# Bearing Health Monitoring Using Relief-F-Based Feature Relevance Analysis and HMM
## 3 Experimental Setup
### SFS Training  
For concrete testing, our SFS approach includes 56 multi-domain features: 18, 23, and 12 for time, frequency, and time-frequency domains, respectively. To favor reliable estimations regarding the frequency-based features, the spectrum vector size is set at \(K=4096\) to compute the parameters showed in Tables 1 and 2. To calculate the \(TF\) parameters, we utilized a filter bank with \(R=\lfloor 3\cdot\log(F_{\text{s}})\rfloor=32\) Mel-filters applied over each spectrum holding \(K=512\) values in frequency domain. In turn, we constructed the multi-domain feature set \(\{\mathbf{X}_{n}\in\mathbb{R}^{W\times M}\}_{n=1}^{N}\) holding \(N=5610\) samples with \(W=7\) windows and \(M=53\) features. Later, a vector concatenation was applied over \(\mathbf{X}_{n}\) to yield a vector \(\mathbf{\Gamma}\in\mathbb{R}^{Q}\), with \(Q=W\times M=371\). In this sense, two parameters were selected to compute the relevance vector, in concordance with Equation (2): \(\varphi=1\) and \(d(\gamma_{n}^{q},\gamma_{n^{\prime}}^{q})=|\gamma_{n}^{q}-\gamma_{n^{\prime}} ^{q}|/(\max_{n}(\gamma_{n}^{q})-\min_{n}(\gamma_{n}^{q}))\)[57,57]. Then, we constructed the feature relevant matrix \(\mathbf{\Delta}\) using Equation (3), and next, we estimated the relevance vector \(\mathbf{\rho}\) as shown in Equation (4). Due to the high number of samples, we consider a representative subset of data by using a clustering approach based on the well-known k-means algorithm [67]. In particular, \(N_{\text{c4}}=3413\) and \(N_{\text{c10}}=4141\) samples are preserved for the \(C=4\) and \(C=10\) classes, respectively. Besides, an HMM-based classifier is trained to fix the  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{\(C=4\) Problem} & \multicolumn{6}{c}{\(C=10\) Problem} \\ \hline
**Id** & \begin{tabular}{c} **Bearing** \\ **State** \\ \end{tabular} & **Id** & \begin{tabular}{c} **Bearing** \\ **State** \\ \end{tabular} & **\begin{tabular}{c} **Fault** \\ **Diameter** \\ \end{tabular} & **Id** & \
\begin{tabular}{c} **Bearing** \\ **State** \\ \end{tabular} & **
\begin{tabular}{c} **Fault** \\ **Diameter** \\ \end{tabular} \\ \hline N & Normal & N & Normal & - & IR2 & Fault in inner race & 0.014\({}^{\prime\prime}\) \\ B & Fault in rolling element & B1 & Fault in rolling element & 0.007\({}^{\prime\prime}\) & IR3 & Fault in inner race & 0.021\({}^{\prime\prime}\) \\ IR & Fault in inner race & B2 & Fault in rolling element & 0.014\({}^{\prime\prime}\) & OR1 & Fault in outer race & 0.007\({}^{\prime\prime}\) \\ OR & Fault in outer race & B3 & Fault in rolling element & 0.021\({}^{\prime\prime}\) & OR2 & Fault in outer race & 0.014\({}^{\prime\prime}\) \\  & IR1 & Fault in inner race & 0.007\({}^{\prime\prime}\) & OR3 & Fault in outer race & 0.021\({}^{\prime\prime}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Description of the different diagnosis/detection scenarios considered for bearing condition monitoring.  
number of states and Gaussian mixtures to \(V=2\), according to [54], and \(U=2\) (value empirically selected). Lastly, we calculated the classification accuracy over the testing set by using a nested 10-fold cross-validation scheme, by adding, one-by-one, the features ranked based on the relevance values in \(\mathbf{\rho}\) (learning curve).