# A Vehicle Velocity Prediction Method with Kinematic Segment Recognition
## 2 Data Preprocessing
### Principal Component Analysis  
If 10 feature parameters are directly used for clustering, the dimensionality disaster will increase the calculation amount. In addition, the large number of features and the linear correlation and information overlap among parameters will affect the clustering results [27]. Therefore, principal component analysis is adopted to reduce the dimensionality of multi-feature parameters to improve the velocity and accuracy of clustering samples with multi-feature. Principal component analysis converts the related initial variables into unrelated synthetic variables through orthogonal transformation and ensures that the synthetic variables can retain as much information as possible about the initial variables [28]. The key to the transformation and solution of the variable matrix is to obtain a low-dimensional scalar value that can describe the original information through a certain transformation of the original multi-dimensional characteristic parameters through the feature vector. The specific implementation steps are as follows:  
1. The initial variable matrix \(X\) is constructed by using the characteristic parameter information of the kinematic fragment above, where l is the number of characteristic parameters 10, and n is the number of samples 317. \[X=\left[ \begin{array}{cccc}x_{11}&x_{12}&\cdots&x_{1l}\\ x_{21}&x_{22}&\cdots&x_{2l}\\ \vdots&\vdots&\ddots&\vdots\\ x_{n1}&x_{n2}&\cdots&x_{nl}\end{array} \right] \tag{11}\]  
The normalized matrix \(Z\) is obtained by using the normalized transformation formula shown in the above publicized formula. \[z_{ij}=\frac{x_{ij}-\overline{x}_{j}}{s_{j}},x_{j}=\frac{\sum\limits_{i=1}^{n} x_{ij}}{n},s^{2}_{j}=\frac{\sum\limits_{i=1}^{n}\left(x_{ij}-\overline{x}_{j} \right)^{2}}{n-1} \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Fragment**} & \(v_{ave}\) & \(v_{\max}\) & \multirow{2}{*}{\(\sigma\)} & \multirow{2}{*}{\(\eta_{a}\)} & \multirow{2}{*}{\(\eta_{d}\)} & \(a_{\max}\) & \(a_{\min}\) & \(a_{a\_{ave}}\) & \(a_{d\_{ave}}\) & \(S\) \\  & (km/h) & (km/h) & & & & & & (m/s2) & (m/s2) & (m/s2) & (m/s2) & (m/s2) \\ \hline
1 & 31.44 & 51.8 & 19.37 & 0.38 & 0.56 & 7.2 & \(-5\) & 2.94 & \(-1.99\) & 1603.6 \\
2 & 18.05 & 30.3 & 11.08 & 0.62 & 0.38 & 4.2 & \(-5.8\) & 2.33 & \(-3.79\) & 397.2 \\
317 & 10.47 & 18.4 & 6.56 & 0.53 & 0.32 & 3.17 & \(-4.44\) & 1.84 & \(-3.07\) & 209.45 \\ \hline \end{tabular}
\end{table} 
Table 1: Characteristic parameters of the kinematic segments. where \(x_{ij}\) is the parameter value of row \(i\) and column \(j\) in the initial variable matrix (\(i=1\), \(2\), \(\ldots\), \(n\); \(j=1\), \(2\), \(\ldots\), \(l\)), and \(\overline{x}_{j}\) and \(s_{j}^{2}\) are the mean and variance of column \(j\), respectively. \[Z= \begin{bmatrix}z_{11}&z_{12}&\cdots&z_{1l}\\ z_{21}&z_{22}&\cdots&z_{2l}\\ \vdots&\vdots&\ddots&\vdots\\ z_{n1}&z_{n2}&\cdots&z_{nl}\end{bmatrix} \] (13) 2. The correlation coefficient matrix \(R\) is calculated based on the correlation of the elements in the matrix. \[R= \begin{bmatrix}r_{11}&r_{12}&\cdots&r_{1l}\\ r_{21}&r_{22}&\cdots&r_{2l}\\ \vdots&\vdots&\ddots&\vdots\\ r_{n1}&r_{n2}&\cdots&r_{nl}\end{bmatrix} \] (14) The element \(r_{\bar{\eta}}\) calculation process is expressed as: \[r_{ij}=\frac{\sum\limits_{k=1}^{n}\left(z_{ki}-\overline{z_{i}}\right)\left( z_{kj}-\overline{z_{j}}\right)}{\sqrt{\sum\limits_{k=1}^{n}\left(z_{ki}- \overline{z_{i}}\right)\sum\limits_{k=1}^{n}\left(z_{kj}-\overline{z_{j}} \right)}}\] (15) 3. The eigenvalues and eigenvectors of the correlation coefficient matrix can be obtained by solving the eigenequation of the correlation matrix R. \[\left|\lambda E-R\right|=0\] (16) 4. The principal component contribution rate is calculated from the eigenvalue, which belongs to a part of all data features contained in the sample and is related to the sample features. The larger the eigenvalue, the more features are covered when the eigenvalue is used to describe the original information.