## 3 Results and Discussion
### Machine Learning Results and Discussion
#### 3.2.1 Comparison of Machine Learning Models  
This section provides a concise comparative analysis of various machine learning models used in this study, with a specific focus on their performance across training validation and testing datasets. The evaluation includes both quantitative metrics, as presented in Table 5, and visual representations depicted in Figures 14 and 15. Additionally, Figure 16A-F and Figure 17A-F provide graphical illustrations that collectively enhance the understanding of the relative predictive capabilities of these models. Based on the metrics presented in Table 5 and supported by Figures 14 and 15, ANN and XGBoost had superior performance measures. However, ANN performed better than XGBoost in predicting \(\sigma_{min}\). ANN and XGBoost achieved the highest R-squared values and the lowest RMSE and MAE, establishing themselves as the top performers in predicting applied stress and dry unit weight. GBR followed closely in second place, showcasing strong predictive capabilities, while SVR and RFR ranked third in terms of predictive accuracy. However, DTR demonstrated a less favorable performance, indicated by its performance metrics and instances of overestimation for specific data points. It is important to note that while SVR achieved metrics similar to GBR, its lower ranking was due to the presence of mispredicted points that deviated significantly from the best fit trend line (ideal line) as shown in Figures 16 and 17. As a result, in predicting \(\sigma_{min}\) and \(\gamma_{max}\) for the expansive soil under consideration, ANN and XGBoost outperformed GBR, RFR, SVR, and DTR. It should be highlighted that the generalization capacity of an ML model is measured by how well it predicts the testing (unseen) dataset. Metric scores in the testing stage unsurprisingly show a decrease in performance for all models, albeit somewhat, due to the unknown data, when compared to the training stage. To quantify this performance gap, Equation (4) [86] defines the degradation rate, or \(r_{d}\):  
\[r_{d}=\left|\frac{m_{train}-m_{test}}{m_{train}}\right|\times 100\% \tag{4}\]  
where \(m_{train}\) and \(m_{test}\) are the values of a specific measure throughout the testing and training stages. All performance measurements reveal that among all models, the ANN produced the least degree of performance degradation. The SVR comes next, although its metrics were not the best. The performance of the other algorithms fell dramatically, especially when it came to the RMSE and MAE for DTR. The ANN produced the most consistent results across both performance phases (training and testing), while the XGBoost and ANN produced the most accurate predictions and highest metrics as shown in Figures 14 and 15. Figure 14: Comparison of RMSE and MAE values in testing sets across various models of \(\sigma_{init}\) predictions. Figure 15: Comparison of RMSE and MAE values in testing sets across various models of \(\gamma_{max}\) predictions. So, ANN is recommended as the best model as it achieved a high performance with the least degree of performance degradation. Figure 16: Comparison of actual and predicted targets for \(\sigma_{min}\) models; (**A**) DR; (**B**) RFR; (**C**) GBR; (**D**) XGBoost; (**E**) SVR; (**F**) ANN. The visualization in Figures 16 and 17 provides a thorough analysis of the models' predictive capabilities of \(\sigma_{min}\) and \(\gamma_{max}\), respectively. Figures 16 and 17 depict the comparison between actual and predicted values for training, validation, and test sets. In the training set, most of the models demonstrated excellent accuracy, achieving an R-squared value of almost 1 and a low RMSE and MAE. This indicates that it effectively captured the variability in the training data and made minimal prediction errors. However, it is worth noting that some data points notably deviated from the best fit trend line (deal line), indicating limitations of the SVR and DTR models in specific situations.