## 4 Multivariate Nonlinear Regression Prediction Model for Rock Deformation  
Multivariate nonlinear regression analysis [28; 29] is a statistical method used to analyze the nonlinear relationships between two or more independent variables and one dependent variable. This type of analysis is particularly useful in dealing with practical problems, especially when the relationship between the dependent variable and independent variables is complex and cannot be adequately described by simple linear models.  
The specific steps are as follows:  
1 In multivariate nonlinear regression, the first step is to determine an appropriate mathematical model based on the research background and data characteristics. This model may include various nonlinear terms, such as powers, exponentials, logarithms, etc. The general form of the model can be expressed as:  
\[y=f(x_{1},x_{2},\ldots,x_{i};\beta_{0},\beta_{1},\ldots,\beta_{P})+\epsilon \tag{7}\]  
In the formula, \(y\) is the dependent variable; \(x_{1},x_{2},\ldots,x_{i}\) are the independent variables; \(\beta_{0},\beta_{1},\ldots,\beta_{P}\) are the model parameters; \(f\) represents the nonlinear function relating the independent variables and parameters; \(\epsilon\) is the error term, typically assumed to follow a normal distribution.  
2 The objective is to minimize the sum of the squared errors across all data points, and this optimization problem can be expressed as:  
\[S(\beta)=\sum_{j=1}^{n}\bigl{(}y_{i}-f\bigl{(}x_{1j},x_{2j},\ldots,x_{ij}; \beta_{0},\beta_{1},\ldots,\beta_{P}\bigr{)}\bigr{)}^{2} \tag{8}\]  
3 The Jacobian matrix and the Hessian matrix are then calculated to minimize \(S(\beta)\):  
\[J_{jk}=\frac{\partial f\bigl{(}x_{1j},x_{2j},\ldots,x_{ij};\beta_{0},\beta_{1},\ldots,\beta_{P}\bigr{)}}{\partial\beta_{k}} \tag{9}\]  
\[H=J^{T}J=\biggl{(}\sum_{i=1}^{n}\frac{\partial f}{\partial\beta_{0}}\frac{ \partial f}{\partial\beta_{0}},\ldots,\sum_{i=1}^{n}\frac{\partial f}{\partial \beta_{0}}\frac{\partial f}{\partial\beta_{P}},\ldots,\sum_{i=1}^{n}\frac{ \partial f}{\partial\beta_{P}}\frac{\partial f}{\partial\beta_{P}}\biggr{)} \tag{10}\]  
Figure 9: Impact of SRC on rock deformation.  
4 The Newton-Raphson method can then be used to update the parameters:  
\[\mathbf{r}=[y_{1}-f(x_{11},x_{21},\ldots,x_{i1};\mathbf{\beta}),\ldots,y_{n}-f(x_{1n},x_ {2n},\ldots,x_{in};\mathbf{\beta})] \tag{11}\]  
\[\
abla S=J^{T}\mathbf{r} \tag{12}\]  
\[\beta_{new}=\beta_{old}-\gamma\Big{(}H^{-1}\
abla S\Big{)} \tag{13}\]  
In the formulas, \(\
abla S\) represents the gradient vector; \(\mathbf{r}\) is the residual vector; and \(\gamma\) is the learning rate, a positive real number that controls the step size.  
This article uses SRC = III as an example, using numerical methods to obtain the total displacement D of the surrounding rock in the roadway under different orthogonal conditions (P, H, B, W). During the process of multivariate nonlinear regression analysis [30], the impact of various factors on the stability of the roadway was particularly considered. According to research needs and previous theoretical analyses, the model prioritizes roadway height (H) and pore water pressure (P), as these two factors are directly related to the structural safety and hydraulic conditions of the roadway, playing a decisive role in the stability of the surrounding rock. Roadway width (W) is also appropriately considered because it influences the stress state and deformation of the roadway to some extent. By contrast, the impact of roadway burial depth (B) is minimized; although it also affects the pressure environment and geological conditions of the roadway to some extent, its impact is considered relatively minor in this study. After a series of data collection and preliminary analyses, a quadratic polynomial regression model was ultimately chosen to fit the results of