## 3 Results
### Assessment of the Proposed Models
\begin{table}
\begin{tabular}{c|c|c c c c c c c c c c c|c} \hline \hline  & & \multicolumn{2}{c|}{**R2 \({}^{*}\)**} & \multicolumn{2}{c|}{**RMSE \({}^{**}\)**} & \multicolumn{2}{c|}{**VAF \({}^{***}\)**} & \multicolumn{2}{c|}{**MAE \({}^{***}\)**} & \multicolumn{2}{c|}{**a20-Index**} & \multicolumn{2}{c|}{**Total Rank**} & \multicolumn{1}{c}{**Final Rank**} \\ \hline
**Model** & **Dataset** & **Value** & **Rank** & **Value** & **Rank** & **Value** & **Rank** & **Value** & **Rank** & **Value** & **Rank** & **Rank** \\ \hline \multirow{2}{*}{RF} & TR & 0.94 & 5 & 0.77 & 5 & 92.97 & 5 & 0.62 & 4 & 0.73 & 4 & 23 & \multirow{2}{*}{41} \\  & TE & 0.83 & 3 & 1.46 & 4 & 82.17 & 3 & 1.19 & 3 & 0.55 & 5 & 18 & \\ \hline \multirow{2}{*}{CART} & TR & 0.67 & 1 & 1.67 & 1 & 67.03 & 1 & 1.32 & 1 & 0.44 & 1 & 5 & \multirow{2}{*}{10} \\  & TE & 0.56 & 1 & 2.39 & 1 & 54.6 & 1 & 1.84 & 1 & 0.42 & 1 & 5 & \\ \hline \multirow{2}{*}{CHAID} & TR & 0.91 & 4 & 0.86 & 4 & 91.3 & 4 & 0.54 & 5 & 0.77 & 5 & 22 & \multirow{2}{*}{32} \\  & TE & 0.68 & 2 & 1.9 & 2 & 67.79 & 2 & 1.47 & 2 & 0.43 & 2 & 10 & \\ \hline \multirow{2}{*}{ANN} & TR & 0.89 & 3 & 0.96 & 3 & 89.14 & 3 & 0.75 & 3 & 0.66 & 2 & 14 & \multirow{2}{*}{35} \\  & TE & 0.84 & 4 & 1.41 & 5 & 83.71 & 4 & 1.13 & 5 & 0.52 & 3 & 21 & \\ \hline \multirow{2}{*}{SVM} & TR & 0.88 & 2 & 1.02 & 2 & 88.48 & 2 & 0.77 & 2 & 0.67 & 3 & 11 & \multirow{2}{*}{32} \\  & TE & 0.85 & 5 & 1.5 & 3 & 84.54 & 5 & 1.17 & 4 & 0.53 & 4 & 21 & \\ \hline \multicolumn{10}{l}{\({}^{*}\) Perfect coefficient of variation (\(R^{2}\)) = 1; \({}^{**}\) perfect root-mean-squared error (RMSE) = 0; *** perfect variance} \\ \multicolumn{10}{l}{accounted for (VAF) = 100%; \({}^{***}\) perfect mean absolute error (MAE) = 0; \({}^{***}\) perfect a20-index = 1; training dataset = TR; testing dataset = TE; CHAID = chi-squared automatic interaction detection.} \\ \end{tabular}
\end{table} 
Table 5: Obtained results of performance indices for all developed models in estimating PPV. probability of a PPV of greater than 5.59 mm/s. As we went through 68% of the data, the CART model showed that we identified 100% of the samples with the probability of a PPV of greater than 5.59 mm/s. The gain chart illustrates that the performance of the RF model was better than other proposed models for the training dataset since the performance of this model was closer to the perfect model compared to the other proposed models. For the testing dataset, the ANN and SVM models performed slightly better than the RF model and other proposed models. The authors provide a comparison between the achieved \(R^{2}\) values in previous studies that were listed in Table 1 and the current study. Figure 7 shows the average number of input variables and average \(R^{2}\) values for several types of models. This figure shows that the ANFIS models with two input variables obtained an average \(R^{2}\) value of 0.98, which was the highest value among other models. The ANFIS was followed by SVM and hybrid models (PSO-ANN and ICA-ANN) that had 4.5 and eight input variables on average and achieved an average \(R^{2}\) value of 0.93. In addition, ANN and FIS models that had 2.6 and 3.3 input variables on average could obtain an average \(R^{2}\) value of 0.92. Turning to this present study, the authors used a hybrid approach to develop several models using six input variables. The FS-RF model developed in this study showed a higher \(R^{2}\) compared to the previous hybrid models, including PSO-ANN and ICA-ANN, while a lower number of input variables were used to develop this model. The other models developed in this study achieved a lower \(R^{2}\) compared to the models previously developed. Comparing the results of this study with similar studies in terms of predictive techniques, it was found that the present study has some advantages. For example, concerning the study by [PERSON] et al. [78], they only developed a CART model (one of the implemented techniques of this study) to predict PPV induced by blasting using MC and D parameters as inputs.