## 4 Results and Discussion
### Improved RF-Based Models
Therefore, it can be inferred that the RF-WOA has a better generalization ability than other models, which can be verified as it has a better performance on the testing set. Additionally, we can also conclude that all of the refined weights RF models have a good performance on the testing set compared with the regular RF model, which means the refined techniques proposed in this paper can effectively enhance the performance of the regular RF model. Figure 8: Results of the evaluation metrics of the developed models on the training set. In a different way, the Taylor diagram is presented in this work, which may be used to graphically summarize how closely a set of patterns fits observations [61]. The similarities between the patterns and observations are quantified by utilizing their correlation coefficients, centered root-mean-square errors, and standard deviations, as shown in Equation (40) [62]:  
\[E^{\prime 2}=\sigma_{p}^{2}+\sigma_{a}^{2}-2\times\sigma_{p}\times\sigma_{a}\times R \tag{40}\]  
where \(E^{\prime}\) is the centered root-mean-square error between the predicted and measured values, \(\sigma_{p}^{2}\) and \(\sigma_{a}^{2}\) are the variances of the predicted and measured values, respectively, and \(R\) is the correlation coefficient between the predicted and measured values. Figures 10 and 11 depict how closely the six developed models match the training and testing sets, respectively, as references. In the Taylor diagram, the standard deviation is shown by the distance between the circles representing the models and the origin point on the \(x\)-axis, and the ticks on the clockwise arc represent the correlation coefficient. The actual PPV values are shown by the star-shaped point 'REF' on the \(x\)-axis, and the distance between the other circles and the point 'REF' reflects the centered RMSE (i.e., the grey arc). Figure 10: Taylor diagram of the training results for the six developed models. Figure 9: Results of the evaluation metrics of the developed models on the testing set. In Figure 10, it can be found that the standard deviations of all six models are less than the standard deviations of the measured values, and the standard deviations of the RF-WOA model, RF-GWO model, and RF-TSA model are closer to the standard deviation of the actual values compared with the models of the RF, RF-R\({}^{2}\), and RF-RMSE. Moreover, the same applies to the correlation coefficient, i.e., the correlation coefficients of the RF-WOA model, RF-GWO model, and RF-TSA model are closer to that of the actual values, but they differ little from each other. For the centered RMSE, the results indicate that the RF-WOA model, RF-GWO model, and RF-TSA model have smaller errors than those of the RF, RF-R\({}^{2}\), and RF-RMSE models. From a holistic perspective, the calculation results of the three metaheuristic algorithms on the training set are closer to the actual PPV values, while the RF-R\({}^{2}\) and RF-RMSE models perform slightly better on the training set than the RF model. In Figure 11, it can be inferred that the standard deviations of the three metaheuristic algorithms on the testing set are close to the standard deviation of the actual PPV and slightly greater. With regard to the centered RMSE and correlation coefficient, the calculation results of the RF-WOA models are better than that of the RF-TSA and RF-GWO models. Overall, based on the distance between these circles and the point 'REF', it can be determined that the ranking of the model's superiority on the testing set is as RF-WOA > RF-TSA > RF-GWO > RF-RMSE > RF-R\({}^{2}\) > RF. Compared with the results on the training set, the results on the testing set can more clearly reflect the performance differences among these models, especially for the results obtained by the metaheuristic algorithms. To sum up, the RF-WOA is the optimal model that is recommended for refining the weights of decision trees of the RF model in this study.