## 2 Grey Correlation Analysis  
As a part of the grey system, grey correlation analysis finds out their correlation and main influencing factors by processing and analyzing various factors to be studied, which is particularly suitable for the analysis and evaluation of influencing factors of roof mid\(-\)span displacement, velocity, and acceleration of civil air defense construction. The flow chart of grey correlation analysis is shown in Figure 1. The specific calculation steps of this method are as follows:  
1. List comparison data matrix and reference data matrix:  
The n factors (elastic modulus of concrete, thickness of overburden layer, density of soil layer, elastic modulus of soil layer, vehicle speed, and vehicle mass) that affect the mid\(-\)span displacement, velocity, and acceleration of civil air defense construction are taken as comparison series \(X\), \(X=(X_{1}X_{2}\cdots X_{n})^{T}\), and the corresponding mid\(-\)span displacement, velocity, and acceleration of roof are taken as reference series \(Y\), \(Y=(Y_{1}Y_{2}\).... \(Y_{n})^{T}\). Each factor of columns \(X\) and \(Y\) has several values:  
\[X_{i}=(X_{i}(1)X_{i}(2)X_{i}(3)X_{i}(4)) \tag{1}\]  
\[Y_{i}=(Y_{i}(1)Y_{i}(2)Y_{i}(3)Y_{i}(4)) \tag{2}\]  
Columns \(X\) and \(Y\) are written in matrix form:  
\[X=\begin{pmatrix}X_{1}\\ X_{2}\\ \vdots\\ X_{n}\end{pmatrix}=\begin{pmatrix}X_{1}(1)&X_{1}(2)&\cdots&X_{1}(4)\\ X_{2}(1)&X_{2}(2)&\cdots&X_{2}(4)\\ \vdots&\vdots&&\vdots\\ X_{n}(1)&X_{n}(2)&\cdots&X_{n}(4)\end{pmatrix} \tag{3}\]  
\[Y=\begin{pmatrix}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{n}\end{pmatrix}=\begin{pmatrix}Y_{1}(1)&Y_{1}(2)&\cdots&Y_{1}(4)\\ Y_{2}(1)&Y_{2}(2)&\cdots&Y_{2}(4)\\ \vdots&\vdots&&\vdots\\ Y_{n}(1)&Y_{n}(2)&\cdots&Y_{n}(4)\end{pmatrix} \tag{4}\]
2. Dimensionless treatment of matrix:  
The dimensionless methods of data are divided into initial value, average value, interval relative value, and normalization methods. \(X\) and \(Y\) are dimensionless through interval relative value.  
\[X^{\prime}{}_{i}=(X^{\prime}{}_{i}(1)X^{\prime}{}_{i}(2)X^{\prime}{}_{i}(3)X^{ \prime}{}_{i}(4)) \tag{5}\]  
where  
\[X^{\prime}{}_{i}(j)=\frac{X_{i}(j)-\underset{j}{\text{min}}X_{i}(j)}{\underset {j}{\text{max}}X_{i}(j)-\underset{j}{\text{min}}X_{i}(j)} \tag{6}\]At the same time, the reference column \(Y_{i}\) is dimensionless.
3. Solving the grey correlation discernibility information of matrix: The discernibility matrix \(\Delta\) is solved by the following formula: \[\Delta_{ij}=\left|Y^{\prime}{}_{i}(j)-X^{\prime}{}_{i}(j)\right|\] (7) Select the maximum and minimum values in \(\Delta\): \[\left\{\begin{aligned} &\Delta_{max}=max(\Delta_{ij})\\ &\Delta_{min}=min(\Delta_{ij})\end{aligned}\right.\] (8)
4. Solving the grey correlation coefficient matrix and grey correlation degree: The calculation equation of grey correlation coefficient is: \[\gamma_{ij}=\frac{\Delta_{min}+\xi\Delta_{max}}{\Delta_{ij}+\xi\Delta_{max}}\] (9) where \(\xi\) is the resolution coefficient, \(\xi\in[0,1]\), and generally \(\xi=0.5\).  
By calculating the average value of the grey correlation coefficient as the correlation degree, the problem of large and scattered grey correlation coefficients is solved. The calculation equation of the grey correlation degree is:  
\[A_{i}=\frac{1}{m}\sum_{j=1}^{m}\gamma_{ij} \tag{10}\]  
The value range of the grey correlation degree is [0, 1]. The ranking of the grey correlation degree value reflects the sensitivity of the influencing factors. The larger the grey correlation degree value, the stronger the correlation between the comparison factors and the reference factors.