## Appendix B  
(1) The realization of the EM algorithm  
The joint likelihood function for the EM algorithm can be expressed as follows:  
\[\begin{array}{ll}&\ln p(X_{0:k},\theta_{k}|\Theta)\\ &=\ln p(X_{0:k}|\theta_{k},\Theta)+\ln p(\theta_{k}|\Theta)\\ &=\frac{1}{2}\ln\sigma_{\theta}^{2}-\frac{(\theta_{0}-\mu_{\theta})^{2}}{2 \omega_{\theta}^{2}}-\frac{k}{2}\ln Q-\sum\limits_{j=1}^{k}\frac{(\theta_{j}- \theta_{j-1})^{2}}{2Q}-\\ &\frac{k}{2}\ln\sigma^{2}-\sum\limits_{j=1}^{k}\frac{(X_{j}-X_{j-1}-\theta_{j- 1}(T_{j}-T_{j-1}))^{2}}{2\sigma^{2}(T_{j}-T_{j-1})}\end{array} \tag{1}\]From Equations (13) and (15), Equation (A1) is formulated by:  
\[\begin{array}{l}\ell(\Theta|\hat{\Theta}_{k}^{(i)})=\mathbb{E}_{\theta_{k}|X_{0 k},\hat{\Theta}_{k}^{(i)}}(\ln p(X_{0:k},\theta_{k}|\Theta))\\ =\frac{1}{2}\ln\sigma_{0}^{2}-\frac{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}{ 2\sigma_{0}^{2}}\frac{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}{2\sigma_{0}^ {2}}-\\ \frac{k}{2}\ln Q-\sum\limits_{j=1}^{k}\frac{[\mathbb{E}_{\theta_{k}|X_{0k}, \hat{\Theta}_{k}^{(i)}}(\theta_{j}^{2})-2\mathbb{E}_{\theta_{k}|X_{0k},\hat{ \Theta}_{k}^{(i)}}(\theta_{j}\beta_{j-1})+\mathbb{E}_{\theta_{k}|X_{0k},\hat{ \Theta}_{k}^{(i)}}(\theta_{j-1}^{2})]}{2\sigma_{0}^{2}}-\\ \frac{k}{2}\ln\sigma^{2}-\sum\limits_{j=1}^{k}\frac{[(X_{j}-X_{j-1})^{2}-2 \hat{\theta}_{j-1k}(X_{j}-X_{j-1})(T_{j}-T_{j-1})+(T_{j}-T_{j-1})^{2}\mathbb{E} _{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j-1}^{2})]}{2\sigma^{2}(T _{j}-T_{j-1})}\end{array}\] (A2)  
For Equation (A2), let \(\partial\ell(\Theta|\hat{\Theta}_{k}^{(i)})/\partial\Theta=0\) The optimal result of the estimated parameter \(\hat{\Theta}_{k}^{(i+1)}\) of the step \(i\)+1 is:  
\[\begin{array}{l}(\mu_{\theta})_{k}^{i+1}=\hat{\theta}_{0k}\\ (\sigma_{0}^{2})_{k}^{(i+1)}=P_{0 jk}\\ (\mathcal{Q})_{k}^{(i+1)}=\frac{1}{k}\sum\limits_{j=1}^{k}\left[\mathbb{E}_{ \theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j}^{2})-2\mathbb{E}_{\theta_ {k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j}\beta_{j-1})+\mathbb{E}_{\theta_{ k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j-1}^{2})\right]\\ (\sigma^{2})_{k}^{(i+1)}=\frac{1}{k}\sum\limits_{j=1}^{k}\frac{[(X_{j}-X_{j-1 })^{2}-2\theta_{j-1k}(X_{j}-X_{j-1})(T_{j}-T_{j-1})+(T_{j}-T_{j-1})^{2}\mathbb{E }_{\theta_{k}|X_{0k},\hat{\[PERSON]}_{k}^{(i)}}(\theta_{j-1}^{2})]}{(T_{j}-T_{j- 1})}\end{array}\] (A3)  
In Equation (A2), \(\mathbb{E}_{\theta_{k}|X_{0k},\hat{\[PERSON]}_{k}^{(i)}}(\theta_{j})\), \(\mathbb{E}_{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j}^{2})\) and \(\mathbb{E}_{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j}\theta_{j-1})\) need to be calculated. The Rauch Tung Striebel (RTS) smoothing algorithm is used to achieve the calculation; the specific steps are as follows.  
1. RTS smoothing algorithm
2. Step 1: Forward iteration by the KF algorithm to get the optimal state estimation \(\theta_{k}|X_{0k}\sim N(\hat{\theta}_{k},P_{ik})\). Step 2: Backward iteration: \[\begin{array}{l}H_{j}=P_{jk}|P_{j+1|j}^{-1}\\ \hat{\theta}_{jk}=\hat{\theta}_{jj}+H_{j}(\hat{\theta}_{j+1|k}-\hat{\theta}_{ j+1|j})\\ P_{jk}=P_{jj}+H_{j}^{2}(P_{j+1|k}-P_{j+1|j})\end{array}\] (A4) Step 3: Initialize: \[M_{\text{kjk}}=(1-K_{k}(S_{j}-S_{j-1}))P_{k-1|k-1}\] (A5) Step 4: Backward iteration of covariance: \[M_{\text{jjk}}=P_{jj}H_{j-1}+H_{j}(M_{j+1|k}-P_{jj})H_{j-1}\] (A6)  
where \(M_{\text{jjk}}=\text{cov}(\theta_{j},\theta_{j-1}|X_{0:k})\) the conditional expectation can be derived:  
\[\begin{array}{l}\mathbb{E}_{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_ {j})=\hat{\theta}_{\text{jjk}}\\ \mathbb{E}_{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j}^{2})=\hat{ \theta}_{\text{jjk}}+P_{jk}\\ \mathbb{E}_{\theta_{k}|X_{0k},\hat{\Theta}_{k}^{(i)}}(\theta_{j}\beta_{j-1})= \hat{\theta}_{jk}\hat{\theta}_{j-1|k}+M_{\text{jjk}}\end{array}\] (A7)  
The Equation (A7) is brought into the Equation (A3), after the wear data of any monitoring mileage is obtained, and the above parameters can be used to estimate and update the model parameters adaptively.