## 2 Data Processing
### Missing Data Handling
\begin{pmatrix}x_{1}(1)&x_{2}(1)&\cdots&x_{m}(1)\\ x_{1}(2)&x_{2}(2)&\cdots&x_{m}(2)\\ \vdots&\vdots&&\vdots\\ x_{1}(N)&x_{2}(N)&\cdots&x_{m}(N)\\ \end{pmatrix}  \tag{1}\]  
where each row represents the axle temperature data collected at the same time for different bearing temperature variables, while each column denotes the axle temperature data collected at different times for the same bearing temperature variable. Assuming \(x_{i}(k)\) to \(x_{i}(k+\partial)\) for \(0<\partial<N-k\) is represented as a segment of missing data. Firstly, locate the nearest data segments before and after the missing data \(x_{i}(k^{\prime})\) and \(x_{i}(k^{\prime\prime})(k^{\prime}<k,k+\partial<k^{\prime\prime})\). \[x_{i,L}(k)=x\big{(}k^{\prime}\big{)}+\frac{x_{i}(k^{\prime\prime})-x(k^{ \prime})}{k^{\prime\prime}-k^{\prime}}\big{(}k-k^{\prime}\big{)} \tag{2}\]  
where \(x_{i,L}(k)\) denotes the estimated value of the missing data segment from \(x_{i}(k)\) to \(x_{i}(k+\partial)\), which is the result value that should be inserted into the time period with missing data. The data matrix after the interpolation of the missing data segments is denoted by \(\mathbf{X}_{\text{s}}\). The experimental data in this article are derived from the on-board axle temperature monitoring system. During transmission, data defects may occur at points with missing data due to weak network signals when the train passes through such areas. Figure 1 shows the shaft temperature data of the vehicle after startup and before storage. A data preprocessing method based on linear interpolation is adopted to properly fill in the missing data points in the bearing temperature data. The missing values in the bearing temperature data exhibit characteristics of sparse points. Interpolation results of missing data are shown in Figure 1.  
After interpolation is completed, it is also necessary to verify whether the interpolated data are significantly different from the original data. At this time, the Kolmogorov-Smirnov test (K-S test) is a commonly utilized non-parametric statistical method used to compare whether the distribution of the two sets of data is significantly different. The detailed step of the K-S one-sample test method is to exploit the actual cumulative distribution of sample sampling with the comparison of the hypothetical theoretical distribution, calculate the cumulative distribution function of the two functions, and then evaluate the maximum value D of the distance between the distribution function, and check the D-value distribution table to determine the confidence interval of the D value. If the D value is placed in the corresponding confidence interval, that is, when the maximum difference value D is in the specified numerical range, you can determine the data sample by approximately obeying or obeying the distribution of the hypothesis. However, when the difference exceeds the specified range when testing that the two data samples have a remarkable difference, the detected data samples do not meet the requirements. In the K-S test, the primary task is to calculate the cumulative empirical distribution function of the two sets of observed data. Assuming that the samples of the datasets are expressed as (\(x_{1}\), \(x_{2}\), \(x_{3}\), \(\ldots\), \(x_{n}\)), the cumulative distribution function is expressed as:  
\[F_{n}(x)=\frac{1}{n}\sum_{i=1}^{i}I_{[-\infty,x]}(x_{i}) \tag{3}\]  
where \(I\) represents the indicator function, expressed by:  
\[I_{[-\infty,x]}(x_{i})=\left\{ \begin{aligned} & 1,x_{i}\leq x;\\ & 0,x_{i}>x;\end{aligned} \right. \tag{4}\]  
Figure 1: The time-varying temperature of various axes in 2 data samples: (**a**) axle 1; (**b**) axle 2; (**c**) axle 3; (**d**) axle 4; (**e**) axle 5; (**f**) axle 6; (**g**) axle 7; (**h**) axle 8. The K-S test generates a p-value for judging the distribution of the two sets of data. If the \(p\)-value is large, it means that the interpolation possesses less influence on the data distribution, and the interpolated data can be taken as consistent with the original data.