## 5 Results and Discussion
### Performance Comparison between the Proposed and Other Models  
To compare the predictive performance with the proposed hybrid models, five common ML models, named the ANN, SVR, extreme learning machine (ELM), kernel-extreme learning machine (KELM) and generalized regression neural network (GRNN), and one empirical formula, were also developed to predict the rock EM. The definition and the hyperparameter settings of the five ML models can be found in [61, 62, 63, 64, 65]. The multivariate-quadratic equation (MQE) of MR was used to generate an empirical formula as expressed using Equation (23). \[ \begin{array}{llllllllllll}\text{EM}&=&-136.609&+&7.864\text{PLI}&-&1.056 \text{PLI}^{2}&-&17.038\text{P}_{n}&+&1.378\text{P}_{n}^{2}\\ &+&4.265\text{V}_{\text{P}}&+&1.084\text{V}_{\text{P}}^{2}&+&5.297\text{SHRN}&-& 0.042\text{SHRN}^{2}\end{array}  \tag{23}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multicolumn{8}{c}{**Performance Indices and Ranking Scores**} & \multirow{2}{*}{**Total**} \\ \cline{2-2} \cline{4-9}  & \(\mathbf{R^{2}}\) & **Score** & **RMSE** & **Score** & **MAE** & **Score** & **WI** & **Score** \\ \hline BSA-RF & 0.9322 & 3 & 10.8902 & 3 & 7.2155 & 3 & 0.9812 & 3 & 12 \\ MVO-RF & 0.9236 & 2 & 11.5567 & 2 & 7.5280 & 2 & 0.9785 & 2 & 8 \\ GEO-RF & 0.9123 & 1 & 12.3826 & 1 & 7.9739 & 1 & 0.9746 & 1 & 4 \\ PRO-RF & 0.9410 & 4 & 10.1548 & 4 & 6.0423 & 4 & 0.9840 & 4 & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance and ranking results of four hybrid RF models in the testing phase. Figure 7: Regression diagrams of the four hybrid RF models using the train set: (**a**) BSA-RF model; (**b**) MVO-RF model; (**c**) GEO-RF model; (**d**) PRO-RF model. The optimal hyperparameter combination of each model and the corresponding performance results are shown in Table 5. The GRNN obtained better performance indices (R\({}^{2}\): 0.9010; RMSE: 13.1593; MAE: 8.2674 and WI: 0.9717) than the other models using the test set. After this model, the prediction accuracy of the KELM and the ELM models is superior to the ANN and the SVR models. The worst prediction model is the MQE; its index values of R\({}^{2}\), RMSE, MAE and WI are, respectively, equal to 0.8318, 17.1476, 13.5849 and 0.9497. Figure 9 shows the rank scores of all comparison models. It can be obviously observed that the performance ranking of the six models is GRNN (24), KELM (20), ELM (16), ANN (12), SVR (8) and MQE (4). The regression diagrams of the six comparison models in the testing phase are presented in Figure 10. As it can be seen in these diagrams, all models have a finite number of rock EM data points near the diagonal line. Compared with the other five models, the GRNN model obtained a good EM regression distribution in the range of 70 to 90 (Figure 10e). For the MQE model, most of the rock EM data points are away from the diagonal, which also indicates a large error between the EM value predicted by this model and the measured EM value. As a result, the GRNN model is the best prediction model  
Figure 8: Regression diagrams of the four hybrid RF models using the test set: (**a**) BSA-RF model; (**b**) MVO-RF model; (**c**) GEO-RF model; (**d**) PRO-RF model.