## 4 Health Prognostics
### Empirical Degradation Model
#### 4.3.2 Machine Learning Method
Define a set of input data \(\{x_{i},t_{i}\}_{i=1}^{N}\) where \(x_{i}\) is the input variable vector, \(t_{i}\) is the target value, \(N\) is the length of the data, the RVM regression can be indicated as  
\[t(x)=\sum_{i=1}^{N}\omega_{i}K(x,x_{i})+\omega_{0}+\varepsilon_{n}, \tag{25}\]  
where \(\mathbf{\omega}=[\mathbf{\omega}_{1},\mathbf{\omega}_{2},\cdots,\mathbf{\omega}_{N}]\) is the weight, \(\omega_{0}\) is the bias, \(K(x,x_{i})\) is kernel function, and \(\varepsilon_{n}=N\big{(}0,\sigma^{2}\big{)}\) is error term with zero mean Gaussian process and variance \(\sigma^{2}\). The Gaussian kernel function is usually preferred and given by  
\[K(x,x_{i})=\exp\Bigl{[}-(x-x_{i})^{\mathsf{T}}(x-x_{i})/2S^{2}\Bigr{]}, \tag{26}\]  
where \(S^{2}\) is the width. More details of the algorithm can be found in the Ref. [274]. There are mainly three different RVM training algorithms: sequential sparse Bayesian learning, MacKay iterative learning, and Expectation-Maximization (EM) iterative learning algorithm. An EM-RVM algorithm is used to predict the RUL of lithium-ion batteries in hybrid vessels, which can monitor its health and prevent the occurrence of failure [213]. [PERSON] et al. used RVM to establish the prediction degradation curve in the statistical and sparse form [159]. [PERSON] et al. applied the data running to fail to predict the survival probability of a single unit of machine components through the RVM model [275]. Relevance vector machine (RVM) is a sparse probability model based on a Bayesian training framework, which can overcome the shortcomings of SVM. At the same time, RVM can deal with high-dimensional, nonlinear, and small samples and provide probability prediction. It has the advantages of good sparsity, generalization ability, and high prediction accuracy. These advantages make it more suitable for solving the problems of prediction with multiple uncertainties in MSAE and difficult to obtain training samples. Gaussian process regression (GPR)  
GPR is a nonparametric method that does not need to assume the candidate model structure in advance. Otherwise, the GPR method has advantages in life prediction when data are unreliable, noisy, or missing. Up till now, this model has been mainly applied in the prediction of bearing and battery life. [PERSON] et al. used Renyi entropy to extract the degradation characteristics of the bearing and input it into the GPR model to predict the RUL of the bearing [276]. [PERSON] firstly decomposes the battery capacity curve by empirical mode decomposition (EMD), then uses different GPR and logistic regression for the decomposed natural mode function and residual part, respectively, and finally adds the results of the two parts to predict the remaining life of the battery [277]. [PERSON] et al. selected the signal peak value and its position feature after wavelet transforms as the input of GPR and effectively predicted the battery's remaining life [278]. [PERSON] et al. combined radial basis function KPCA and GPR to predict RUL [279]. [PERSON] et al. used the improved GPR method to realize multi-step ahead RUL prediction [280]. In general, the machine learning method relies on massive historical information to establish a model and predict RUL. It has high flexibility, does not need a lot of professional knowledge and detailed understanding of the exact degradation law, and has low requirements for the physical knowledge of the inherent fault mechanism of the system. It is suitable for complex equipment with a limited understanding of the physical level of the system, such as diesel engines in MSAE. Although machine learning models have some advantages in practical application, there are also several problems in the implementation of these models. First, it is difficult to accurately define the fault threshold due to the nonlinear nature of degradation caused by the uncertainty of the environment, load, and self-degradation. Second, the training machine learning algorithm is facing the challenge of a lack of running to the fault data set.