## 3 Models' Application
### SVM Models with Different Kernels  
This study applied four SVM models with four different Kernels, including RBF, polynomial, sigmoid, and linear. We used four parameters, including stemming, powder factor, the maximum charge per delay, and distance, which were identified as the most critical and relevant parameters for developing the SVM models. The research team used several considerations for developing these models. Stopping criteria were set as \(1.0\times 10^{-3}\); the regularization parameter (C) was established as 10; and the regression precision (epsilon) was developed as 0.1. Before the models' development, the data were split into train and test partitions using a ratio of 80:20. Thus, 104 samples were used in the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable (Category)** & **Unit** & **Minimum** & **Maximum** & **Mean** & **Standard Deviation** \\ \hline Burden to spacing (input) & - & 0.41 & 0.91 & 0.75 & 0.103 \\ Distance from the blast-face (input) & m & 65 & 710 & 329 & 142.961 \\ Maximum charge per delay (input) & kg & 69.79 & 309.09 & 202.44 & 64.534 \\ Powder factor (input) & kg/m\({}^{3}\) & 0.24 & 0.98 & 0.69 & 0.197 \\ Spacing (input) & m & 2.2 & 5.1 & 3.65 & 0.721 \\ Stemming length (input) & m & 1.4 & 4 & 2.87 & 0.619 \\ \hline PPV (target) & mm/s & 1.21 & 37.44 & 14.4 & 8.673 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of parameters in the projecting models. Figure 6: Measured PPV values and their frequency distribution. training phase, and 36 samples were used for the testing phase. The measured PPV values and predicted values by all four models are shown in Figure 7.  
This research used two commonly used criteria for assessing the models' performance. These criteria included the Pearson's correlation coefficient (\(R\)) and the mean absolute error (\(MAE\)). In addition, a gain chart also was used to illustrate the performance of the models graphically:  
\[R=\frac{\sqrt{\sum_{i=1}^{n}\left(y_{im}-y_{ip}\right)^{2}}}{\sqrt{\sum_{i=1}^{ n}\left(y_{im}-\overline{y}_{im}\right)^{2}}} \tag{1}\]  
\[MAE=\frac{1}{N}\sum_{j=1}^{N}\left|y_{im}-y_{ip}\right| \tag{2}\]  
where \(y_{im}\), \(y_{ip}\), and \(\overline{y}_{im}\) indicate the measured, predicted and the mean of measured values, \(n\) represents the total number of data. A simplistic ranking system that rates the performance of the models for each partition was also developed. In this system, training and testing rankings were assigned to each model. Also, an accumulative ranking was produced, which was the total of the training and testing ranks. The formula for computing the accumulative ranking for each model is presented below:  
\[A\text{-}R=\sum(\alpha_{tr}+\beta_{tr})+(\alpha_{te}+\beta_{te}) \tag{3}\]  
where, \(A\text{-}R\) is the accumulative ranking of each model, \(\alpha\) denotes the ranking of \(R\), \(\beta\) shows the ranking of MAE, \"\(tr\)\" means the training ranking, and \"\(te\)\" signifies the testing ranking. The performances of the models developed in this study are shown in Table 3. As can be seen, for the training phase, the BC-SVMRBF model achieved the highest ranks of \(R\) and \(MAE\) compared to other models. On the other hand, the lowest rankings of \(R\) and \(MAE\) belonged to the BC-SVMSIG model in the training phase. For the testing phase, the BC-SVMRBF model outperformed other models in terms of R; however, regarding the \(MAE\), the BC-SVMLIN achieved the highest ranking. Again, the BC-SVMSIG model achieved the lowest ranking in the testing phase comparing with other models. With regards to the accumulative ranking, the BC-SVMRBF model achieved the highest ranking (\(A\text{-}R\) ranking = 15), followed by the BC-SVMLIN model (\(A\text{-}R\) ranking = 12). Alternatively, BC-SVMSIG had the worst performance and consequently had the lowest accumulative ranking. Figure 7: Predicted PPV values of the SVM models with different Kernels. We also employed a gain chart to compare the models developed in this study.