## 2 Data Processing
### Missing Data Handling  
In cases where there are low-density missing values in the data, common approaches include missing data removal, manual completion, regression estimation completion, or data interpolation for handling. Removing missing data can ensure data integrity but may lead to a significant waste of data resources and affect the objectivity and accuracy of the results. Manual completion is labor-intensive and not very accurate, while completing the regression estimate is complex for data with low densities of missing values. Therefore, we herein choose to use linear interpolation, a data interpolation method, as a suitable approach to deal with missing values. Common data interpolation methodologies include spline interpolation, mean imputation, multiple imputation, and maximum likelihood estimation imputation. In the present paper, linear interpolation is employed to interpolate the missing data. The main idea is that, after obtaining the historical bearing temperature data, the axial temperature data and the previous moment of the missing data are selected to estimate the missing data value. In practical applications, the reported temperature typically represents an average or maximum value, which provides a general indication of the thermal state of the axle without delving into the complexities of spatial variations. However, it is crucial to recognize the significance of temperature gradients along the axle. These gradients can cause differential expansion and contraction, leading to additional stresses and potential damage over time [18]. Studies have shown that temperature variations can indeed impact the mechanical properties and longevity of axles [19; 20]. Furthermore, the temperature field along the axles is generally both time-varying and spatial-varying. The temperature field is able to vary significantly across the diameter and length of the axle, leading to gradients that may induce forces and stresses, potentially affecting the service life of the axle. These spatial and temporal variations are critical for understanding the thermal behavior of the axle but have not been accurately taken into account in the linear interpolation model used for handling missing data. Linear interpolation, while effective for simple data gaps, does not capture the complex variations in the temperature field. Future research could focus on developing more sophisticated models that account for these spatial temperature variations to enhance predictive maintenance strategies. Advanced interpolation techniques or models that consider both time-varying and spatial-varying characteristics could improve the accuracy and reliability of temperature data reconstruction, thereby providing a more comprehensive understanding of the axle's thermal behavior and contributing to better maintenance and safety strategies. The experimental data for this study were obtained from an on-board axle temperature monitoring system. We collected axle temperature data using 36 sensors. Data from the first 10 days were utilized to train a neural network model, and 1 day was randomly selected from eight typical locations for missing data processing. Initially, the axle temperature data collected from the temperature sensors on the train bearings were transmitted to the axle temperature monitoring system and then submitted to the ground system through network transmission. Let us assume that the train operating temperature data are represented as a data matrix \(\mathbf{X}\)\(\in\)R\({}^{N\times m}\), where \(N\) represents the number of samples, and \(m\) denotes the number of bearing temperature variables. In addition, the period of missing data is denoted by \(t\)\(\in\)[\(t_{1}\),\(t_{2}\)], where \(t_{1}\) represents the start time of the missing data period, and \(t_{2}\) denotes the end time of the missing data period. The sampled values of bearing temperature within the period of missing data are empty. Let \(\mathbf{X}\) be denoted as follows:  
\[\mathbf{X}=