## 3 Methodology
### Principal Component Analysis
\right) \tag{5}\]  
In order to orthogonalize any two variables, the covariance matrix should have 0 elements other than the diagonal positions, which means \(Cov(A,\,B)=0\). Therefore, the covariance matrix is diagonalized, which the process of the eigenvalue decomposition. Based on the eigenvalue and eigenvector properties of the matrix, the following relationship equation is obtained:  
\[T\overset{\rightarrow}{\alpha_{i}}=\lambda_{i}\overset{\rightarrow}{\alpha_ {i}} \tag{6}\]  
The set of vectors is orthogonalized to obtain a set of unit vectors, and the matrix \(T\) is decomposed into the product of the diagonal matrix and the matrix of eigenvectors:  
\[T=Q\sum\limits_{i=1}^{n}{Q^{-1}} \tag{7}\]where \(\Sigma\) is the diagonal matrix of eigenvalues \(\lambda_{i}\) arranged by size and \(Q\) is the matrix of corresponding eigenvectors \(\overrightarrow{\alpha_{i}}\). In general, for a matrix that is not square, \(T_{m^{*}n}\), the singular value decomposition method can be used:  
\[T_{m*n}=U_{m*m}\sum_{m*n}V_{n*n}^{T} \tag{8}\]  
where the matrices \(U_{m^{*}m}\) and \(V_{n^{*}n}\) are orthogonal matrices and \(\Sigma_{m^{*}n}\) is a matrix with all elements except the main diagonal element being singular. Finally, to construct irrelevant variables and the representation of changes, use a matrix approach:  
\[F=AX \tag{9}\]  
where X is the original data representation and vector \(F\) is the new data representation. \[\left( \begin{array}{c}\overrightarrow{\alpha_{1}}\\ \overrightarrow{\alpha_{2}}\\ \vdots\\ \overrightarrow{\alpha_{m}}\end{array} \right)\left( \begin{array}{c} \overrightarrow{x}_{1},\overrightarrow{x}_{2},\ldots,\overrightarrow{x}_{n} \\ \vdots\\ \overrightarrow{\alpha_{m}}\end{array} \right)=\left( \begin{array}{c} \overrightarrow{\alpha_{1}}\overrightarrow{x}_{1},\overrightarrow{x}_{1} \overrightarrow{x}_{2},\ldots,\overrightarrow{x}_{1}\overrightarrow{x}_{n} \\ \overrightarrow{\alpha_{2}}\overrightarrow{x}_{1},\overrightarrow{\alpha_{2}} \overrightarrow{x}_{2},\ldots,\overrightarrow{\alpha_{2}}\overrightarrow{x} _{n}\\ \vdots\\ \overrightarrow{\alpha_{m}}\end{array} \right) \tag{10}\]  
Among the functions, \(i=1,\,2,\,\ldots\,,\,m\), represents the \(i\)th mutually orthogonal basis and \(j=1,\,2,\,\ldots\,,\,n\), represents the original data and their dimensions are \(t\). The next steps are to sort according to the corresponding eigenvalues, select according to the principle that the contribution rate needs to reach more than 85%, and obtain the final orthogonal variables for analysis, where \(m\) is the first \(m\) principal components selected. After the PCA procedure, the principal component regression equation is required to be constructed according to the results. In the research, \(m\) characteristic indicators of the vehicle during the three steps of the lane-changing process are considered. When handling its correlation, a linear model is adopted to express this, which is represented by:  
\[H=\sum_{i}c_{i}x_{i},i=1,2,3,\ldots,m. \tag{11}\]  
where \(H\) indicates the longitudinal driving distance of the lane-changing process, \(x_{i}\) indicates the influencing variables in the lane-changing process, and c\({}_{i}\) indicates the weight of each variable. After the principal component analysis, the principal component regression equation for the longitudinal driving distance was constructed according to the percentage of the contribution of the principal components:  
\[H=\sum_{i}c_{i}\,\widehat{x}_{i},i=1,2,3,\ldots,n,n\leq m. \tag{12}\]