# Open-Pit Bench Blasting Fragmentation Prediction Based on Stacking Integrated Strategy
## 4 Results and Discussion
### Model Performance Analysis and Discussion  
During the research process, four evaluation indicators were used to assess the predictive performance of the model. Table 3 presents the results of the metrics for the training and testing sets of the stacking integrated model. From the results in the table, it is evident that the MSE and RMSE values of the integrated model on the testing set are lower than those on the training set, while the MAE values on the testing set are slightly higher than those on the training set, though they are very close to each other. A comparison of the  
Figure 7: Comparison of the prediction results between the training and testing sets of the four models: (**a**) random forest; (**b**) SVR; (**c**) XGBoost; (**d**) stacking integrated model. indicators of the four models is shown in Figure 8. It is evident that the \(R^{2}\) value of the stacking integrated model is the highest, at 0.943. The \(R^{2}\) value of the random forest model is relatively close to that of the XGBoost model, with values of 0.797 and 0.758, respectively. The \(R^{2}\) value of the SVR model is the lowest, at 0.578. The stacking ensemble algorithm demonstrates significant improvements in \(R^{2}\), achieving increases of 0.146, 0.365, and 0.185 compared to the respective base algorithms. The comparison of the MSE, RMSE, and MAE results of the four prediction models, shown in Figure 9, reveals that the MSE values of the four models are relatively close, with the stacking integrated model having the lowest value of 0.00197. The MSE values of the random forest and XGBoost models are 0.00903 and 0.01081, respectively. The SVR model has the highest MSE value, at 0.01883. The distribution of RMSE and MAE values is very similar to MSE, with values of 0.04436 and 0.03688 for the stacking integrated model, 0.09503 and 0.0668 for the random forest model, 0.10397 and 0.05264 for the XGBoost model, and 0.13722 and 0.11388 for the SVM model. Figure 9 clearly shows that the MSE, RMSE, and MAE results of the stacking integrated model are the lowest among the models, proving that the integrated model has the best predictive performance. By integrating models, the issues of insufficient prediction accuracy in base algorithms like random forest and SVR, as well as the overfitting problem in XGBoost on the training set, can be effectively addressed and avoided. In summary, using the stacking strategy to construct an integrated model for predicting \(X_{50}\) can effectively improve predictive performance and better avoid overfitting. Feature importance analysis can effectively assess the degree of influence of input variables on \(X_{50}\) prediction. In this study, the importance assessment tools in random forest and XGBoost were used to extract the importance of input variables of the stacking integrated model and compare the assessment results of the two. The importance assessment of input features in the random forest algorithm is based on impurity calculation, and the features are sorted before taking the average value. The final result also requires normalizing the scores of all features, ensuring that the sum of the importance values of the features equals 1. The XGBoost algorithm determines the importance of features based  
Figure 8: Comparison of \(R^{2}\) values of four models. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **MSE** & **RMSE** & **MAE** \\ \hline Training set & 0.00269 & 0.05187 & 0.03320 \\ Testing set & 0.00197 & 0.04435 & 0.03687 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of stacking integrated model evaluation indicators. on their ability to split data at nodes. The stronger the splitting ability, the greater the contribution to the final result. Figure 10 illustrates the extraction results of the two methods. The input parameter \(E\) is identified as the most important feature variable for predicting \(X_{50}\) in both methods, with importance values of 0.7129 and 0.4608, respectively. \(X_{B}\) and \(T/B\) follow, exhibiting similar importance values.