## 4 Results and Discussion
### Improved RF-Based Models
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Training**} & \multicolumn{10}{c}{**Testing**} & \multicolumn{10}{c}{**Score Summation**} \\ \cline{2-13}
**Models** & \(\mathbf{R^{2}}\) & **Score** & **RMSE** & **Score** & **MAE** & **Score** & **VAF** & **Score** & **R2** & **Score** & **RMSE** & **Score** & **MAE** & **Score** & **VAF** & **Score** \\ \hline RF-WOA & 0.986 & 6 & 0.118 & 5 & 0.093 & 6 & 98.603 & 5 & 0.932 & 6 & 0.246 & 6 & 0.188 & 6 & 95.032 & 6 & 46 \\ RF-OVO & 0.986 & 6 & 0.116 & 6 & 0.093 & 6 & 98.635 & 6 & 0.929 & 4 & 0.251 & 4 & 0.191 & 5 & 94.528 & 4 & 41 \\ RF-STA & 0.986 & 6 & 0.118 & 5 & 0.093 & 6 & 98.598 & 4 & 0.931 & 5 & 0.248 & 5 & 0.191 & 5 & 94.743 & 5 & 41 \\ RF & 0.973 & 3 & 0.164 & 2 & 0.123 & 3 & 97.298 & 1 & 0.923 & 1 & 0.262 & 1 & 0.209 & 2 & 93.994 & 1 & 14 \\ RF-R2 & 0.974 & 4 & 0.161 & 3 & 0.121 & 4 & 97.380 & 2 & 0.924 & 2 & 0.261 & 2 & 0.207 & 3 & 94.078 & 2 & 22 \\ RF-RMSE & 0.975 & 5 & 0.156 & 4 & 0.119 & 5 & 97.530 & 3 & 0.925 & 3 & 0.258 & 3 & 0.203 & 4 & 94.219 & 3 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance of the RF and improved RF models on predicting the PPV. After obtaining the best model for individual metaheuristic algorithms in their respective model comparisons, the next step is to perform further comparative analysis of them and the two previously mentioned improved weighting methods based on the accuracy (i.e., R\({}^{2}\) and the RMSE) to identify the optimal improved RF model. Concerning the improved weights based on the R\({}^{2}\) and RMSE of every single decision tree, the weights of the decision trees are computed through Equations (28) and (30), respectively. For the metaheuristic algorithms, the RF-WOA with 100 swarm sizes, the RF-GWO with 200 swarm sizes, and the RF-TSA with 50 swarm sizes are chosen because of their best performance. At the same time, the regular RF model is also used for comparative analysis in this section. Then, the results of the performance of these six models are tabulated in Table 6. As can be seen from Table 6, the model with the highest score is the RF-WOA, whose scoring is 46, followed by the RF-GWO and RF-TSA, whose scorings are both 41. As for the RF-RMSE and RF-R\({}^{2}\), their scores were 30 and 22, respectively, which indicates the performance of the RF-RMSE is slightly better than that of the RF-R\({}^{2}\). In general, these five models all outperform the regular RF model with a scoring of 14. Moreover, the ranking of these five models is as follows: RF-WOA > RF-GWO = RF-TSA > RF-RMSE > RF-R\({}^{2}\) > RF. To intuitively observe the differences among these models, we divide the four evaluation metrics into two groups, one for the VAF and R\({}^{2}\)\(\times\) 100, where larger values indicate better model performance, and the other one for the RMSE and MAE, where smaller values indicate better models. As depicted in Figure 8, for the group of the VAF and R\({}^{2}\)\(\times\) 100 (the left one in Figure 8), the accuracies of the RF-GWO, RF-WOA, and RF-TSA are quite close and are all better than the other three models (i.e., RF-RMSE, RF-R\({}^{2}\), and RF), whereas, for the group of the RMSE and MAE (the right one in Figure 8), the error of the RF-GWO is slightly lower than those of the RF-WOA and RF-TSA. Likewise, the errors of the three metaheuristic-based RFs are all lower than those of the RF-RMSE, RF-R\({}^{2}\), and RF. Moreover, it can be concluded that all of the refined weights RF models have a good performance on the training set compared with the regular RF model. As for the testing set, the results are shown in Figure 9. For the group of the VAF and R\({}^{2}\)\(\times\) 100 (the left one in Figure 9), the ranking of the accuracy of these six models is shown as RF-WOA > RF-TSA > RF-GWO > RF-RMSE > RF-R\({}^{2}\) > RF. The same situation occurs in the group of the RMSE and MAE (the right one in Figure 9), which indicates the ranking of error of these six models is RF-WOA < RF-TSA < RF-GWO < RF-RMSE < RF-R\({}^{2}\) < RF.