## 4 Results and Discussion
### Improved RF-Based Models  
In this section, five methods were used to optimize the regular RF model, namely the refined weights of decision trees based on their predictive capabilities that are characterized by R\({}^{2}\) and the RMSE, and the refined weights of decision trees based on the optimization solution of three metaheuristic algorithms (i.e., GWO, WOA, and TSA). The main goal of the five techniques is to obtain the optimal weight set of decision trees that can minimize the generalization error of the improved RF models. Moreover, to validate the models' performance, each model's accuracy and error on the training set and testing set are both utilized. In this way, the optimal improved RF model can be effectively obtained. Figure 5: MSE with respect to the number of trees in a forest. Figure 6: MSE with respect to the maximum depth of trees in a forest. Figure 7 presents the calculation results of the three metaheuristic algorithms on the training set. It can be seen that there are some differences in the calculation results due to the swarm sizes. For example, for the RF-WOA model, the final fitness value is 200 swarms sizes \(<\) 150 swarm sizes \(<\) 100 swarm sizes \(<\) 50 swarm sizes; for the RF-GWO model, the final fitness value is 100 swarm sizes \(<\) 200 swarm sizes \(<\) 150 of swarm sizes \(<\) 50 swarm sizes; and for the RF-TSA model, the final fitness value is 150 of swarm sizes \(<\) 200 swarm sizes \(<\) 50 of swarm sizes \(<\) 100 swarm sizes. It should be noted that the above results are obtained based on the training set. For a clearer identification of the performance of each model, it should be combined with its performance on the testing set. For this purpose, a scoring evaluation method was employed to select the best model [60]. The principle of this method is that the higher the score, the better the performance, and vice versa. Consequently, the final scoring of the RF-WOA, RF-GWO, and RF-TSA models with different swarm sizes on both training and test sets is obtained in Tables 3-5. According to the results in Tables 3-5, it can be concluded that the best RF-WOA model with a total score of 26 is obtained when the swarm sizes are set to 100, the best RF-GWO model with a total score of 28 is obtained when the swarm sizes are set to 200, and the best RF-TSA model with a total score of 27 is obtained when the swarm sizes are set to 50. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multicolumn{6}{c}{**Training**} & \multicolumn{6}{c}{**Testing**} \\ \cline{2-13}  & **Swarm Size** & \(\mathbf{R^{2}}\) & **Score** & **RMSE** & **Score** & **MAE** & **Score** & **VAF** & **Score** & **R\(\mathbf{^{2}}\)** & **Score** & **RMSE** & **Score** & **MAE** & **Score** & **VAF** & **Score** \\ \hline \multirow{3}{*}{RF-WOA} & 50 & 0.986 & 4 & 0.116 & 4 & 0.092 & 4 & 98.584 & 1 & 0.930 & 4 & 0.250 & 4 & 0.192 & 3 & 94,456.3 & 27 \\  & 100 & 0.986 & 4 & 0.116 & 4 & 0.092 & 4 & 98.633 & 2 & 0.929 & 3 & 0.251 & 3 & 0.192 & 3 & 94,4751 & 24 \\  & 150 & 0.986 & 4 & 0.116 & 4 & 0.092 & 4 & 98.637 & 4 & 0.929 & 3 & 0.252 & 2 & 0.193 & 2 & 94,488.2 & 25 \\  & 200 & 0.986 & 4 & 0.116 & 4 & 0.093 & 3 & 98.635 & 3 & 0.929 & 3 & 0.251 & 3 & 0.191 & 4 & 94,528.4 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of different swarm sizes of the RF-GWO models on predicting the PPV. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multicolumn{6}{c}{**Training**} & \multicolumn{6}{c}{**Testing**} \\ \cline{2-13}  & **Swarm Size** & \(\mathbf{R^{2}}\) & **Score** & **RMSE** & **Score** & **MAE** & **Score** & **VAF** & **Score** & **R\(\mathbf{^{2}}\)** & **Score** & **RMSE** & **Score** & **MAE** & **Score** & **VAF** & **Score** \\ \hline \multirow{3}{*}{RF-WOA} & 50 & 0.986 & 4 & 0.119 & 1 & 0.094 & 2 & 98.584 & 1 & 0.932 & 4 & 0.246 & 4 & 0.187 & 4 & 94,723 & 3 & 23 \\  & 100 & 0.986 & 4 & 0.118 & 2 & 0.093 & 3 & 98.603 & 2 & 0.932 & 4 & 0.246 & 4 & 0.188 & 3 &