## 3 Results and Discussion
### Machine Learning to Predict \(q_{u}\) and \(E_{t}\)
#### 3.4.1 Comparison of Machine Learning Models  
This section presents a comparison of the models used in this study. The metrics derived from k-fold and TVT cross-validation techniques are shown in Tables 8 and 9, respectively. Tables 8 and 9 show that ANN-GMDH, ANN and SVR have superior performance metrics, with SVR closely matching ANN's performance. As a result, ANN-GMDH, ANN and SVR outperform XGBoost, GBR, RFR, and DTR in predicting \(E_{t}\) and \(q_{u}\) for the expansive soil under consideration. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Metrics**} & \multicolumn{4}{c}{**\(E_{t}\)**} & \multicolumn{4}{c}{\(q_{u}\)} \\ \cline{2-10}  & & **Training** & **Validation** & **Testing** & **r\({}_{d}\) (\%)** & **Training** & **Validation** & **Testing** & **r\({}_{d}\) (\%)** \\ \hline \multirow{3}{*}{DTR} & R\({}^{2}\) & 0.9975 & 0.8243 & 0.8372 & 16.07 & 0.99971 & 0.8132 & 0.9065 & 9.32 \\  & RMSE & 102.41 & 808.88 & 787.88 & 669.34 & 6.88 & 140.64 & 113.09 & 1543.75 \\  & MAE & 22.73 & 634.1 & 622.84 & 2640.17 & 1.4963 & 91.589 & 75.847 & 4968.97 \\ \hline \multirow{3}{*}{RFR} & R\({}^{2}\) & 0.9857 & 0.9088 & 0.898 & 8.90 & 0.982 & 0.83 & 0.883 & 10.08 \\  & RMSE & 244.33 & 582.83 & 623.55 & 155.21 & 54.21 & 134.144 & 126.724 & 133.76 \\  & MAE & 184.3 & 483.757 & 484.45 & 162.86 & 36.84 & 89.92 & 77.85 & 111.32 \\ \hline \multirow{3}{*}{GBR} & R\({}^{2}\) & 0.9991 & 0.985 & 0.9764 & 2.27 & 0.99886 & 0.9133 & 0.9495 & 4.94 \\  & RMSE & 59.39 & 236.44 & 300 & 405.14 & 13.637 & 95.806 & 83.11 & 509.44 \\  & MAE & 48 & 164.36 & 224.48 & 367.67 & 10.462 & 55.105 & 49.745 & 375.48 \\ \hline \multirow{3}{*}{XGBoost} & R\({}^{2}\) & 0.99824 & 0.988 & 0.9853 & 1.30 & 0.99902 & 0.9568 & 0.9557 & 4.34 \\  & RMSE & 85.81 & 211.14 & 237.2 & 176.42 & 12.664 & 67.644 & 77.818 & 514.48 \\  & MAE & 66.1 & 148.44 & 169.82 & 156.91 & 9.155 & 44.433 & 51.099 & 458.15 \\ \hline \multirow{3}{*}{SVR} & R\({}^{2}\) & 0.996 & 0.9947 & 0.9909 & 0.51 & 0.9972 & 0.99655 & 0.98873 & 0.85 \\  & RMSE & 131.02 & 140.61 & 186.27 & 42.17 & 21.5785 & 19.124 & 39.265 & 81.96 \\  & MAE & 62.83 & 110.55 & 128.37 & 104.31 & 10.99 & 13.27 & 22.374 & 103.59 \\ \hline \multirow{3}{*}{ANN} & R\({}^{2}\) & 0.9951 & 0.9955 & 0.9924 & 0.27 & 0.995 & 0.9963 & 0.9944 & 0.06 \\  & RMSE & 145.4 & 129.15 & 170.03 & 16.94 & 28.54 & 19.8 & 27.63 & 3.19 \\  & MAE & 109.116 & 108.792 & 115.408 & 5.77 & 19.471 & 15.83 & 18.9 & 2.93 \\ \hline \multirow{3}{*}{ANN-GMDH} & R\({}^{2}\) & 0.9963 & 0.9956 & 0.9935 & 0.28 & 0.996 & 0.9968 & 0.9949 & 0.11 \\  & RMSE & 125.67 & 128.24 & 157.31 & 25.18 & 25.54 & 18.43 & 26.37 & 3.25 \\ \cline{1-1}  & MAE & 95.66 & 96.658 & 120.86 & 26.34 & 17.73 & 14.24 & 18.45 & 4.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The study’s metrics for the various models using the Train/Validation/Test Split cross-validation technique. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Regation**} & \multirow{2}{*}{**Metrics**} & \multicolumn{4}{c}{**Training**} & \multicolumn{1}{c}{**Testing**} \\ \cline{3-8}  & & **Training** & **Validation** & **Testing** & **r\({}_{d}\) (\%)** & **Training** & **Validation** & **Testing** & **r\({}_{d}\) (\%)** \\ \hline \multirow{3}{*}{ \begin{tabular}{c} Linear \\ regression \\ \end{tabular}  } & R\({}^{2}\) & 0.9975 & 0.8243 & 0.8372 & 16.07 & 0.99971 & 0.8132 & 0.9065 & 9.32 \\  & RMSE & 102.41 & 808.88 & 787.88 & 669.34 & 6.88 & 140.64 & 113.09 & 1543.75 \\  & MAE & 22.73 & 634.1 & 622.84 & 2640.17 & 1.4963 & 9A comprehensive assessment of machine learning models was undertaken to evaluate their effectiveness in predicting \(E_{t}\) and \(q_{u}\), with results presented across mean absolute error (MAE), root-mean=square error (RMSE), and R-squared (R\({}^{2}\)) metrics in Tables 8 and 9. Decision tree regression (DTR) demonstrates strong performance on the training set but struggles with overfitting on validation and testing sets, limiting its extrapolative capability.