## 2 Methodology
### Principal Component Analysis  
Principal component analysis (PCA) is a method that reduces unimportant dimensions by analyzing the covariance structure of the original dataset and describes the dataset using a few irrelevant comprehensive indicators [20]. By converting the data from one coordinate system to another coordinate system, this ensures the first coordinate after conversion contains the largest variance in the dataset, the second coordinate contains the second largest variance, and so on, to achieve the purpose of dimensionality reduction. In this process, principal component analysis retains the main information of the dataset, ignores the dimensions that contribute little to the variance of the dataset, and simplifies the data.  
Assuming that the original dataset \(c\) contains m observed samples and each sample has \(n\) features, that is, matrix \(c\in R_{m\times n}\), the main steps of the principal component analysis based on matrix \(c\) are as follows:  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Formula** & **Parameter** \\ \hline Linear kernel function & \(k(x,z)=x\cdot z\) & \\ Polynomial kernel function & \(k(x,z)=\left(\gamma x\cdot z+r\right)^{d}\) & \(\gamma,r,d\) \\ Gaussian kernel function & \(k(x,z)=exp(-\gamma\left\|x-z\right\|^{2})\) & \(\gamma\) \\ Sigmoid kernel function & \(k(x,z)=\tanh(\gamma x\cdot z+r)\) & \(\gamma,r\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Support vector regression kernel function classification.  
Step 1: Normalize matrix \(c\) and calculate the normalized matrix \(S_{ij}\):  
\[S_{ij}=\frac{x_{ij}-\mu_{j}}{\sigma_{j}},\ 1\leq i\leq m;1\leq j\leq n, \tag{6}\]  
where \(\sigma_{j}\) and \(\mu_{j}\) denote the standard deviation and mean of the \(j\)-th variable, respectively.  
Step 2: The covariance matrix \(C\) of matrix \(X\) is determined by singular value decomposition  
\[C=\frac{1}{m-1}SS^{\rm T}, \tag{7}\]  
Step 3: Calculate the eigenvalues \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}...\geq\lambda_{n}\) of the covariance matrix \(C\) and the corresponding eigenvectors \(P_{1}\), \(P_{2}\), \(P_{3}...\), \(P_{l}\).  
Step 4: Select the principal component with a higher cumulative variance contribution to preserve the main information of the data.  
\[\Big{\{}\min(k)|\sum_{i=1}^{k}\lambda_{i}/\sum_{i=1}^{n}\lambda_{i}\geq T_{\rm C }\Big{\}}, \tag{8}\]  
where \(\sum_{i=1}^{k}\lambda_{i}/\sum_{i=1}^{n}\lambda_{i}\) denotes the contribution rate of the first \(k\) principal components, and \(T_{c}\) denotes the contribution threshold.  
Step 5: Construct the dataset after dimensionality reduction:  
\[c^{\prime}=\sum_{j=1}^{n}\alpha_{ij}\sqrt{\lambda_{j}}P_{j}. \tag{9}\]  
The PCA-SVR approach distinguishes itself from existing hybrid methods in tunneling engineering by offering enhanced predictive accuracy and interpretability. Specifically, it integrates PCA for dimensionality reduction with the robust regression of SVR, resulting in a streamlined and efficient model that captures complex tunneling data relationships with fewer parameters. This unique combination not only delivers a superior generalization performance compared to traditional hybrid models, but also provides deeper insights into the underlying factors influencing tunneling processes, ultimately facilitating more informed decision making in engineering practice.