## 5 Model Results
### Evaluation of MARS Models on Unseen Dataset  
The following two equations were derived from the final optimal models after hyper-parameter tuning for \(w_{opt}\) and \(\rho_{\rm{dmax}}\). The equations are as follows:  
\[ \begin{array}{ll}w_{opt}&=19.3773983\\ &-0.0639987\times\max(97-CF,0)\\ &-5.1538333\times\max(CF-97,0)\\ &-0.0898632\times\max(64-LL,0)\\ &-0.6415284\times\max(21-PL,0)\\ &+0.4288445\times\max(PL-21,0)\\ &+0.0025993\times\max(1347-\text{EkJ}/\text{m}^{3},0)\\ &+0.2425873\times\max(CF-97,0)\times PL\\ &-0.0000061\times LL\times\max(\text{EkJ}/\text{m}^{3}-1347,0)\\ &+0.0578447\times\max(7.6-CG,0)\times\max(21-PL,0)\\ &-0.0093695\times\max(70-CF,0)\times\max(PL-21,0).\end{array}  \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Performance Measures**} \\ \cline{2-4}
**Folds** & **RMSE (Mg/m\({}^{3}\))** & **R\({}^{2}\)** & **MAE (Mg/m\({}^{3}\))** \\ \hline Fold 1 & 0.068 & 0.860 & 0.051 \\ Fold 2 & 0.067 & 0.908 & 0.050 \\ Fold 3 & 0.048 & 0.948 & 0.040 \\ Fold 4 & 0.071 & 0.870 & 0.057 \\ Fold 5 & 0.065 & 0.909 & 0.049 \\ Average & 0.064 & 0.899 & 0.050 \\ SD & 0.009 & 0.035 & 0.006 \\ CoV (\%) & 0.141 & 0.039 & 0.120 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results derived from a five-fold cross-validation for the \(\rho_{\rm{dmax}}\) model. \[ \begin{array}{ll}\rho_{dmax}&=1.39385529\\ &+0.00261487\times\max(33.1-CS,0)\\ &+0.00621111\times\max(98-CF,0)\\ &+0.00505340\times\max(LL-64,0)\\ &+0.00337354\times\max(130-LL,0)\\ &-0.00526625\times\max(LL-130,0)\\ &-0.01028165\times\max(PL-11,0)\\ &-0.00019175\times\max(1347-\text{EkJ}/\text{m}^{3},0)\\ &-0.00188667\times\max(CF-98,0)\times PL\\ &-0.00005026\times\max(CS-53,0)\times\max(98-CF,0)\\ &+0.00002616\times\max(CF-30,0)\times\max(130-LL,0).\end{array}  \tag{10}\]  
The cross plots presented in the subplots (a) and (b) of Figure 6 portray the application of two previous MARS-based models in predicting two crucial parameters for compacted soil: the optimum water content (\(w_{opt}\)) and maximum dry density (\(\rho_{\text{dmax}}\)). These subplots showcase the comparison between the models' predictions and the actual values of \(w_{opt}\) and \(\rho_{\text{dmax}}\) for the testing datasets. The plots illustrate that a significant accumulation of data points clustering near the \(Y=X\) line signifies the strength and reliability of the established models. Clearly, the implemented models yield predictions that align remarkably well with the unit slope line. As a result, the proposed models can confidently be considered highly reliable for estimating the \(w_{opt}\) and \(\rho_{\text{dmax}}\) of compacted soil throughout the testing and training phases. Figure 7 showcases the relative error of the MARS-based models compared to the actual measurements of \(w_{opt}\) and \(\rho_{\text{dmax}}\). The subplots in Figure 7 demonstrate that the outputs of the proposed MARS-based models exhibit an acceptable deviation from the real measured data. A significant portion of the data falls within the range of \(-10\%\) to \(+10\%\) for \(w_{opt}\) and \(-5\%\) to \(5\%\) for \(\rho_{\text{dmax}}\). Additionally, the statistical evaluation of the proposed models is presented in Table 4, providing a comprehensive summary of the assessment. The table includes the MAE, R\({}^{2}\), and RMSE values for the training, testing, and overall datasets. The statistical evaluation clearly indicates that both MARS-based models exhibit outstanding prediction performance. The overall RMSE values for \(w_{opt}\) and \(\rho_{\text{dmax}}\) are 1.428 and 0.052, respectively, highlighting the models' accuracy. The combined findings from Figures 6 and 7, along with the statistical  
Figure 6: Actual vs. predicted values using the MARS model for the testing dataset: (**a**) \(\rho_{\text{dmax}}\) and (**b**) \(w_{opt}\). results in Table 4, demonstrate the effectiveness of our newly proposed correlations.