## 2 Prognostic Approach
### Estimating Initial Parameters by an Inverse Kalman Filter  
Although the algorithm by [PERSON] et al. [17] has self-adaption update capability, even if the initial parameters are not set correctly, it will approach the accurate value as the new data is acquired. However, a practical problem is that the initial parameters are difficult to determine without a large amount of historical information, and it is also unknown where monitoring data starts from the whole life cycle. A progressively updating algorithm of parameter \(\theta\) is proposed by [PERSON] et al. [17]. If the initial parameter is set to be more inconsistent with the actual, such errors will affect the subsequent estimation process, and its convergence rate will be slow, resulting in that a large amount of monitoring data is needed to complete the parameter convergence. Even more, the algorithm has not completed convergence, experimental object has been damaged and the inaccurate life prediction results are obtained in its life cycle. To this end, this section will discuss an inverse Kalman filter to update the initial parameters in real time. The initial parameters are updated at the source, thus that the convergence speed of the estimation is accelerated. The reason for this application of an inverse Kalman filter is that the drift parameter \(\theta\) is unobservable, where the nonlinear form adopted is an unknown problem. Because the degradation is more stable in the early phase, when the sampling interval is not long, it is reasonable for \(\mu_{\theta,k}\) to obey stochastic Gauss distribution around \(\mu_{\theta,k-1}\) based on large sample statistical theory [25]. At the same time, nonlinear modeling methods (such as exponential models) can be used, and model parameters can be updated with the Bayesian updating and EM. After model parameters are determined, the initial parameters will be estimated by an inverse Kalman filter. In this way, the algorithm will be very complex and computation time will be greatly increased. More importantly, although the initial phase of degradation is relatively stable, it does not necessarily satisfy the overall degradation model. For example, the initial degradation process data in this paper does not agree with the overall degradation trend. That is, the model that conforms to the overall data may not necessarily satisfy the initial degradation process, it is related to the time point of when the monitoring begins. Therefore, it is difficult to estimate the exact initial parameters by using the determined nonlinear model. It is often the case with actual degradation data: The initial degradation phase is stable, and then the volatility of data becomes more pronounced until the system fails, such as the data from this paper and reference [26]. Therefore, it is more and more difficult to update the initial parameters by normal methods. Initial parameters are difficult to converge, even if the initial parameters are convergent, they are not the exact values. If the initial parameters are updated via recursively in reverse, it will be found that degradation data will be smooth gradually, updating the true value of the initial parameters \(\mu_{\theta,k},\sigma_{\theta,k}^{2}\) that are expected to be improved, and converge to the exact value. For the actual operation of the system, the monitoring data corresponding to the current time \(t_{k}\) is \(x_{k}\), and the observation data are \(X_{0:k}=\{x_{0},x_{1},x_{2},\ldots,x_{k}\}\). In order to show the [PERSON] updated process is more intuitive, the order of the elements are flipped as \(X^{\prime}_{0:k}=\{x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2},\ldots,x^{ \prime}_{k}\}\), where \(x^{\prime}_{0}=x_{k},x^{\prime}_{1}=x_{k-1},\cdots,x^{\prime}_{k}=x_{0}\). \(T_{0:k}\) has been rewritten as \(T^{\prime}_{0:k}=\{t^{\prime}_{0},t^{\prime}_{1},t^{\prime}_{2},\ldots,t^{ \prime}_{k}\}\) by the same method.