## 4 Health Prognostics
### Empirical Degradation Model
#### 4.3.2 Prognostics Method Fusion  
Prognostics method fusion has achieved good application in the prediction of battery remaining life. [PERSON] et al. presented a hybrid neural network model to determine the hidden degradation function and predict the RUL of lithium-ion batteries [287]. [PERSON] et al. used the particle filter (PF) method to dynamically update the hyperparameters of the SVR model, overcome the disadvantage of poor prediction ability when there are significant differences between the training set and test set, and improve the robustness and generalization ability of the model [288]. In addition, it has also achieved good results in the fields of air compressor, bearing and so on. [PERSON] et al. integrated SOM, Multiple Linear Regression (MLR), and Polynomial Regression (PR) models to estimate the RUL of reciprocating compressor [175]. [PERSON] et al. used a neural network to fuse the features extracted from the signal at the feature level, then used smoothing and wavelet decomposition to denoise, and used multivariate nonlinear model regression to predict the time series when the deterioration curve triggered the alarm threshold, to realize the HCM and RUL prediction of the compressor [289]. [PERSON] et al. proposed a kernel regression depth transfer metric learning method and successfully applied it to RUL prediction of bearings under multiple working conditions [290]. The above method successfully solves the single-point prediction of RUL. However, because the prediction method will be affected by various sources of uncertainty, such as operation uncertainty, modeling uncertainty, and prediction method uncertainty, the method based on point estimation alone cannot provide sufficient confidence for the prediction results, resulting in inaccurate results. Therefore, in the prediction method of MSAE, it is very necessary to explain and quantify the prediction uncertainty and improve the prediction results. To solve this problem, [PERSON] et al. proposed the fusion of two Bayesian models to provide a probability distribution for the uncertainty in the final prediction results, and the framework is shown in Figure 12[143]. The Bayesian theorem can be expressed as  
\[p(w|X,Y)=\frac{p(w)p(Y,X|w)}{p(Y|X)} \tag{27}\]  
where \((X,Y)\) is the observed variables, \(w\) is the unobserved parameters. Firstly, the diesel engine HCM data are preprocessed to extract the required training set and test set data. Secondly, through the indirect estimation method, Bayesian neural network (BNN) is used to model the relationship between HCM data (IAS signals) and HI data (IMEP data), so as to realize the diesel engine health monitoring and prediction process. The model consists of an input layer, two hidden layers, and an output layer. There are \(n\) sets of HCM data \(\mathbf{D}=\mathbf{X},\mathbf{Y}\) corresponding to the input features of IAS signals \(\mathbf{X}=\left\{x^{i}\right\}_{j=1}^{n},x^{i}\in R^{1\times m}\) and the output value of IMEP \(\mathbf{Y}=\left\{y^{i}\right\}_{i=1}^{n},y^{i}\in R\). The parameters \(z\) of BNN model are then trained by \(x^{i}\) and \(y^{i}\). The new output value \(\hat{y}^{i+1}\) could be expressed as  
\[\hat{y}^{i+1}=\int f\left(x^{i+1},z\right)p(z|D)dz \tag{28}\]  
According to the VI algorithm, a variational distribution \(q(z)\) is approximate to the true posterior \(p(z|D)\), so the \(\hat{y}\) could be written as  
\[\hat{y}^{i+1}\approx\int f\left(x^{i+1},z\right)q(z)dz \tag{29}\]  
where \(f\) is the nonlinear activation function. Then Bayesian logistic regression (BLR) model is used to model the predicted HI data, and the degradation performance with uncertainty is evaluated and quantified with  
Figure 12: Application of hybrid performance prognostics to diesel engine. a confidence interval to generate a series of possible final results rather than a single prediction value. The input of BLR model is the IMEP value \(\dot{y}^{i}\), and output \(c^{i}\) is a Bernoulli variable where the normal condition is 1 and the fault condition is 0.