## 1 Introduction
According to whether the data directly reflect the performance or health status of the system, the method is subdivided into methods based on direct monitoring data and those based on indirect monitoring data. The residual life prediction methods driven by direct monitoring data can be divided into those based on time series modeling and those based on stochastic process modeling. Commonly used time series models include time series analysis techniques ([PERSON] and [PERSON], 1991), gray models ([PERSON] and [PERSON], 1991), artificial neural networks ([PERSON] and [PERSON], 1991), support vector machines ([PERSON] and [PERSON], 1991), and combined prediction models of these methods ([PERSON] and [PERSON], 1991). However, this method can only obtain the magnitude of the residual life span, not its distribution form, and thus does not reflect the uncertainty of the life span prediction results well. The residual life prediction methods based on stochastic processes mainly include those based on Gamma process modeling, inverse Gaussian process modeling, Wiener process modeling and Markov chain modeling, etc. The Gamma process is usually only applicable to modeling the degradation trajectory of monotonic data ([PERSON] and [PERSON], 1991; [PERSON], 1991), such as metal wear and crack growth. However, in the actual working environment the system operation will be disturbed by noise and environment, etc., and its degradation process is not strictly monotonic and will show certain random perturbations. The inverse Gaussian process is also only applicable to describe the monotonic degradation process ([PERSON], 2000), but compared with the Gamma process, the inverse Gaussian process is mathematically easier to derive and implement and more flexible. The advantages of the Wiener model in mathematical and physical sense make it a widely used degradation modeling and residual life prediction method [12], but due to the introduction of Brownian motion, it will have a direct impact on the system residual life prediction results directly due to the introduction of Brownian motion, in other words, it will increase the uncertainty of the results. Markov chain-based modeling methods often require a large amount of expert knowledge or data for training in order to ensure the accuracy of the prediction results [13]. For the continuous degradation process, a discrete approximation is required, which can also cause certain errors. Indirect monitoring data-driven methods for residual life prediction include stochastic filtering models [14,15], Hidden Markov Chain Models (HMM) [16, 17, 18, 19], Hidden Semi-Markov Models (HSMM) [20, 21, 22], and Proportional Hazard Models (PHM) [23]. The random filtering model can absorb the risk of the model in a single step. The stochastic filtering model can absorb the latest monitoring data time by time, thus realizing the real-time prediction of residual life and making the prediction results more and more accurate without setting the failure threshold in advance. However, it is also limited by the amount of operational monitoring data, and the lack of data can lead to inaccurate prediction results. HMM as a probabilistic statistical method with good stochasticity characterization ability and potential structural relationship description ability is widely used in the field of mechanical product modeling. However, to be used directly for the residual life prediction of the system, HSMM is to add the time factor to the HMM structure, which can well describe the intrinsic relationship between the system state and the external observations, but this method can only obtain the expected value of the residual life, which is not convenient for the later health management of the system. It can effectively predict the residual life of the system. However, to ensure the effectiveness of PHM, it must assume that the risk ratio is fixed, i.e., the influence of covariates on the survival probability does not change with time.