## 5 Results and Discussion
### B. Regression Model  
To determine the best regression model for the normalized training set, we used regression feature selection methods such as forward selection, backward elimination, and the best subsets. All these methods unanimously selected the second-order regression model with two explanatory variables, I\({}_{\text{d2}}\) and G\({}_{\text{s}}\). All the parameters were highly significant (see Table 3), and the coefficient of determination R\({}^{2}\) and adjusted R\({}^{2}\) were 51.8% and 50.5%, respectively. Figure 5 shows the predicted values from the interpolated regression model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Coeff.** & **T-Value** & _p-Value_ & **VIF** \\ \hline Constant & 2.5878 & 36.39 & \(p\) \textless{} 0.001 & \\ I\({}_{\text{d2}}\) & 0.4622 & 6.00 & \(p\) \textless{} 0.001 & 1.14 \\ G\({}_{\text{s}}\) & 0.3006 & \(-\)3.91 & \(p\) \textless{} 0.001 & 1.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MLR Parameter Estimates. Figure 4: MSE and MAE as a function of number of epochs. The normality test of the residuals is shown in Figure 6. The \(p\)-value of the Kolmogorov-Smirnov test exceeded 15%, which clearly shows that there was no deviation from normality. Besides, the variance inflation factor (VIF) was very low (1.14), indicating that multicollinearity was not detected. VIF values exceeding 10 were regarded as indicative of multicollinearity. Figure 7 shows a diagnostic plot for the residuals of the model. The residual plot in Figure 7 did not show any pattern of heteroscedasticity in the \"nonconstant error variance\". To test the correlation among the residuals, the Durbin-Watson test was performed, and a test statistic of d = 2.50 was obtained. At the 5% significance level, the upper critical value of the test was du, 0.025 = 1.51; clearly, the observed value of the test statistic was larger than both du, 0.025 and 4-du, 0.025, these critical values support the claim that those errors are not correlated. Figure 5: BTS values predicted from interpolation. Figure 6: Residuals normality plot. The family of penalized regression techniques was an alternative to MLR models and examples of these family include Ridge, Lasso, and Elastic Net. Lasso and Elastic Net are feature selection tools as well as predictive modeling techniques. These models apply regularized constraints (\(\lambda\), \(\alpha\)) to the model coefficients and shrink some of them to zero. To determine the optimal regularization parameter \(\lambda\) for these models, the cv.glmnet, and glmnet R packages were used. These functions use penalized maximum likelihood method to fit generalized linear models. The Ridge, Lasso, and the Elastic Net model paths were fitted using the mean-squared error CV criterion. The workflow of the methodology of the study is summarized in Figure 8.  
First, a sequence of n lambdas was generated, and the training dataset was divided into K = 10 folds. The model was cross-validated n times using nine subsamples as the training set and the remaining sample as the test set. Each time, as MSE was computed, a fold was removed and a different one was chosen. The lambda value with the smallest MSE was chosen, and the best model was fitted. The upper panels of Figure 9 show CV MSE as a function of log (\(\lambda\)). The vertical dashed lines in these plots represent the log (\(\lambda\))  
Figure 8: Methodology workflow. Figure 7: Regression residual plots. value with the minimum MSE and the largest value of log (\(\lambda\)) within one standard error to that of log (\(\lambda\)) with the minimum MSE for the Ridge and Lasso models, respectively. The plots in the lower panels of Figure 9 show the shrinking of the coefficients of the Ridge and Lasso models as a function of log (\(\lambda\)). All the coefficients of the Ridge and Lasso models approach zero at log (\(\lambda\)) = 4 and \(-1.5\), respectively, whereas those of the Elastic Net approach zero at log (\(\lambda\)) = 1.