# A Tutorial for Feature Engineering in the Prognostics and Health Management of Gears and Bearings
## 3 Feature Extraction
### Features for Gears
At this step, the gear specific features are extracted as shown in Figure 3, which are the figure of merits zero (FM0) and sideband energy ratio (SER). FM0 serves to detect changes in gear engagement patterns as an indicator of the gear's main fault:  
\[\textit{FM0}=\frac{\textit{PP}_{x}}{\sum_{i=1}^{N_{\textit{var}}}\textit{P}_{ i}}. \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Abbreviation** & **Full Name** & **Brief Explanation** & **Formula** & \begin{tabular}{c} **Matlab** \\ **Functions** \\ \end{tabular} \\ \hline MEAN & Mean & Average & \(\frac{\sum X}{N}\) & mean(x) \\ RMS & Root mean square & \begin{tabular}{c} Value that generally tends to get bigger as the \\ degree of fault in the bearing increases \\ \end{tabular} & \begin{tabular}{c} \(\sqrt{\frac{\sum X^{2}}{N}}\) \\ \(\sqrt{\frac{\sum(X-\overline{X})^{2}}{N-1}}\) \\ \end{tabular} & rms(x) \\ STD & Standard deviation & Value representing the dispersion of a signal & \(\sqrt{\frac{\sum(X-\overline{X})^{2}}{N-1}}\) & std(x) \\ PEAK & Peak & \begin{tabular}{c} Maximum value of signal absolute value \\ The asymmetry of the probability density \\ function of the vibration signal \\ \end{tabular} & \begin{tabular}{c} max(\(|X|\)) \\ \(\frac{\frac{1}{N}\sum(X-\overline{X})^{3}}{STD^{3}}\) \\ \end{tabular} & \begin{tabular}{c} max(abs(x)) \\ skewness(x) \\ \end{tabular} \\ KUR & Kurtosis & \begin{tabular}{c} The sharpness of the probability distribution of \\ the vibration signal, and if this value is close to \\ 3, it is closer to normal distribution \\ \end{tabular} & kurtosis(x) \\ CF & Cres factor & The ratio of peak values to the RMS of a signal & \begin{tabular}{c} \(\frac{\textit{PEK}}{\textit{PEK}}\) \\ \(\frac{\max(|X|)}{\left(\frac{\sum x}{N}\right)}\) \\ \end{tabular} & - \\ CL & Clearance factor & Peak value divided by the square of root mean & \begin{tabular}{c} - \\ \(\frac{\max(|X|)}{\left(\frac{\sum x}{N}\right)}\) \\ \end{tabular} & - \\ SF & Shape factor & RMS divided by mean & \begin{tabular}{c} - \\ \(\frac{\textit{BMS}}{\textit{MEAN}}\) \\ \end{tabular} & - \\ IF & Impulse factor & The ratio of peak values to the mean of a signal & \
\begin{tabular}{c} - \\ \(\frac{\max(X)}{\left(\frac{\sum x}{N}\right)}\) \\ \end{tabular} & - \\ P2P & Peak to peak &
\begin{tabular}{c} The difference between maximum and \\ minimum values of the signal \\ \end{tabular} & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: General time domain features [17; 22]. where \(PP_{x}\) is the difference between the maximum and minimum of the time domain signal, \(P_{i}\) is the amplitude of the \(i\)th harmonic frequencies of GMF, and \(N_{har}\) is the number of harmonic frequencies. The GMF is the frequency caused by the engagement of gear teeth, given by the product of shaft frequency and number of teeth. In the frequency domain of gear signal, sidebands occur at both sides of the GMF and its harmonics with the interval of shaft frequency. SER is the ratio of the sum of sideband frequency amplitudes to that of the first harmonic of GMF:  
\[SER=\frac{\sum_{i=1}^{N_{sh}}\!\left(S_{i}^{+}+S_{i}^{-}\right)}{P_{1}}, \tag{2}\]  
where \(P_{1}\) is the amplitude at the first harmonic of GMF, \(N_{sb}\) is the number of sidebands, which is usually 3 [23], and \(S_{i}^{+}\) and \(S_{i}^{-}\) denote the amplitudes of the \(i\)th sideband at the first harmonic of GMF. The next step is to obtain the residual signal by removing the components of GMF and their harmonics, which are those not related with the fault. From the residual signal, the feature NA4 is obtained, which is to detect progress of defects in gears.