# Risk Assessment of Drilling and Blasting Method Based on Nonlinear FAHP and Combination Weighting
## 3 Risk Assessment Methodology
### Determination of Weight Vectors
#### 3.2.2 Calculation of Objective Weights Using PCA  
In multi-attribute evaluation, multiple indicators are typically considered, but as their number grows, complexity and information overlap increase. Principal component analysis (PCA) addresses this by reducing dimensionality to a few uncorrelated principal components, which enhances efficiency by focusing on the most critical factors. The sample data obtained from the expert scoring method is dimensionally reduced to improve the concentration of sample information and obtain the weights of each principal component. The specific steps are as follows:  
For n samples of m dimensions x\({}_{\text{i}}\) = (x\({}_{\text{i1}}\), x\({}_{\text{i2}}\), \(\ldots\), x\({}_{\text{ip}}\)), form the sample matrix X = (x\({}_{\text{ij}}\))\({}_{n\times\text{p}}\). Step 1. Standardize the sample data to eliminate the influence of dimensions as follows:  
\[\widetilde{x}_{\text{ij}}=\frac{x_{ij}-\overline{x}_{j}}{s_{j}} \tag{7}\]  
where \(\widetilde{x}_{\text{ij}}\) is the standardized indicator (i = 1, 2, \(\ldots\), n; j = 1, 2, \(\ldots\), m). \[\overline{x}_{j}=\frac{1}{n}\sum_{i=1}^{n}x_{ij} \tag{8}\]  
\[s_{j}=\frac{1}{n-1}\sum_{i=1}^{n}\left(x_{ij}-\overline{x}_{j}\right)^{2} \tag{9}\]  
where \(\overline{x}_{j}\) and \(s_{j}\) are the sample mean and sample standard deviation of the j-th indicator. \[\widetilde{x}_{i}=\frac{x_{i}-\widetilde{x}_{i}}{s_{i}} \tag{10}\]  
where \(\widetilde{x}_{i}\) is the standardized indicator variable (i = 1, 2, \(\ldots\), m). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline RI & 0.00 & 0.00 & 0.58 & 0.90 & 1.12 & 1.24 & 1.32 & 1.41 & 1.46 \\ \hline \end{tabular}
\end{table} 
Table 3: The values of the random consistency index. Step 2. Obtain the covariance matrix \(Z\) from the standardized data as follows:  
\[\mathbf{Z}=\left(z_{ij}\right)_{m\times m}= \begin{bmatrix}z_{11}&z_{12}&\cdots&z_{1m} \\ z_{21}&z_{22}&\cdots&z_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ z_{m1}&z_{m2}&\cdots&z_{mm}\\ \end{bmatrix}  \tag{11}\]  
where \(z_{ij}\) is the element of the covariance matrix \(Z\) (i = 1, 2,..., m; j = 1, 2,..., m). Step 3. Solve the characteristic equation of the covariance matrix \(Z\) to obtain its eigenvalues \(\lambda^{\prime}=\left[\lambda^{\prime}{}_{1},\lambda^{\prime}{}_{2},\cdots \lambda^{\prime}{}_{m}\right]\) and eigenvectors \(\mathbf{v}=\left[v_{1},v_{2},\cdots,v_{m}\right]^{T}\), where \(v_{j}=\left(v_{1j},v_{2j},\cdots,v_{nj}\right)^{T}\). Select principal components with eigenvalues greater than 1 for analysis. Step 4. Determine each principal component \(Y=\left[y_{1},y_{2},\cdots,y_{p}\right]\) and calculate the variance contribution rate \(b_{j}\) and cumulative variance contribution rate \(a_{p}\) of each eigenvalue:  
\[\left\{ \begin{array}{l}y_{1}=v_{11}\widetilde{x}_{1}+v_{21}\widetilde{x}_{2 }+\cdots+v_{m1}\widetilde{x}_{m}\\ y_{2}=v_{12}\widetilde{x}_{1}+v_{22}\widetilde{x}_{2}+\cdots+v_{m2} \widetilde{x}_{m}\\ \vdots\\ y_{p}=v_{1m}\widetilde{x}_{1}+v_{2m}\widetilde{x}_{2}+\cdots+v_{mp} \widetilde{x}_{m}\\ \end{array} \right. \tag{12}\]  
where p is the number of principal components. The variance contribution rate of each principal component \(y_{j}\) is as follows:  
\[b_{j}=\frac{\lambda^{\prime}{}_{j}}{\sum\limits_{k=1}^{m}\lambda^{\prime}{}_{ k}} \tag{13}\]  
The cumulative variance contribution rate \(a_{p}\) of principal component Y is as follows:  
\[a_{p}=\frac{\sum\limits_{k=1}^{p}\lambda^{\prime}{}_{k}}{\sum\limits_{k=1}^{ m}\lambda^{\prime}{}_{k}} \tag{14}\]  
where j, k = 1, 2,..., m.  
Step 5. Calculate the principal component loadings as follows:  
\[q=\sqrt{\lambda^{\prime}{}_{r}}v_{rj} \tag{15}\]  
where r = 1, 2,..., p; j = 1, 2,..., m; \(\lambda^{\prime}{}_{r}\) is the eigenvalue corresponding to the r-th principal component, and \(v_{ij}\) is the j-th coefficient of the eigenvector corresponding to the r-th principal component. Step 6.