## 3 Results and Discussion  
To check the performance of the developed PSO-ELM model, two well-known soft computing models (PSO-ANN, and KELM) were also tested for comparison and performance validation. Table 2 shows the goodness of fit measures of the investigated three models. It is evident from the table that the PSO-ELM model was superior in prediction accuracy to the other two models, PSO-ANN and KELM. During the training phase, the highest \(r^{2}\) was observed for PSO-ELM (r\({}^{2}\) = 0.981) followed by KELM (r\({}^{2}\) = 0.693) and PSO-ANN (r\({}^{2}\) = 0.64). Meanwhile, during the testing phase, the prediction accuracy was slightly lower than that obtained during training phase for all of the investigated models. In terms of RMSE, a lower value of RMSE was observed for PSO-ELM (RMSE = 369.592) when compared to KELM (RMSE = 1075.378) and PSO-ANN (RMSE = 1184.155). Figure 3 shows the relationship between the observed and predicted resilient moduli for the investigated three models. The slopes of linear fitting of the PSO-ELM model were 0.98 and 1.01 for the training and testing stages, respectively. Comparison between the linear fitting slopes of the investigated three models shows that the slope of the linear fitting of data for the PSO-ELM model was closer to unity than that of the other models, indicating lower bias. It is evident from both Table 2 and Figure 3 that the PSO-ELM model showed the best fit of the M\({}_{\rm r}\) data along the equality line among the three models during both the training (Figure 3a) and testing (Figure 3b) phases. In addition, in order to study the influence of input variables on predicting M\({}_{\rm r}\) values, four different models with changing input variables were built and studied in the training stage, as presented in Table 3. Little significant effect was observed on the prediction of M\({}_{\rm r}\) as a result of changing the input variables, indicating that all input variables could be considered when modelling M\({}_{\rm r}\). The influence of correlation between M\({}_{\rm r}\) and input variables depends on the variation in the used data, test protocol stress states, material type, and sample preparation and conditioning. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **r\({}^{2}\)** & **RMSE** & **MAE** \\ \hline PSO-ANN (Train) & 0.640 & 1117.367 & 881.90 \\ PSO-ANN (Test) & 0.597 & 1184.155 & 929.18 \\ \hline KELM (Train) & 0.692 & 1064.782 & 804.90 \\ KELM (Test) & 0.674 & 1075.378 & 815.94 \\ \hline PSO-ELM (Train) & 0.981 & 253.439 & 191.66 \\ PSO-ELM (Test) & 0.963 & 369.592 & 280.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance measures for the investigated models. For better representation, a mathematical graphical Taylor diagram was also plotted, as seen in Figure 4, using R software [51]. It evaluates the degree of correspondence between the observed and predicted M\({}_{\text{r}}\) values in terms of the r\({}^{2}\), standard deviation, and RMSE in a single plot [52]. From Figure 4, it can be seen that the developed model (PSO-ELM) had the highest prediction accuracy by a significant margin, when compared with the other investigated models. Furthermore, a new generation violin plot was also drawn, as can be seen in Figure 5, in order to understand the model prediction distribution behavior or variation over the predicted M\({}_{\text{r}}\) data. This violin plot is a hybrid of  
Figure 3: Scatter plot of the three investigated models: (**a**) training, (**b**) testing. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Input Variables** & \(\textbf{r}^{2}\) & **RMSE** & **MAE** \\ \hline
1 & W-D, CSAFR, DMR, \(\sigma_{3}\) and \(\sigma_{d}\) & 0.981 & 253.439 & 191.66 \\
2 & W-D, CSAFR, DMR and \(\sigma_{3}\) & 0.948 & 415.554 & 299.43 \\
3 & W-D, CSAFR, DMR and \(\sigma_{d}\) & 0.973 & 304.451 & 204.98 \\
4 & W-D, CSAFR and DMR & 0.921 & 521.08 & 378.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input variable effect on the PSO-ELM model.