLE) of 0.017 and the Root Mean Squared Logarithmic Error (RMSLE) of 0.129. In the testing dataset, the CART model exhibited a slightly higher MAE of 23.583 kPa, indicating a slightly lower predictive accuracy on unseen data compared to the training dataset. Nevertheless, the model's predictive ability extended well to the testing data, as evidenced from the high \(R^{2}\) value of 0.921, suggesting its effectiveness in explaining the variance in the testing dataset. Although the MSE and RMSE values of 981.006 and 31.321, respectively, were higher in the testing dataset, the model still maintained good performance, as indicated by the relatively low MSLE of 0.031 and RMSLE of 0.177. Overall, the results in Table 7 validate the reliability of the best CART model in predicting the bearing capacity of shallow foundations and underscore its potential as a valuable tool for practical applications in foundation engineering, enabling more accurate and informed decision making. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Metric** & **Training Dataset** & **Testing Dataset** \\ \hline MAE & 18.388 & 23.583 \\ MSE & 568.105 & 981.006 \\ RMSE & 23.835 & 31.321 \\ MSLE & 0.017 & 0.031 \\ RMSLE & 0.129 & 0.177 \\ \(R^{2}\) & 0.951 & 0.921 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Overall performance of the best CART model to predict the ultimate bearing capacity of shallow foundations for both training and testing databases. Figure 7: Results of proposed CART based on (**a**) training database and (**b**) testing database.