## 3 Case Study
This was consistent with the above-mentioned explanations. Figure 8: Correlation test between degradation rate and degradation diffusion. (**a**) Correlation test result of the 1-st wheel tread and (**b**) correlation test result of the 2-nd wheel tread. Figure 3:0 Extracted degradation features and phase changing times (PCTs) of the two wheel treads. (**a**) Extracted root mean square (RMS) values and PCT of the 1-st wheel tread and (**b**) extracted RMS values and PCT of the 2-nd wheel tread. To identify the effectiveness of the proposed model, we predicted the degradation state and the RUL distribution at different monitoring points of the two wheel treads. Figure 10 compares the predicted degradation path with the original degradation path. We can observe that the prediction results matched the original degradation path well except for the first twenty points, which means that our proposed model had a better goodness-of-fit with vibration monitoring data of the wheel treads. Figure 11 presents the PDFs of the RUL of the wheel treads at different monitoring points calculated by our proposed model. We found that our proposed model could cover actual RULs with an acceptable precision. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\mu_{\lambda,\theta}\) & \(\sigma^{2}_{\lambda,\theta}\) & \(\theta_{0}\) & \(m_{0}\) & \(\sigma^{2}_{m,\theta}\) \\ \hline PCT of the 1-st wheel tread & \(1.13\times 10^{-8}\) & \(6.40\times 10^{-10}\) & 15.62 & 176.11 & \(1.44\times 10^{-2}\) \\ PCT of the 2-nd wheel tread & \(1.07\times 10^{-4}\) & \(3.40\times 10^{-3}\) & 2.80 & 44.12 & \(9.60\times 10^{-3}\) \\ \(t_{stripping}\) of the 2-nd wheel tread & \(1.80\times 10^{-3}\) & \(2.70\times 10^{-3}\) & 0.61 & 6.56 & \(1.07\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter initialization results of the two wheel treads. Figure 10: The predicted degradation paths of the two wheel treads. (**a**) The predicted RMS values of the 1-st wheel tread and (**b**) the predicted RMS values of the 2-nd wheel tread. Figure 9: Parameter updating procedure based on Kalman filter (KF) and expectation maximization (EM) algorithm. (**a**) The updating procedure of the mean and variance of drift parameter for the 1-st wheel tread and (**b**) the updating procedure of the mean and variance of drift parameter for the 2-nd wheel tread. For comparison, we further compared the expected RULs derived by our model (defined as Model 1) and the following model (defined as Model 2), i.e.,  
\[x(t)=x_{0}+\lambda\int_{0}^{t}\varphi(\tau;\theta)d\tau+\sigma B(t)+\varepsilon, \varepsilon\sim N\Big{(}0,\sigma_{m}^{2}\Big{)}, \tag{20}\]  
which does not consider the quantitative relationship between degradation rate and degradation variation, with the actual RULs at various monitoring points. Table 2 shows the relative errors (REs) of the predicted results. Noticeably, our proposed model incurred smaller REs compared with the model in Equation (20). Therefore, the results could demonstrate that our proposed model could help to improve the accuracy of degradation path tracking and RUL estimation. In the meanwhile, it should be mentioned that the REs at the first two monitoring points of the 2-nd wheel tread were a little big, which means that the predicted results were not quite satisfactory. This is reasonable because the two points were particularly close to the stripping point of the wheel tread and there was not much degradation information available for RUL prediction. With the increasing of degradation data, we could see that the REs of the predicted results became smaller and smaller, especially for our proposed model.