## 5 Analysis of Correlation among Impact Factors  
In conducting research on prediction models, especially for complex engineering issues like roadway deformation, relying solely on traditional regression models may not fully reveal the essence of the problem. This is because such models often assume that the factors are independent, neglecting the complex interactions and correlated responses that may exist between them. In reality, the phenomenon of roadway deformation is the result of the combined effects of various geological, structural, engineering, and environmental factors, where the interactions between these factors can significantly impact the stability and safety of the roadway. To more comprehensively understand how these factors jointly influence roadway deformation, the entire prediction model can be viewed as a dynamic grey system [31, 32]. In grey system theory, part of the system information is known, while part is unknown, making it suitable for dealing with complex systems with incomplete information. By introducing grey relational analysis (GRA), we can not only identify the correlations between factors but also assess the impact of these factors on roadway deformation and their relative importance. Grey relational analysis will help us to determine which variables are most closely related to roadway deformation, which factors are dominant, and which might be secondary or indirect. This analytical method provides  
Figure 10: Regression model prediction validation. (**a**) Data points vary with changes in pore water pressure; (**b**) data points vary with changes in roadway burial depth; (**c**) data points vary with changes in roadway height; (**d**) data points vary with changes in roadway width. a quantified way to measure the strength of interdependencies and interactions among various factors, allowing the model to reflect not only the independent effects of factors but also their combined effects. Additionally, models based on grey system theory are adaptable to the incompleteness and uncertainty of data, which is particularly important in practical engineering applications where data may be missing or of low quality. By using grey relational analysis, we can optimize model design, improve the accuracy and reliability of predictions, and thereby provide a more scientific basis for roadway support design and construction [33; 34]. Grey relational analysis (GRA) is a method used to identify the degrees of correlation among factors, suitable for situations with small sample sizes and high data uncertainty. Here are the basic computational steps of grey relational analysis. 1 Normalize the data to eliminate the impact of different units and magnitudes among data indices, transforming the data into the [0, 1] range:  
\[x_{ij}^{*}=\frac{x_{ij}-min\big{(}x_{j}\big{)}}{max\big{(}x_{j}\big{)}-min \big{(}x_{j}\big{)}} \tag{15}\]  
In the formula, \(x_{ij}\) represents the original data, \(x_{ij}^{*}\) represents the normalized data, \(i\) denotes the ith data point, and \(j\) denotes the jth indicator. 2 Calculate the difference between each comparison sequence and the reference sequence. The difference sequence is defined as:  
\[\Delta_{ij}=\big{|}x_{0}^{*}(j)-x_{i}^{*}(j)\big{|} \tag{16}\]  
In the formula, \(x_{0}^{*}(j)\) is the normalized value of the jth indicator in the reference sequence, and \(x_{i}^{*}(j)\) is the normalized value of the jth indicator in the ith comparison sequence. 3 Calculate the correlation coefficient. The correlation coefficient is a measure of the degree of similarity between two sequences. The calculation formula is:  
\[\xi_{ij}=\frac{min_{ij}\Delta_{ij}+\rho max_{ij}\Delta_{ij}}{\Delta_{ij}+\rho max _{ij}\Delta_{ij}} \tag{17}\]  
In the formula, \(\Delta_{ij}\) is the difference value between the ith comparison sequence and the reference sequence on the jth indicator. \(\rho\) is the distinguishing coefficient, typically ranging between [0, 1], with a common value being 0.5, the role of this coefficient is to adjust the sensitivity of the correlation degree.