## 5 Vibration Velocity Prediction of Long Hole Blasting  
### Peak Particle Velocity Prediction Using the Support Vector Regression Model  
To process the 41 sets of blasting test data, 35 sets derived from the first seven long hole blasting events constituted the training set, and the remaining six sets from the eighth and ninth events were designated as the test set, all of which underwent normalization. The radial basis function was chosen as the kernel function, and the MSE served as the initial fitness metric for the model. Employing the PSO intelligent algorithm with parameters detailed in Table 2, 5-fold cross-validation technology was applied. The 35 training datasets were randomly split into five groups, with four groups utilized for training and one group for model verification. This process was repeated five times, optimizing the model using the \(C\) and \(g\) resulting in the smallest MSE. The PSO-SVR model was employed for predictions involving 12 mass point peak vibration velocities. The optimized model yielded \(C\) ranging from 1.8746 to 170.8970 and \(g\) ranging from 0.1957 to 4.2967 (Table 3). Interestingly, penalty factor \(C\) and kernel function deviation \(g\) exhibited a negative correlation, with an \(R^{2}\) of 0.6011. Utilizing these optimized parameters, an SVR model was constructed. Predictions were made for both the training and test sets, and comparisons were drawn against measured values. The MAE, MSE, and \(R^{2}\) for the predicted and measured peak vibration velocities of the mass point in the training set were 0.0796, 0.0172, and 0.8525, respectively. The main indicator curves demonstrated relative stability (Figure 7). Correspondingly, the mean MAE, MSE, and \(R^{2}\) for the predicted and measured peak vibration velocities of the mass point in the testing set were 0.1426, 0.0416, and 0.8023, respectively. Notably, the fourth dataset exhibited superior performance (testing set, MAE: 0.1022, MSE: 0.0252, and \(R^{2}\): 0.9645). In this study, an SVR model with a penalty factor \(C\) of 15.2756 and kernel function deviation \(g\) of 0.6742 was employed, which is determined by the result of the fourth dataset. In contrast, utilizing the fitting equation based on the USBM formula resulted in test set indices of 0.0948, 0.0275, and 0.7877, respectively (Table 4). Comparing the results, the MAE increased by 7.78%, the MSE decreased by 8.37%, and the \(R^{2}\) increased by 22.44%. Figure 8 illustrates the curve representing the predicted and measured peak vibration velocities of the testing set, while Figure 9 depicts the comparison curve between the predicted and measured peak vibration velocities, with an impressive \(R^{2}\) value of 0.9645. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Number of iterations (\(m\)) & 1000 \\ Number of particles (\(n\)) & 500 \\ Penalty factor \(C_{1}\) & 2 \\ Penalty factor \(C_{2}\) & 2 \\ Inertia weight (\(\omega_{\text{line}}\)) & 0.8 \\ Constrained weight (\(\omega_{\text{con}}\)) & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main parameters of the PSO. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**MAE** & **MSE** & \(\mathbf{R^{2}}\) & **Model** \\ \hline
0.0948 & 0.0275 & 0.7877 & The fitting equation based on USBM formula \\
7.78\% & \(-8.37\%\) & 22.44\% & PSO-SVR \\
7.99\% & \(-0.52\%\) & 2.17\% & PSO-RF \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Key indicators in predicting PPVs for testing datasets. Figure 8: The relation between PPVs predicted by the PSO-SVR model and measured PPVs for testing datasets.