## 2 Prognostic Approach
### Estimating Initial Parameters by an Inverse Kalman Filter
According to the conventional Kalman filter [25; 27], all monitoring data of the system is recursively incorporated into parameter \(\theta\) by an inverse Kalman filter technology. The specific calculation process is as follows:  
Step 1: Establish system equation  
\[\mu_{k}=\mu_{k-1}+\eta^{\prime} \tag{22}\]  
\[x^{\prime}_{k}-x^{\prime}_{k-1}=\left(t^{\prime}_{k}-t^{\prime}_{k-1}\right) \cdot\mu_{k}+\sigma\epsilon_{k} \tag{23}\]  
where \(\eta\sim N(0,Q)\), \(\sigma\epsilon_{k}\sim N\big{(}0,\sigma^{2}\big{(}t^{\prime}_{k}-t^{\prime}_{ k-1}\big{)}\big{)}\), \(Q\) is the forecast variance. Step 2: Time updating  
\[\hat{\mu}_{k}^{-}=\hat{\mu}_{k-1} \tag{24}\]  
\[P_{k|k-1}=P_{k-1|k-1}+Q \tag{25}\]  
where \(\hat{\mu}_{0}=-\mu_{0}\), \(P_{0|0}=\sigma_{0}^{2}\). Step 3: Status updating  
\[K_{k}=\frac{P_{k|k-1}\cdot\left(t_{k}^{\prime}-t_{k-1}^{\prime}\right)}{\left(t _{k}^{\prime}-t_{k-1}^{\prime}\right)^{2}\cdot P_{k|k-1}+\sigma^{2}\left(t_{k} ^{\prime}-t_{k-1}^{\prime}\right)} \tag{26}\]  
\[\hat{\mu}_{k}=\hat{\mu}_{k}^{-}+K_{k}\big{(}\big{(}x_{k}^{\prime}-x_{k-1}^{ \prime}\big{)}-\big{(}t_{k}^{\prime}-t_{k-1}^{\prime}\big{)}\hat{\mu}_{k}^{-} \big{)} \tag{27}\]  
\[P_{k|k}=\big{(}1-K_{k}\cdot\big{(}t_{k}^{\prime}-t_{k-1}^{\prime}\big{)}\big{)} P_{k|k-1} \tag{28}\]  
where \(R\) is the system error, \(R=\sigma^{2}\big{(}t_{1}^{\prime}-t_{0}^{\prime}\big{)}\). Step 4: Update results of the initial parameter  
\[\mu_{\theta,k}=-\hat{\mu}_{k} \tag{29}\]  
\[\sigma_{\theta,k}^{2}=P_{k|k} \tag{30}\]  
The parameters of an inverse Kalman filter are set as \(R=0.025\ Q=0.50\), which can make the algorithm more dependent on system measurements. **Remark 1**. : _For an inverse Kalman filter, the difference from the conventional Kalman filter [25; 27] is that the order of the elements \(X_{0:k}\) are flipped as \(X_{0:k}^{\prime}=\big{\{}x_{0^{\prime}}^{\prime},x_{1}^{\prime},x_{2}^{\prime },\ldots,x_{k}^{\prime}\big{\}}\),it is accounted for \(x_{0}^{\prime}=x_{k},x_{1}^{\prime}=x_{k-1},\cdots,x_{k}^{\prime}=x_{0}\), \(x_{k}\) is the last monitoring data at the current time \(t_{k}\),\(x_{0}\) is the initial monitoring data at the current time \(t_{0}\). The flipped data as \(X_{0:k}^{\prime}=\big{\{}x_{0^{\prime}}^{\prime},x_{1}^{\prime},x_{2}^{\prime },\ldots,x_{k}^{\prime}\big{\}}\) guarantees the iteratively updated forward of the estimation from the last monitoring point when it starts to filter._  
**Remark 2**. : _The consequence of applied an inverse Kalman filter reveals that it can be obtained by the optimal estimation \(\mu_{\theta,k}\)._  
**Remark 3**. : _It is obvious that the steps according to an inverse Kalman filter are similar with the conventional Kalman filter. However, a practical problem is that the initial parameters are difficult to determine without a large amount of historical information, thus they are indeterminate and often set with errors. If we take advantage of the conventional Kalman filter to solve the errors belonging to the initial parameters, it will not be sensitive to them. In contrast, the last parameters are fixed and reliable relatively accounts for a number of historical information. The advantage of an inverse Kalman filter is that, when we start to filter from the last monitoring data \(x_{k}\), the accuracy of the filtering is improved for the initial monitoring data especially._