# Predictive Modelling for Blasting-Induced Vibrations from Open-Pit Excavations
## 4 Prediction Models  
### Artificial Neural Network  
Trial-and-error analysis of hyper-parameters is required to obtain the optimal prediction model which has the lowest validation loss. In this analysis, it was carried out with a different number of hidden layers, nodes, normalization methods, and activation  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rock Type** & **P-Wave Velocity (m/s)** & **Reference** \\ \hline Shale & 3500 & & [23] \\ Andesite & 5121 & & [24] \\ Rybolite & 4100 & & [25] \\ Tuff & 2750 & & [26] \\ \hline \hline \end{tabular}
\end{table}
Table 2: \(Cont.\)  
Figure 3: Correlations between initial influential factors.  
functions; one and two hidden layers; 3, 5, 7, 11, 14, 15, 21, 28 and 35 nodes for the hidden layer; min-max and standard scalings; and three activation functions, sigmoid, hyperbolic tangent, and ReLU. In other words, 54 (2 \(\times\) 9 \(\times\) 3) and 486 (2 \(\times\) 9 \(\times\) 9 \(\times\) 3) structures were assessed on 1 and 2 hidden layers, respectively. The number of nodes was determined by Table 4. Here, \(N_{i}\) and \(N_{o}\) mean number of input and output parameters, respectively. We added some equations in the final row of Table 4 to analyze many structures. The Adam optimizer [27] was used to reduce the loss with a learning rate of 0.001. Also, we used an early stopping to avoid overfitting and to obtain the best-fitted model. Every structure of the ANN model was trained with the 714 training datasets and validated by the 179 validation datasets. Every ANN model was developed with the software PYTHON Version 3.7.6.  
In the results of trial-and-error analysis, the average validation loss of 540 structures was 0.126 cm/s. Among the 540 ANN models, the structure composed of two hidden layers with 21 and 28 nodes, normalized by min-max scaling and combined with ReLU showed the lowest validation loss of 0.115 cm/s. Therefore, we selected the ANN model, which has the 7-21-28-1 structure depicted in Figure 4 as an optimal ANN model for a PPV prediction. The training of this model was stopped at 4208 epochs by early stopping. Table 5 summarizes the characteristics of the selected ANN model. This model is represented by Equations (10)-(12). PPV is calculated by Equation (10). Equations (11) and (12) represent hidden layers 1 and 2, respectively.  
\[\text{PPV}=[\text{H}_{2}]\cdot[\text{W}_{3}]+[\text{b}_{3}] \tag{10}\]  
\[[\text{H}_{2}]=\text{R}([\text{H}_{1}]\cdot[\text{W}_{2}]+[\text{b}_{2}]) \tag{11}\]  
\[[\text{H}_{1}]=\text{R}([\text{m}[\text{I}]]\cdot[\text{W}_{1}]+[\text{b}_{1}]) \tag{12}\]  
Figure 4: Structure of the artificial neural network (ANN) model developed.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Equation** & **Number of Nodes** & **Reference** \\ \hline \(\sqrt{N_{i}\times N_{o}}\) & 3 & [28] \\ \(\left(4N_{i}^{2}+3\right)/\left(N_{i}^{2}-8\right)\) & 5 & [29] \\ \(3N_{i}/2\) & 11 & [30] \\ \(2N_{i}+1\) & 15 & [31] \\ \(3N_{i}\) & 21 & [32] \\ \(N_{i}\), \(2N_{i}\), \(4N_{i}\), \(5N_{i}\) & 7, 14, 28, 35 & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Equations for determination of the number of nodes.  
In these equations, [I] is the matrix of input data sets, [W] is the matrix of weights, and [b] is the matrix of biases. The weight and bias matrices are constants that were obtained from the ANN training. Here, [W\({}_{1}\)], [W\({}_{2}\)], [W\({}_{3}\)], [b\({}_{1}\)], [b\({}_{2}\)], and [b\({}_{3}\)] are 7 \(\times\) 21, 21 \(\times\) 28, 28 \(\times\) 1, 1 \(\times\) 21, 1 \(\times\) 28, and 1 \(\times\) 1 matrices. When predicting \(i\) PPVs, [I] is an \(i\times 7\) matrix. R is a ReLU function expressed by Equation (3), m is a min-max scaling expressed by Equation (4).