# Bearing Health Monitoring Using Relief-F-Based Feature Relevance Analysis and HMM
## 2 Stochastic Feature Selection
### Multi-Domain Feature Estimation  
Since vibration signals hide complex and non-stationary dynamics, the following time, frequency, and time-frequency parameters are extracted from each segment \(\mathbf{z}\in\mathbf{Z}\).  
Time-based parameters (T): Time-domain features are computed as the natural representation of the input vibration signal [45; 52]. So, we consider the statistical descriptors presented in Table 1.  
Frequency-based parameters (F): Frequency-based features are extracted based on the well-known fast Fourier transform (FFT) to code harmonic patterns [45; 53]. Hence, a spectrum vector \(\mathbf{s}\in\mathbb{C}^{K}\) is computed as follows: \(s_{k}=|\sum_{t=1}^{T}z_{t}e^{-i2\pi kt/T}|\), where \(z_{t}\in\mathbf{z}\), \(\mathbf{\lambda}\in\mathbb{R}^{K}\) is a frequency index vector, \(\lambda_{k}=kF/2K\), and \(F\in\mathbb{R}\) is the sampling frequency (\(k\in\{0,1,\ldots,K\}\)). Then, we compute the statistical parameters shown in Tables 1 and 2.  
Time-frequency-based parameters (TF): Mel-frequency cepstral coefficients (MFCC) are used to highlight both linear and nonlinear patterns of the vibration signals [54; 55]. Thus, from the FFT-based spectrum \(\mathbf{s}\), the log-energy output vector \(\mathbf{\vartheta}\in\mathbb{R}^{R}\) is computed towards a filter bank \(\{\mathbf{\eta}_{r}\in\mathbb{R}^{K}\}_{r=1}^{R}\) of size \(R\) in the Mel-scale, yielding: \(\vartheta_{r}=\log(\mathbf{s}\mathbf{\eta}_{r}^{\top})\). Next, \(H\) MFCC are stored in the vector \(\mathbf{\iota}\in\mathbb{R}^{H}\), which is computed as the discrete cosine transform (DCT) of the logarithm Mel spectrum:  
\[\iota_{h}=\sum_{r=1}^{R}\vartheta_{r}\cos\bigg{(}h\frac{\pi}{R}\left(r+\frac{ 1}{2}\right)\bigg{)}. \tag{1}\]  
Lastly, once all the aforementioned parameters are computed for all provided segments, a multi-domain feature set \(\{\mathbf{X}_{n},y_{n}\}_{n=1}^{N}\) is built after concatenation of the \(T\), \(F\), and \(TF\) parameters of each \(\mathbf{z}_{w,n}\) in the row vector \(\mathbf{x}_{w,n}\in\mathbb{R}^{M}\) of \(\mathbf{X}_{n}\in\mathbb{R}^{W\times M}\), being \(M\) the number of multi-domain features extracted.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Estimation** & **Parameter** & **Estimation** \\ \hline Mean (\(\mu_{z}\)) & \(\sum\limits_{t=1}^{T}\frac{z_{t}}{T}\) & Skewness & \(\sum\limits_{t=1}^{T}\frac{z_{t}^{3}}{T\mathbf{z}_{t}^{3}}\) \\ Median & median(\(z\)) & Max value & \(\max\limits_{t}(z_{t})\) \\ Standard deviation & \(\left(\sum\limits_{t=1}^{T}\frac{(z_{t}-\mu_{z})^{2}}{T}\right)^{1/2}\) & Min value & \(\min\limits_{t}(z_{t})\) \\ Root mean square (\(Y_{z}\)) & \(\left(\sum\limits_{t=1}^{T}\frac{z_{t}^{2}}{T}\right)^{1/2}\) & Range & \(|\max\limits_{t}(z_{t})-\min\limits_{t}(z_{t})|\) \\ Peak (\(\beta_{z}\)) & \(\max\limits_{t}(|z_{t}|)\) & Interquartile range & iqr(\(z\)) \\ Shape factor & \(\frac{TY_{z}}{T}\) & Kurtosis (\(\kappa\)) & \(\sum\limits_{t=1}^{T}\frac{z_{t}^{4}}{T\mathbf{z}_{t}^{4}}\) \\  & \(\frac{\beta_{z}}{Y_{z}}\) & Speed kurtosis & \(\kappa\{z^{\prime}\}\) \\ Impulse factor & \(\frac{T\beta_{z}}{\sum\limits_{t=1}^{T}|z_{t}|}\) & Acceleration kurtosis & \(\kappa\{z^{\prime\prime}\}\) \\ Clearance factor & \(\frac{T^{1/2}\beta_{z}}{\sum\limits_{t=1}^{T}|z_{t}|^{1/2}}\) & Acceleration kurtosis derivative & \(\kappa\{z^{\prime\prime\prime}\}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Time domain features (\(T\)). Time-based parameters are computed for each vibration segment \(\mathbf{z}\).