# Prediction of Blast-Induced Ground Vibration Using an Adaptive Network-Based Fuzzy Inference System
## 5 ANFIS Model for Vibration Prediction Called ANFISBLAST
### ANFIS Structure  
Our ANFIS, called ANFISBLAST, comprises different mathematical models \(f\) for \(f\in F\), where \(F=\{PPV,\)_FREQUENCY_\(\}\)--i.e., two prediction models--as follows:  
1. _PPV_: The ANFIS model for peak particle velocity;
2. _FREQUENCY_: The ANFIS model for frequency calculation. The structure of the \(f\) model, for each \(f\in F\), is presented in Figure 6. The nodes on the left side represent the input variables and the node on the right represents the output. Each model contains two input variables, namely the charge \(Q\) (kg) and the distance \(d\) (m) from the point of measurement to the center of gravity of the blast, as well as a single output. In this way, two ANFIS models predict two different outputs, such as the peak particle velocity (mm/s) and the frequency (Hz). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Name of predictor**} & \multirow{2}{*}{**Equation**} & \multicolumn{2}{c}{**Site Coefficient**} & \multicolumn{3}{c}{**Goodness of Fit**} \\  & & **B** & **K** & **SSE** & **R-Squared** & **Adjusted** & **RMSE** \\  & & **B** & **(mm/s)** & **R-Squared** & **R-Squared** & **(mm/s)** \\ \hline USBM [34] & \(PPV=K\left(\frac{d}{\sqrt{\text{Q}_{MAX}}}\right)^{-B}\) & 1.033 & 53.18 & 159.12 & 0.62 & 0.61 & 2.05 \\ Ambraseyés-Hendron [35] & \(PPV=K\left(\frac{d}{\text{Q}_{MAX}^{1/3}}\right)^{-B}\) & 1.047 & 107.7 & 165.1 & 0.61 & 0.60 & 2.08 \\ [PERSON] [36] & \(PPV=K\left(\frac{\sqrt{\text{Q}_{MAX}}}{d^{9/3}}\right)^{B}\) & 2.652 & 1.255 & 151.49 & 0.64 & 0.63 & 2.00 \\ \hline \hline \end{tabular} Note: SSE, sum of squares due to error; RMSE, root mean square error.  
\end{table} 
Table 5: Estimated site coefficients based on the training data and goodness of fit of each prediction equation. Figure 6: The fuzzy inference system included in the structure of adaptive network of the ANFISBLAST. The structure of both models is the same, but they were constructed on the basis of different input data. For a Sugeno-type fuzzy model [39], a set of rules with \(i\), \(i\)\(\in\)\(I\), \(I\) = {1, 2}, and fuzzy \"if-then\" rules, is specified by Equations (5) and (6):  
1. _If Q is \(A_{1}\) and \(d\) is \(B_{1}\) then \(f_{1}=a_{0}^{1,f}+a_{1}^{1,f}\cdot Q+a_{2}^{1,f}\cdot d\)_ (5)
2. _If Q is \(A_{2}\) and \(d\) is \(B_{2}\) then \(f_{2}=a_{0}^{2,f}+a_{1}^{2,f}\cdot Q+a_{2}^{2,f}\cdot d\)_ (6)  
where \(a_{0}^{i,f}\), \(a_{1}^{i,f}\), and \(a_{2}^{i,f}\) are the consequent parameters, and \(Q\) and \(d\) are the input variables. For each\(f\)\(\in\)\(F\), for the \(f\) model, different parameters were calculated. The general computation procedure of the ANFIS models is presented below [41]:  
1. The degree of membership of a fuzzy set (\(A_{i}\), \(B_{i}\)) is computed;
2. The product of membership function for every rule is computed;
3. The proportion between the \(i\)th rule's strength and the sum of all of the rules' strengths is computed;
4. The output of every rule is computed;
5. The weighted average of every rule's output is computed. In the first step, the degree of membership of the fuzzy set (\(A_{i}\), \(B_{i}\)) was computed using Equations (7) and (8):  
\[\mu_{A_{i}}^{f}(Q)=exp\left[-\left(\frac{Q-c_{A_{i}}^{f}}{\sqrt{2}\cdot c_{A_{ i}}^{f}}\right)^{2}\right] \tag{7}\]  
\[\mu_{B_{i}}^{f}(d)=exp\left[-\left(\frac{d-c_{B_{i}}^{f}}{\sqrt{2}\cdot c_{B_{ i}}^{f}}\right)^{2}\right] \tag{8}\]  
where \(Q\) and \(d\) are the inputs to membership functions (Gaussian), and parameters \(c_{A_{i}}^{f}\), \(c_{B_{i}}^{f}\), \(\sigma_{A_{i}}^{f}\), and \(\sigma_{B_{i}}^{f}\) are the premise parameters. The products of the membership functions for each rule were also computed:  
\[w_{1}^{f}=\mu_{A_{1}}^{f}(Q)\cdot\mu_{B_{1}}^{f}(d) \tag{9}\]  
\[w_{2}^{f}=\mu_{A_{2}}^{f}(Q)\cdot\mu_{B_{2}}^{f}(d) \tag{10}\]  
where \(\underline{w_{i}}\) represents the strength of the \(i\)th rule.