## 2 Prognostic Approach
### Adding Fading Factor Based on the STF to Enhance Robustness  
In this section, we discuss how to adjust parameters in time and guarantee the accuracy of estimation when the model parameters and real-time data are mismatched.  
It can be proved from Equations (4) and (7):  
\[\begin{array}{ll}\sigma_{\theta,k-1}^{2}=\sigma_{0}^{2},\\ \sigma_{\theta,k}^{2}=\frac{\sigma^{2}\sigma_{\theta,k-1}^{2}}{t_{k}\sigma_{ \theta,k-1}^{2}+\sigma^{2}}=\sigma_{\theta,k-1}^{2}\cdot\frac{\sigma^{2}}{t_{k }\sigma_{\theta,k-1}^{2}+\sigma^{2}}<\sigma_{\theta,k-1}^{2}\end{array}\] (8 It is easy to prove that \(\sigma^{2}_{\theta,k}\) will approach to 0 when \(t_{k}\rightarrow\infty\). And because \(\sigma^{2}_{\theta,k-1}\) in the Equation (9) is also monotonically decreasing, hyper-parameter \(\sigma^{2}_{\theta,k}\) will decay faster. When \(\sigma^{2}_{\theta,k}\to 0\), it can be seen that:  
\[\lim_{\sigma^{2}_{\theta,k}\to 0}\mu_{\theta,k}=\lim_{\sigma^{2}_{ \theta,k}\to 0}\frac{\left(\mu_{\theta,k-1}\sigma^{2}+x_{k}\sigma^{2}_{ \theta,k}\right)}{\left(t_{k}\sigma^{2}_{\theta,k}+\sigma^{2}\right)}=\mu_{ \theta,k-1} \tag{10}\]  
\(\mu_{\theta,k}\) will not change as new data is acquired from Equation (10), that is, the stochastic degradation parameter \(\theta\) will not change with the acquisition of new data. [PERSON] et al. [17] directly uses Bayesian updating and EM algorithm to estimate unknown parameters \(\Theta\). When the degraded data is stationary, it can be well estimated. However, when parameters are convergent, it will not obtain good estimation results if the newly acquired data is different from the model parameter.  
The reason for the parameter no longer being updated is \(\sigma^{2}_{\theta,k}\to 0\). Considering that the Kalman filter algorithm is obtained in the Bayesian framework, it has some similarities with the Bayesian updating. The Kalman filter results in the fact that the \(K\) value tends to the minimum, thus that it is no longer sensitive to the prediction error. The STF solves the problem of mutational degradation data on the basis of the Kalman filter [24]. It is not practical that \(\sigma^{2}_{\theta,k}\) is rapidly approaching 0, inspired by reference [24], we added the prediction error \(Q\) in each step update to ensure the update capability of the algorithm, then we added a fading factor to adjust the prediction variance in real time, thus that \(\sigma^{2}_{\theta,k}\) could be sensitive to the prediction error. The specific calculation steps are as follows:  
Step 1: Establish system equation  
\[\mu_{\theta,k}=\mu_{\theta,k-1}+\eta\ \,\ \ \eta\sim N\Big{(}0,\sigma^{2}_{ \theta,k-1}\Big{)} \tag{11}\]  
\[\begin{array}{c}x_{k}-x_{k-1}=\mu_{\theta,k}(t_{k}-t_{k-1})+\sigma_{k}\epsilon ^{\prime}(t_{k}),\\ \epsilon^{\prime}(t_{k})\sim N\big{(}0,\sigma^{2}_{k}(t_{k}-t_{k-1})\big{)} \end{array} \tag{12}\]  
Step 2: Set initial parameters \(\mu_{0},\sigma_{0},\alpha,\rho\)  
Step 3: Calculated fading factor \(v(t_{k})\)  
\[\gamma(t_{k})=x_{k}-x_{k-1}-\mu_{\theta,k}(t_{k}-t_{k-1}) \tag{13}\]  
\[V_{0}(t_{k})=\left\{\begin{array}{c}\gamma^{2}(t_{1}),k=1\\ \frac{\rho V_{0}(t_{k-1})+\gamma^{2}(t_{k})}{1+\rho},k>1\end{array}\right. \tag{14}\]  
\[B(t_{k})=V_{0}(t_{k})-\sigma^{2}_{\theta,k}(t_{k}-t_{k-1})^{2}-\alpha\sigma^{ 2}_{k}(t_{k}-t_{k-1}) \tag{15}\]  
\[C(t_{k})=P_{k-1|k-1}(t_{k}-t_{k-1})^{2},v_{0}=\frac{B(t_{k})}{C(t_{k})} \tag{16}\]  
\[\
u(t_{k})=\left\{\begin{array}{c}v_{0},v_{0}\geq 1\\ 1,v_{0}<1\end{array}\right. \tag{17}\]  
Step 4: Status updates  
\[P_{k-1|k}=\
u(t_{k})P_{k-1|k-1}+\sigma^{2}_{\theta,k} \tag{18}\]  
\[K_{k}=\frac{P_{k|k-1}(t_{k}-t_{k-1})}{\left(t_{k}-t_{k-1}\right)^{2}P_{k|k-1}+ \sigma^{2}_{k}(t_{k}-t_{k-1})} \tag{19}\]  
Model parameter updating:  
\[\hat{\mu}_{\theta,k}=\mu_{\theta,k}+K_{k}\cdot\gamma(t_{k}) \tag{20}\]  
Estimation variance updating:  
\[P_{k|k}=(1-K_{k}\cdot t_{k})P_{k|k-1} \tag{21}\]Usually, the forget factor is set to \(\rho=0.95\), the softening coefficient is \(\alpha=1.1\), and the prediction error is \(Q=0.5\).