## 2 Prognostic Approach
### Expression of Remaining Useful Life  
Based on the concept of stochastic process lead time, when the failure threshold \(\omega\) is reached for the first time, the system life is considered to be terminated. Based on the observed data \(X_{0:k}=\{x_{0},x_{1},x_{2},\ldots,x_{k}\}\), the RUL \(L_{k}\) of the system at the moment \(t_{k}\) is defined as:  
\[L_{k}=\inf\{l_{k}:X(l_{k}+t_{k})\geq\omega|X_{0:k}\} \tag{31}\]After getting the new data and corresponding updated parameters \(\Theta=[\sigma^{2},\mu_{\theta,k},\sigma_{\theta,k}^{2}]\), According to the literature [17], the remaining life of PDF (probability density function) and CDF (cumulative distribution function) can be obtained respectively:  
\[f_{L_{k}|X_{0k}}(l_{k}|X_{0k})=\frac{\omega-x_{k}}{\sqrt{2\pi l_{k}^{3}\Big{(} \sigma_{\theta,k}^{2}l_{k}+\sigma^{2}\Big{)}}}\exp\Bigg{\{}-\frac{(\omega-x_{ k}-\mu_{\theta,k}l_{k})^{2}}{2l_{k}\Big{(}\sigma_{\theta,k}^{2}l_{k}+\sigma^{2} \Big{)}}\Bigg{\}} \tag{32}\]  
\[F_{L_{k}|X_{0k}}(l_{k}|X_{0k})=1-\Phi\Bigg{(}\frac{\omega-x_{k}-\mu_{\theta,k}l _{k}}{\sqrt{\sigma_{\theta,k}^{2}l_{k}^{2}+\sigma^{2}l_{k}}}\Bigg{)}+\exp \bigg{\{}\frac{2\mu_{\theta,k}(\omega-x_{k})}{\sigma^{2}}+\frac{2\sigma_{ \theta,k}^{2}(\omega-x_{k})^{2}}{\sigma^{4}}\bigg{\}}\times\Bigg{(}-\frac{2 \sigma_{\theta,k}^{2}(\omega-x_{k})l_{k}+\sigma^{2}(\mu_{\theta,k}l_{k}+ \omega-x_{k})}{\sigma^{2}\Big{\{}(}\sigma_{\theta,k}^{2}l_{k}^{2}+\sigma^{2} l_{k}\Big{)}}\Bigg{)} \tag{33}\]  
Here, the life prediction method under the linear stochastic deterioration model has been completed. The following steps are concluded to estimate the RUL of the micro-switches, which belongs to the S826 rail vehicle:  
Step 1: A linear degradation model based on a Wiener process is proposed: \(X(t)=\theta t+\sigma B(t)\), the degradation detection data are described as \(X_{0k}=\{x_{0},x_{1},x_{2},\ldots,x_{k}\}\). Step 2: \(\theta\) is a random parameter and will be updated with the data obtained at the current moment \(t_{k}\). \(\theta\) is distributed as \(\theta\sim N\Big{(}\mu_{\theta,k},\sigma_{\theta,k}^{2}\Big{)}\), in order to estimate the hyper-parameters \(\mu_{\theta,k},\sigma_{\theta,k}^{2}\) in the random parameter \(\theta\), the Bayesian posterior estimation is used in this paper. Finally, calculate the expression of the hyper-parameters \(\mu_{\theta,k},\sigma_{\theta,k}^{2}\). Step 3: Unfortunately, the unknown parameters \(\Theta=[\sigma^{2},\mu_{0},\sigma_{0}^{2}]\) are not updated by Bayesian estimation. In order to reflect the updated characteristics of \(\Theta\) over time, we use the EM algorithm to estimate \(\Theta\) through the monitoring data \(X_{0k}\), and its update results are expressed as \(\Theta_{k}=[\sigma_{k}^{2},\hat{\mu}_{0,k},\hat{\sigma}_{0,k}^{2}]\). The main updated parameter of the EM is \(\sigma^{2}\), other two parameters \(\mu_{\theta,k},\sigma_{\theta,k}^{2}\) are also updated. Initial parameters-based Bayesian estimation are improved after Bayesian updating in the next step. Step 4: When the parameters are convergent, it will not obtain good estimation results if the newly acquired data is different from the model parameter. The STF method solves the problem of mutational degradation data on the basis of the Kalman filter. We add prediction error \(Q\) in each step update to ensure the update capability of the algorithm, then we add the fading factor to adjust the prediction variance in real time, thus that \(\sigma_{\theta,k}^{2}\) can be sensitive to the prediction error. Step 5: The drift parameter \(\theta\) is unobservable, and the nonlinear form is adopted to an unknown problem. The initial parameter is set to be more inconsistent with the actual parameter, and such errors will affect the subsequent estimation process, and its convergence rate will be slower, resulting in a large amount of monitoring data being needed to complete the parameter convergence.