## 2 Methodology
### Parameter Initialization and Updating Algorithm  
RUL prediction starts with offline parameter estimation. A total of \(c\) monitoring time points starting from the PCT, i.e., \(t_{1},t_{2},\cdots,t_{c}\), were selected as the training set, as shown in Figure 2. Maximum-likelihood was chosen as the estimation algorithm. Figure 2: Offline parameter initialization and online remaining useful lifetime (RUL) prediction in phase II. Denote RMS values at these time points as \(\mathbf{y_{c}}=[y_{1},\cdots,y_{c}]^{\prime}\), where \([\cdot]^{\prime}\) represents the vector transposition, and \(y_{i}=y(t_{i})\), \(i=1,2,\cdots,c\). Based on (3), \(y_{i}\) can be expressed as  
\[y_{i}=x_{0}+\lambda\int_{0}^{t_{i}}\varphi(\tau;\theta)d\tau+\sqrt{m\lambda}+\varepsilon. \tag{4}\]  
**Proposition 1**. : _Let \(\mathbf{t_{c}}=\left[\int_{0}^{t_{1}}\varphi(\tau;\theta)d\tau,\cdots,\int_{0}^{t_ {c}}\varphi(\tau;\theta)d\tau\right]^{\prime}\), then \(\mathbf{y_{c}}=[y_{1},\cdots,y_{c}]^{\prime}\) is multivariate normally distributed, characterized by_  
\[\mathbf{y_{c}}\sim\mathrm{MN}\Big{(}\mu_{\lambda}\mathbf{t_{c}},\sigma_{\lambda}^{2} \mathbf{t_{c}}\mathbf{t_{c}}^{\prime}+\mathbf{\Omega}\Big{)}, \tag{5}\]  
_with \(\mathbf{\Omega}=m\mu_{\lambda}\mathbf{Q_{c}}+\sigma_{m}^{2}\mathbf{E_{c}}\), where \(\mathbf{Q_{c}}\)=\(\left[\min\{t_{i},t_{j}\}\right]_{1\leq i,j\leq c}\), \(\mathbf{E_{c}}=\mathrm{diag}\{1,\cdots,1\}_{c\times c}\) and MN represents the multivariate normal distribution._  
Proof of Proposition 1 can be found in Appendix A. Denote unknown model parameters as \(\Theta=\left[\mu_{\lambda},\sigma_{\lambda}^{2},\theta,m,\sigma_{m}^{2}\right]^ {\prime}\), then the log-likelihood function based on the extracted RMS up to \(t_{c}\) is  
\[\ln l\Big{(}\Theta|\mathbf{y_{c}}\Big{)}=-\frac{c}{2}\ln(2\pi)-\frac{1}{2}\ln \left|\sigma_{\lambda}^{2}\mathbf{t_{c}}\mathbf{t_{c}}^{\prime}+\Omega\right|-\frac{1} {2}\Big{(}\mathbf{y_{n}}-\mu_{\lambda}\mathbf{t_{c}}\Big{)}^{\prime}\Big{(}\sigma_{ \lambda}^{2}\mathbf{t_{c}}\mathbf{t_{c}}^{\prime}+\Omega\Big{)}^{-1}\Big{(}\mathbf{y_{c}}- \mu_{\lambda}\mathbf{t_{c}}\Big{)}. \tag{6}\]  
For simplification, a parameter transformation is carried out for \(\mathbf{\Omega}\), i.e.,  
\[\mathbf{\overline{\Omega}}=\mathbf{\Omega}/\sigma_{\lambda}^{2}=\frac{m\mu_{\lambda}} {\sigma_{\lambda}^{2}}\mathbf{Q_{c}}+\frac{\sigma_{m}^{2}}{\sigma_{\lambda}^{ 2}}\mathbf{E_{c}}, \tag{7}\]  
and the log-likelihood function is transformed into  
\[\ln l(\Theta|y_{c})=-\frac{c}{2}\ln(2\pi)-\frac{c}{2}\ln\sigma_{\lambda}^{2}- \frac{1}{2}\ln\left|\mathbf{t_{c}}\mathbf{t_{c}}^{\prime}+\widetilde{\Omega}\right|- \frac{1}{2\sigma_{\lambda}^{2}}\Big{(}\mathbf{y_{c}}-\mu_{\lambda}\mathbf{t_{c}}\Big{)} ^{\prime}\Big{(}\mathbf{t_{c}}\mathbf{t_{c}}^{\prime}+\widetilde{\Omega}\Big{)}^{-1} \Big{(}\mathbf{y_{c}}-\mu_{\lambda}\mathbf{t_{c}}\Big{)}. \tag{8}\]  
The first partial derivatives of \(\ln l\Big{(}\Theta|\mathbf{y_{c}}\Big{)}\) with respect to \(\mu_{\lambda}\) and \(\sigma_{\lambda}^{2}\) are given as  
\[\left\{ \begin{array}{c}\frac{\partial\ln l\big{(}\Theta|\mathbf{y_{c}}\big{)}}{ \partial\mu_{\lambda}}=\frac{\mathbf{t_{c}}^{\prime}\big{(}\mathbf{t_{c}}\mathbf{t_{c}}^{ \prime}+\widetilde{\Omega}\big{)}^{-1}\big{(}\mathbf{y_{c}}-\mu_{\lambda}\mathbf{t_{c }}\big{)}}{\sigma_{\lambda}^{2}}\\ \frac{\partial\ln l\big{(}\Theta|\mathbf{y_{c}}\big{)}}{\partial\sigma_{\lambda}^{ 2}}=-\frac{c}{2\sigma_{\lambda}^{2}}+\frac{1}{2\sigma_{\lambda}^{2}}\Big{(} \mathbf{y_{c}}-\mu_{\lambda}\mathbf{t_{c}}\Big{)}^{\prime}\Big{(}\mathbf{t_{c}}\mathbf{t_{c}}^{ \prime}+\widetilde{\Omega}\Big{)}^{-1}\Big{(}\mathbf{y_{c}}-\mu_{\lambda}\mathbf{t_{c }}\Big{)}\end{array}