# Bearing Health Monitoring Using Relief-F-Based Feature Relevance Analysis and HMM
## 2 Stochastic Feature Selection
### Supervised Relevance Analysis under Stochastic Modeling
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Estimation** & **Parameter** & **Estimation** \\ \hline Mean frequency (\(\mu_{\mathbf{s}}\)) & \(\sum\limits_{k=1}^{K}\frac{\delta_{k}}{\mathbb{R}}\) & Standard deviation frequency & \(\left(\sum\limits_{k=1}^{K}\frac{(\lambda_{k}-\Xi)^{2}s_{k}}{K\mu_{\mathbf{s}}} \right)^{1/2}\) \\ Central frequency (\(\Xi\)) & \(\sum\limits_{k=1}^{K}\frac{\lambda_{k}s_{k}}{K\mu_{\mathbf{s}}}\) & Kurtosis & \(\sum\limits_{k=1}^{K}\frac{\delta_{k}^{4}}{K\mu_{\mathbf{s}}^{2}}\) \\ Root mean square frequency & \(\Xi^{1/2}\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Frequency domain features (\(F\)). Frequency-based parameters are computed for each pair frequency and spectrum vectors \(\mathbf{\lambda},\mathbf{s}\in\mathbb{R}^{K}\). according to the locality preservation capability in \(\rho\), that is, \(\tilde{\mathbf{X}}\) holds the \(M^{\prime}\) features that exhibit the highest \(\rho_{m}\) values, a HMM can be defined by the set of parameters \(\mathbf{\Psi}=\{\mathbf{\Pi},\mathbf{b},\mathbf{p}_{o}\}\) and the set of states \(\mathcal{S}=\{\mathbf{\theta}_{v}\in\mathbb{R}^{M^{\prime}}\}_{v=1}^{V}\). The matrix \(\mathbf{\Pi}\in[0,1]^{V\times V}\) holds the state transition probabilities with \(\Pi_{vv^{\prime}}=P\left(\theta_{j+1}=v^{\prime}|\theta_{j}=v\right);(v,v^{ \prime},j=\{1,2,\ldots,V\}\), \(P(\cdot|\cdot)\) is a conditional probability distribution), and \(\mathbf{B}=\{b_{v}(\cdot)\}_{v=1}^{V}\) is an output probability function set, defined as follows:  
\[b_{v}(\mathbf{\hat{x}}_{w})=\sum_{u=1}^{U}\varpi_{vu}\mathcal{N}(\mathbf{\hat{x}}_{w}| \mathbf{\theta}_{vu},\mathbf{\Sigma}_{vu}), \tag{5}\]  
where \(\mathbf{\hat{x}}_{w}\in\mathbb{R}^{M^{\prime}}\) is the \(w\)-th observation in \(\mathbf{\hat{X}}\), \(U\) is the number of mixtures used to model the overall probability distribution, \(\varpi_{vu}\in[0,1]\) is a mixture weight, \(\mathbf{\theta}_{vu}\in\mathbb{R}^{M^{\prime}}\) and \(\mathbf{\Sigma}_{vu}\in\mathbb{R}^{M^{\prime}\times M^{\prime}}\) are the mean vector and the covariance matrix for the state \(v\) and the mixture \(u\), respectively, and \(\mathcal{N}(\cdot)\) denotes a normal distribution. Moreover, \(\mathbf{p}_{o}\in\mathbb{R}^{V}\) is the vector of initial state probabilities. Successively, a classifier can be trained from each bearing fault class, thereby, a model set \(\Lambda=\{\mathbf{\Psi}_{c}:c\in\{1,\ldots,C\}\}\) is computed as the following maximum likelihood estimation:  
\[\mathbf{\Psi}_{c}=\arg\max_{\mathbf{\Psi}}P(\mathbf{\hat{X}}_{n}|\mathbf{\Psi});\quad\forall\{ \mathbf{X}_{n}:y_{n}=c\}. \tag{6}\]  
Finally, given a new matrix of feature segments \(\mathbf{\hat{X}}_{*}\in\mathbb{R}^{W\times M^{\prime}}\), a HHM-based classification rule is given by:  
\[\hat{c}_{*}=\arg\max_{c}P(\mathbf{\hat{X}}_{*}|\mathbf{\Psi}_{c}). \tag{7}\]  
In practice, the optimization problem in Equation (6) can be solved using an Expectation-Maximization-based approach [60]. Figure 1 summarizes the pipeline of the proposed bearing fault diagnosis system based on our SFS approach. Note that the time complexity of our SFS resides in the Relief-F and HMM required number of operations. Thereby, Relief-F boats a time complexity of \(\mathcal{O}(N\times Q)\) meanwhile the HMM requires \(\mathcal{O}(N^{2}\times Q)\)[61; 62]. Our SFS's MatLab implementation can be found at a GitHub repository ([[https://github.com/SiegE89/KBS-2018](https://github.com/SiegE89/KBS-2018)]([https://github.com/SiegE89/KBS-2018](https://github.com/SiegE89/KBS-2018))).