## 5 Results and Discussion  
### Results of the Proposed Four Hybrid Models  
To determine the optimal solution and corresponding hyperparameter combination (i.e., Nf and Maxdepth) of the RF model, all hybrid models were performed during 200 iterations. The iteration curves of each hybrid RF model with five solutions are shown in Figure 6. As it can be seen in Figure 6a, it is obvious that the BSA-RF with 60 solutions has a lower fitness value than the other four BSA-RF models during the 200 iterations. The solution of 90 is the most suitable for generating the MVO-RF model by means of the lowest fitness value (see Figure 6b). As illustrated in Figure 6c,d, the optimal solutions of the GEO-RF and PRO-RF models are equal to 120 and 90, respectively. Table 2 lists the results and the best hyperparameter combination of the four optimized RF models. The PRO-RF model has the lowest value of RMSE (0.1861) among all models. This optimized RF model shows that the optimal Nf and Maxdepth values are 17 and 2, respectively. Figure 6: Iteration curves of four hybrid RF models for predicting the rock EM. Four hybrid RF models with the optimal hyperparameter combinations were used to predict the rock EM in the training phase. The performance indices of each hybrid RF model are listed in Table 3. As illustrated in this table, four models achieved a good prediction accuracy with high values of R\({}^{2}\) and WI and low values of RMSE and MAE. Compared with the other three models, the PRO-RF is the best prediction model for predicting the EM by means of the best performance indices, i.e., R\({}^{2}\) is equal to 0.9423, RMSE to 10.7420, MAE to 7.6514 and WI is equal to 0.9843. The ranking score results of all models indicated that the GEO-RF model with the lowest total score of 4 is the worst model among all hybrid models for predicting EM in the training phase. The score of the MVO-RF model is the second model in the rank just after the PRO-RF model. As illustrated in Figure 7, the regression distribution of the predicted and the measured EM values represents the performance of the prediction models. The position of each rock EM data in the two-dimension regression diagram is determined by the predicted EM values (horizontal axis, x) and the measured EM values (vertical axis, y). If y = x, it means that the predicted EM value is equal to the measured EM value. The rock EM data are then located on the diagonal line. To this end, the PRO-RF model has obtained the most rock EM data on the diagonal or close to the line among the four hybrid RF models (Figure 7d). After the PRO-RF model, the MVO-RF model and the BSA-RF model have similar regression distributions for both the small and big rock EM data. The MVO-RF model has obtained more data close to the diagonal line than the BSA-RF model in the interval between 20 and 70. To further determine the performance of all hybrid RF models, the performance indices were calculated again using the test set, as shown in Table 4. The best model is still the PRO-RF, which has the lowest values of RMSE and MAE (10.1548 and 6.0423) and the highest values of R\({}^{2}\) and WI (0.9410 and 0.9840). The predictive performance of the BSA-RF model is better than the MVO-RF model in the testing phase, the former not only has the better performance indices but also has a higher score (12) than the latter (8). In addition, the GEO-RF model still does not achieve better predictive performances than the other models using the test set.