## 6 Reliability Prediction Model
### Rvm  
The Relevance Vector Machine (RVM), proposed by [PERSON] in 2001 [PERSON] (2001), is a type of sparse probabilistic model. Its fundamental concept is to achieve model sparsity based on the theory of sparse Bayesian learning, integrated with Markov properties, maximum likelihood estimation, and automatic relevance determination (ARD) priors. During the iterative process of the RVM, the posterior distributions of most feature parameters will converge towards zero, indicating that the information contained in these features is redundant for classification. The features with non-zero posterior distributions are referred to as \"relevance vectors\", which can reflect the most critical information in the data.  
Let \(\{x_{i}\}_{i=1}^{N}\) and \(\{r_{i}\}_{i=1}^{N}\) represent the input and output of the training samples, respectively. It can be obtained by the following formula:  
\[R=\sum_{i=1}^{N}W_{i}\cdot K(x,x_{i})+C+\epsilon_{n} \tag{26}\]  
In the formula, \(\epsilon_{n}\) represents the noise with a mean of zero and variance \(\sigma^{2}\), \(K(x,x_{i})\) is the kernel function, \(C\) is the bias, and \(W_{i}\) is the weight vector.  
The Relevance Vector Machine (RVM), as an efficient algorithm, is capable of mapping original nonlinear data into a high-dimensional feature space, making the data linearly separable in this new space. This mapping not only significantly simplifies the complexity of the problem but also effectively reduces computational costs.  
Due to the complexity of bearing signal data, relying solely on a single kernel function makes it difficult to fully capture its characteristics. To enhance the learning ability of the model, the multi-kernel learning method combines various kernel functions with different characteristics, allowing the model to focus on both global and local features of the data, thereby demonstrating superior performance. When dealing with complex data, multi-kernel functions often outperform single kernel functions. Based on this, this study proposes a weighted combination of polynomial kernel functions with global response characteristics and Gaussian kernel functions with local response characteristics to construct a new mixed kernel function. Combined with this composite kernel function \(K(x_{i},x_{j})\), a Mixed Kernel Relevance Vector Machine (MKRVM) model is designed for bearing performance prediction. Its expression is as follows:  
\[\begin{split} K(x_{i},x_{j})=\lambda k_{1}\big{(}x_{i},x_{j}\big{)}+ (1-\lambda)k_{2}\big{(}x_{i},x_{j}\big{)}\\ k_{1}\big{(}x_{i},x_{j}\big{)}=\exp\!\left(-\frac{x_{j}-x_{i}^{ 2}}{2\delta^{2}}\right)\\ k_{2}\big{(}x_{i},y_{j}\big{)}=\big{(}bx_{i}^{\mathrm{T}}y_{j}+c \big{)}^{d}\end{split} \tag{27}\]  
where \(\lambda\) is the weight parameter used to adjust the influence of different kernel functions; \(k_{1}(x_{i},x_{j})\) and \(k_{2}(x_{i},y_{j})\) represent the Gaussian kernel function and the polynomial kernel function, respectively; \(\delta\), \(b\), \(c\), and \(d\) are the parameters corresponding to the kernels. This model not only fully utilizes the complementary advantages of the polynomial kernel function and the Gaussian kernel function but also effectively enhances the feature extraction and prediction capabilities for complex bearing signals, providing a new approach for accurate performance evaluation.