## 4 Health Prognostics
### Empirical Degradation Model
#### 4.3.2 Prognostics Method Fusion
The transition expressed by percentage from normal to fault can be formulated as  
\[ \begin{array}{l}p(c|y,\theta)=Bernoulli\Big{(}\frac{1}{1+\rho^{y_{i}}}\Big{)} \\ =\prod\limits_{i=1}^{N}\Big{(}\frac{1}{1+\rho^{y_{i}}}\Big{)}^{c_{i}}\Big{(}1- \frac{1}{1+\rho^{y_{i}}}\Big{)}^{1-c_{i}}\end{array}  \tag{30}\]  
where \(\theta y_{i}=\alpha+\beta y_{i}\), \(\alpha\) and \(\beta\) are the process parameters, \(y_{i}\) is the predictd value of IMEP, and the binary output is \(c_{i}\in{0,1}\). Variational inference (VI) and Markov Chain Monte Carlo (MCMC) are used to train and infer the relevant parameters in the two Bayesian methods, respectively. Finally, the new data are used to train BNN and BLR models, and the optimal parameters are retained for further prediction and assessment with uncertainty quantification. This method considers the uncertainty of parameters, which can improve the prediction performance, the robustness, and prevent overfitting of the model. With the understanding of the environment, the decision made by the model becomes more deterministic, plays a significant role in condition-based prognostics, and makes the maintenance decision of marine engines more comprehensive. In addition, another point of interest is the adaptive fusion model of automatic selection machine learning algorithm. [PERSON] et al. summarized the shortcomings of the traditional data-driven prediction method using training data sets to construct multiple candidate algorithms and considered that a single algorithm may lack robustness and waste resources in constructing abandoned alternative methods. Therefore, a multi data-driven integrated prediction method based on three weighting schemes is proposed [153]. The future development direction is how to automatically build alternative data-driven methods and select the fusion of various methods adaptively. (3) Degradation Model and Prognostics Methods Fusion  
[PERSON] et al. used the normal operation data to create the identification model and used the RMS of the residual for the construction of HI, combined Cox's PH model and SVM to build a fusion model of a three-stage method to predict the RUL of the compressor [164]. [PERSON] et al. integrated Bayesian inference-based SOM, logistic regression (LR), and high-order particle filtering (HOPF) to build a data-model-fusion scheme to evaluate and predict the health degradation of the machine [291]. Aiming at the problems of incomplete historical data and lack of prior knowledge in the process of fault prediction, [PERSON] et al. Combined Weibull proportional risk model with the least-squares linear regression function to predict the RUL of marine diesel engines [292]. [PERSON] et al. combined model-based and vibration-based features with data-driven methods to evaluate the current health state of the coupling and predict the RUL [293]. These methods aim to integrate the advantages of the degradation model and prognostics method for reliable machine health assessment and prediction. It not only solves the problem that the degradation model cannot directly reflect the degradation degree, and the complex parameters are difficult to define, make it is difficult to build a real model for machine degradation propagation, and is not suitable for equipment whose physical parameters and failure modes may change under different conditions. At the same time, it also solves the disadvantage that the prognostics model needs to use historical data including normal and fault operation in the training process, which are difficult to obtain in MSAE. These fusion model methods improve the applicability and accuracy of RUL prediction of MSAE.