## 4 Health Prognostics
### Empirical Degradation Model  
The empirical degradation model--also known as the degradation model based on a statistical method--is a degradation mechanism model summarized by technicians in the process of long-term use. It does not necessarily have physical significance. The empirical degradation model can obtain the probability distribution of life or remaining life by modeling the monitoring data compared with the physical degradation model. It is convenient to quantify the uncertainty of life or remaining life [236]. [PERSON] et al. [237] and [PERSON] et al. [152] summarized the statistical data-driven methods used in RUL prediction, divided into a non-random empirical degradation model and stochastic empirical degradation model. The non-random empirical degradation model ignores the internal mechanism of equipment degradation, adopts typical distribution curves, including the Exponential model [238; 239; 240], Weibull model [241], Proportional hazards model [242], and so forth. The advantage of this model is to describe the degradation trend of the equipment through the regression model and predict the RUL of equipment by extrapolation. However, its disadvantage is that the degradation trend model based on experience is arbitrary and uncertain. The stochastic empirical degradation model fully considers the influence of uncertainty in the degradation process [214] and can describe the random time variability in the process of equipment degradation. Many stochastic empirical degradation models have been used for RUL prediction, including Wiener process [243; 244; 245; 246], Gamma process [247; 248], Inverse Gaussian process [249; 250; 251], Markov model [252; 253; 254; 255; 256; 257; 258]. The Wiener process has good physical and mathematical properties and can explain the time variability and nonmonotonic process of the degradation process. However, due to its dependence on the Markov hypothesis, and mainly focuses on the current degradation level and ignores the historical data, resulting in inaccurate RUL prediction. Gamma process and Inverse Gaussian process are limited to Markov condition, which can only simulate monotonic degenerate trajectory, resulting in an impractical model. Markov model is suitable for describing multi-stage transition process in the degradation process, but it is difficult to obtain training samples in different degradation stages in practical application. The advantage of the stochastic empirical degradation model is that it can obtain the analytical expression of RUL. However, it needs to use the advanced stochastic process theory for mathematical derivation, which is not conducive to the application and promotion in MSAE, the model parameter identification is difficult, and the prediction accuracy needs to be improved. In addition, the degradation trend model based on experience is arbitrary, and it is difficult to explain the degradation mechanism when the results are inaccurate. In practical application, the mechanical system usually presents a nonlinear degradation process. Therefore, in addition to considering the uncertain influence in the degradation process, it is necessary to establish a nonlinear degradation model to describe the degradation process of machinery [259; 260]. [PERSON] et al. used an exponential model to represent the nonlinear degradation process, but this method is only effective for the exponential degradation process [206]. [PERSON] et al. responded to the nonlinear degradation process by introducing acceleration factors and predicted RUL combined with the linear autoregressive model [261]. [PERSON] et al. described the nonlinear degradation process using the Paris Erdogan model and predicted RUL using the PF algorithm [262]. [PERSON] proposed an adaptive nonlinear prediction model and considered the influence of uncertainty for RUL prediction [260]. [PERSON] et al. proposed a statistical method to extract features from vibration signals, build a dynamic model and predict its evolution with time to estimate when the gear reaches the critical stage [167].