## 2 Data Processing
### Outlier Treatment  
To ensure the accuracy of the modeling and subsequent monitoring and fault diagnosis, outlier treatment is performed on the transient anomalies in certain axle temperature variable data caused by factors such as weak connections and electromagnetic interference in the collected bearing temperature data. Based on industry standards and our findings, the typical operational limit for axle-bearing temperatures in passenger trains usually ranges from 80 \({}^{\circ}\)C to 120 \({}^{\circ}\)C under normal conditions. However, temperatures could reach up to 140 \({}^{\circ}\)C under extreme situations without immediate risk when cooling systems and monitoring mechanisms are in place. Outliers are usually points that are significantly different from the surrounding data, so that a gradually varying temperature gradient should not be considered as outliers. This can happen by considering the continuity of the data, namely that this phenomenon should not be regarded as abnormal if the temperature value gradually rises or decreases. Smooth methods, such as moving window or sliding average, are used to identify progressive trends and avoid misidentification of temperature gradient changes as anomalies. For the anomaly detected automatically by the program, the anomaly is confirmed by manual secondary verification or a more detailed physical measurement. This can effectively prevent the true temperature change from being misjudged. The outliers present in the data can also be considered as a type of fault. In 2014, [PERSON] et al. [21] proposed a reconstruction-based contribution method based on dynamic principal component analysis (DPCA), namely the multi-directional reconstruction contribution method. Using a continuous stirred-tank reactor (CSTR) as an example, the effectiveness of their proposed method was validated. By combining the dynamic characteristics of the axle temperature data used in this study and the long-term scale axle temperature data, the outliers belong to sparse points. Therefore, the present study also attempts to propose an effective approach for outlier reconstruction by combining DPCA and principal component search, mainly dynamic principal component search (DPCS) [22]. To this end, we commence with integrating the DPCA-based approach with variable delay expansion to extract the dynamic relationships between variables, and the matrix \(\mathbf{X}_{\text{d}}\) of delayed data is obtained from \(\mathbf{X}_{\text{s}}\) based on the delay time \(d\) in the following form:  
\[\mathbf{X}_{d}= \begin{pmatrix}\mathbf{X}_{\text{s}}^{T}(1:N-d+1,1:m)\\ \mathbf{X}_{\text{s}}^{T}(2:N-d+2,1:m)\\ \mathbf{X}_{\text{s}}^{T}(d:N,1:m)\end{pmatrix}  \tag{5}\]  
Based on the delayed data matrix, \(\mathbf{X}_{\text{d}}\) is decomposed into \(\mathbf{X}_{\text{d}}=\)\(\mathbf{L}\)+\(\mathbf{S}\)+\(\mathbf{N}\) as per the formula, where \(\mathbf{L}\),\(\mathbf{S}\), and\(\mathbf{N}\)represent the low-rank data matrix, the sparse outlier data, and noise, respectively. By combining DPCA with DPCS via L\({}_{1}\)-norm and convex optimization [23], the method could reconstruct the outlier data to obtain a matrix \(\mathbf{X}_{\text{r}}\) of normal axle temperature data. The results of the outlier treatment of the axial temperature data used in this experiment are shown in Figures 2 and 3. In order to reduce the experimental error and improve the performance and generalization ability of the machine learning model, this paper selected the k-fold cross-validation method, and the ratio of the training set, validation set, and test set was 7:2:1, and the number of iterations was 50. The original dataset was randomly split into 3 equal-sized sub-cross validation sets, with 2 sub-cross validation sets used for model training and the remaining sub-cross validation set for model testing. The above process was repeated three times, and finally three independent model performance assessments were obtained.