## 2 Prognostic Approach
### Estimation of Unknown Parameters Based on EM Algorithm  
As seen in the previous section, unknown parameters \(\Theta=[\sigma^{2},\mu_{0},\sigma_{0}^{2}]\) are not updated by the Bayesian estimation. The reason for using the EM algorithm instead of the maximum likelihood estimation algorithm is that the unknown parameter \(\Theta\) contains the hidden variable \(\theta\), which cannot be directly estimated by the maximum likelihood estimation. We need to approximate the maximum likelihood estimation of parameters by maximizing the joint likelihood function \(p(X_{0:k},\theta|\Theta_{k})\). In order to reflect the updated characteristics of \(\Theta\) over time, we use the EM algorithm to estimate \(\Theta\) through monitoring data \(X_{0:k}\), its update results are expressed as \(\hat{\Theta}_{k}=[\hat{\sigma}_{k}^{2},\hat{\mu}_{0:k},\hat{\sigma}_{0,k}^{2}]\)[17].  
First, complete logarithm likelihood function \(\ln p(X_{0:k},\theta|\Theta_{k})\) is calculated as:  
\[\begin{array}{ll}\ln p(X_{0:k},\theta|\Theta_{k})&=\ln p(X_{0:k}|\theta, \Theta_{k})+\ln p(\theta|\Theta_{k})\\ &=-\frac{k+1}{2}\ln 2\pi-\frac{1}{2}\sum_{j=1}^{k}\ln\bigl{(}t_{j}-t_{j-1} \bigr{)}-\frac{k}{2}\ln\sigma_{k}^{2}-\\ &\sum_{j=1}^{k}\frac{\bigl{(}x_{j}-x_{j-1}-\theta\bigl{(}t_{j}-t_{j-1}\bigr{)} \bigr{)}^{2}}{2\sigma_{k}^{2}\bigl{(}t_{j}-t_{j-1}\bigr{)}}-\frac{1}{2}\ln \sigma_{0,k}^{2}-\frac{\bigl{(}\theta-\mu_{0:k}\bigr{)}^{2}}{2\sigma_{0,k}^{2 }}\end{array} \tag{5}\]  
E-step: Calculate the expected value \(\ell\Bigl{(}\Theta_{k}\Bigl{|}\hat{\Theta}_{k}^{(i)}\Bigr{)}\) of \(\ln p(X_{0:k},\theta|\Theta_{k})\) which is about \(p(\theta|X_{0:k},\Theta_{k}^{(i)})\)  
\[\begin{array}{ll}\ell\Bigl{(}\Theta_{k}\Bigl{|}\hat{\Theta}_{k}^{(i)} \Bigr{)}=\operatorname{E}_{\theta|X_{0:k},\Theta_{k}^{(i)}}\{\ln p(X_{0:k}, \theta|\Theta_{k})\}\\ =-\frac{k+1}{2}\ln 2\pi-\frac{1}{2}\sum_{j=1}^{k}\ln\bigl{(}t_{j}-t_{j-1} \bigr{)}-\frac{k}{2}\ln\sigma_{k}^{2}\\ -\sum_{j=1}^{k}\frac{\bigl{(}x_{j}-x_{j-1}\bigr{)}^{2}-2\mu_{\theta_{k}} \bigl{(}t_{j}-t_{j-1}\bigr{)}\bigl{(}x_{j}-x_{j-1}\bigr{)}+\bigl{(}t_{j}-t_{j- 1}\bigr{)}^{2}\bigl{(}x_{\theta_{k}^{2}}^{2}+\sigma_{\theta_{k}^{2}}^{2}\bigr{)} }{2\sigma_{k}^{2}\bigl{(}t_{j}-t_{j-1}\bigr{)}}-\frac{1}{2}\ln\sigma_{0,k}^{2 }-\frac{h_{\theta_{k}^{2}+\sigma_{\theta_{k}^{2}}^{2}-2\mu_{\theta_{k}}\mu_{0,k}+\mu_{0,k}^{2}}}{2\sigma_{0,k}^{2}}\end{array} \tag{6}\]  
M-step: Fixed parameter \(\theta\), and take the maximum value of \(\Theta\). \(\partial\ell\Bigl{(}\Theta_{k}\Bigl{|}\hat{\Theta}_{k}^{(i)}\Bigr{)}/\partial \Theta_{k}=0\), \(\hat{\Theta}_{k}^{(i+1)}\) can be expressed as follows:  
\[\begin{array}{ll}\hat{\sigma}_{k}^{2(i+1)}=\frac{1}{k}\sum_{j=1}^{k}\frac{ \bigl{(}x_{j}-x_{j-1}\bigr{)}^{2}-2\mu_{\theta_{k}}\bigl{(}t_{j}-t_{j-1} \bigr{)}\bigl{(}x_{j}-x_{j-1}\bigr{)}+\bigl{(}t_{j}-t_{j-1}\bigr{)}^{2}\bigl{(} x_{\theta_{k}^{2}}^{2}+\sigma_{\theta_{k}^{2}}^{2}\bigr{)}}{\bigl{(}t_{j}-t_{j-1} \bigr{)}},\\ \hat{\mu}_{0,k}^{(i+1)}=\mu_{\theta,k},\;\sigma_{0,k}^{2}=\sigma_{\theta,k}^{ 2}\end{array} \tag{7}\]  
Moreover, the updated results in Equation (7) required only one step to compute the maximum value, which have been given proof by the literature [17]. One step to solve the maximum value greatly reduces the computing time and has a strong practical value.  
From the results of Equation (7), the main updated parameter of the EM is \(\sigma^{2}\). The other two parameters \(\mu_{\theta,k},\sigma_{\theta,k}^{2}\) are also updated. The initial parameters in the Bayesian estimation are improved after Bayesian updating in the next step.