## 4 Discussion
### Model Prediction Performance Evaluation
Figure 8: Distribution of prediction results and histogram of differences based on three hybrid models: (**a**) ALO-RF(Pop = 10, Train set); (**b**) ALO-RF(Pop = 10, Test set); (**c**) WVO-RF(Pop = 100, Train set); (**d**) WVO-RF(Pop = 100, Test set); (**e**) GOA-RF(Pop = 10, Train set); (**f**) GOA-RF(Pop = 10, Test set). After determining the optimal population size for each hybrid model (ALO-RF, WVO-RF, and GOA-RF), further comparisons were made with the unoptimized RF model. Figure 9 depicts Taylor diagrams for four models and lists the corresponding evaluation indices. Taylor diagrams can demonstrate the model fitting performance for solving regression problems. The picture depicts a negative correlation between the azimuth of the points and the correlation coefficient, while the distance between the point and the reference point represents the RMSE. This tool is commonly used to compare the observed results and predicted results by prediction model [95]. The positions of ALO-RF (Pop = 10), WVO-RF (Pop = 100), and GOA-RF (Pop = 10) models are closer to the observed points in the Taylor diagram than the unoptimized RF model, indicating that the three models have a lower RMSE than the unoptimized RF model. In addition, the smaller azimuth showed that ALO-RF, WVO-RF, and GOA-RF have greater correlation coefficients than unoptimized RF sites, which demonstrated that the three optimized algorithms play a significant role in improving the accuracy of \(GS\) prediction. Based on the calculated evaluation indices of all models in the training and testing phases, the R\({}^{2}\) values of ALO-RF (Pop = 10) (Train: R\({}^{2}\) = 0.9338, Test: R\({}^{2}\) = 0.9055), WVO-RF (Pop = 100) (Train: R\({}^{2}\) = 0.9409, Test: R\({}^{2}\) = 0.9202), and GOA-RF (Pop = 10) (Train: R\({}^{2}\) = 0.9487, Test: R\({}^{2}\) = 0.9282) are significantly greater than the unoptimized RF (Train: R\({}^{2}\) = 0.7912, Test: R\({}^{2}\) = 0.7195), indicating that the prediction accuracy of the three hybrid RF models are more accurate than the unoptimized RF model. Furthermore, all hybrid RF models provided significantly lower RMSE, MAE and MAPE values than the unoptimized RF model for the \(GS\) prediction. Figure 7d ranks the evaluation metrics for ALO-RF (Pop = 10), WVO-RF (Pop = 100), GOA-RF (Pop = 10), and RF as shown in Figure 10. The Taylor diagrams in Figure 9 show that the point represented by the GOA-RF model is the closest to the observation point (indicating its smallest RMSE) and its azimuth is the smallest (indicating its largest correlation coefficient), both in the training and testing phases. In conjunction with the comparison and the distribution of Taylor diagrams points shown in Figure 9b, the GOA-RF (Pop = 10) model achieves the optimal \(GS\) prediction performance among all hybrid models. Table 5 displays the results of the predictive performance comparison between the proposed GOA-RF model, the unoptimized RF model, and previous studies. The results indicated that the GOA-RF model has significantly improved the \(GS\) prediction accuracy compared to previous published results. On the other hand, the GOA algorithm has a significant advantage in optimizing the hyperparameters of RF model for the \(GS\) prediction. In addition, there is no significant over- or under-fitting occurred in the GOA-RF (Pop = 10) model using the train and test sets. Therefore, it is recommended to use this model to predict the GS.