# A Novel Rolling Bearing Fault Diagnosis and Severity Analysis Method
## 4 Multiclass Relevance Vector Machine  
### Relevance Vector Machine  
The relevance vector machine (RVM) is a machine-learning technique that uses Bayesian inference to provide probabilistic predictions [22]. RVMs achieve sparsity by producing models that have both a structure and a parameterization process.  
There is a set of training data \(\mathbf{T}=\{\mathbf{x}_{i},t_{i}\}_{i=1}^{N}\), where \(\mathbf{x}_{i}\in R^{D}\) is the input vector and \(t_{i}\) is the corresponding class label. The classification function of the RVM is defined as  
\[y(\mathbf{x},\mathbf{w})=\sum_{i=1}^{N}w_{i}K(\mathbf{x},\mathbf{x}_{i})+w_{0}, \tag{21}\]  
where \(K(\mathbf{x},\mathbf{x}_{i})\) is a kernel function, \(\mathbf{w}=[w_{1},w_{2},\cdots,w_{N}]\) is the weight vector, and \(w_{0}\) is the bias.  
The output target includes the additive noise, so the functional relationship between \(\mathbf{x}_{i}\) and \(t_{i}\) is given as follows:  
\[t_{i}=\sum_{i=1}^{N}W_{i}K(x,x_{i})+w_{0}+\varepsilon_{i}, \tag{22}\]  
where \(\varepsilon_{i}\) is the \(i\)th independent sample from a noise process with zero mean and variance \(\sigma^{2}\). According to the statistical convention, the linear model is generalized by applying the logistic sigmoid function \(\sigma(y)=1/\left(1+e^{-y}\right)\), so \(y(\mathbf{x},\mathbf{w})\) becomes a linear method. Given that \(P(\mathbf{t}|\mathbf{x})\) is the Bernoulli distribution, its probability density function is \(f(k)=p^{k}(1-p)^{1-k}(k=0\text{ or }1)\). The likelihood estimation probability \(P(\mathbf{t}|\mathbf{x})\) is written as  
\[P(\mathbf{t}|\mathbf{w})=\prod_{i=1}^{N}\sigma[y(\mathbf{x}_{i};\mathbf{w})]^{t_{i}}[1-\sigma [y(\mathbf{x}_{i};\mathbf{w})]]^{1-t_{i}}. \tag{23}\]  
However, the weights \(\mathbf{w}\) cannot be analytically obtained in the process of classification because of the non-normal distribution of \(P(\mathbf{t}|\mathbf{w})\). Thus, the closed-form expression \(P(\mathbf{w}|\mathbf{t},\mathbf{\alpha})\) or the marginal likelihood \(P(\mathbf{t}|\mathbf{\alpha})\) cannot be deduced, where \(\mathbf{\alpha}\) is a vector of \(N+1\) hyper-parameters. Therefore, a Laplace-based approximation procedure proposed by [PERSON] [31] is used as follows:  
(1) For a fixed value of \(\mathbf{\alpha}\), because \(P(\mathbf{w}|\mathbf{t},\mathbf{\alpha})\propto P(\mathbf{t}|\mathbf{w})P(\mathbf{w}|\mathbf{\alpha})\), find the maximum posteriori weight through the following equation:  
\[\log[P(\mathbf{t}|\mathbf{w})P(\mathbf{w}|\mathbf{\alpha})]=\sum_{i=1}^{n}[t_{i}\log y_{i}+(1 -t_{i})\log(1-y_{i})]-\frac{1}{2}\mathbf{w}^{T}\mathbf{A}\mathbf{w}, \tag{24}\]  
where \(y_{i}=\sigma[y(\mathbf{x}_{i};\mathbf{w})]\) and \(\mathbf{A}=diag(\alpha_{0},\alpha_{1},\cdots,\alpha_{n})\) are composed of the current values of \(\mathbf{\alpha}\).  
(2) Differentiate the quantity above twice by applying [PERSON]'s method to yield the following equation:  
\[\
abla_{\mathbf{w}}\
abla_{\mathbf{w}}\log p(w|t,\mathbf{\alpha})|_{\mathbf{w}_{MP}}=-(\mathbf{ \Phi}^{T}B\mathbf{\Phi}+\mathbf{A}), \tag{25}\]  
where \(\mathbf{B}=diag(\beta_{1},\beta_{2},\cdots,\beta_{n})\) is a diagonal matrix with \(\beta_{i}=\sigma[y(\mathbf{x}_{i};\mathbf{w}_{MP})][1-\sigma[y(\mathbf{x}_{i};\mathbf{w}_{MP})]]\), \(\