## 2 Vibration Signal Processing Methods
### Multi-Domain Feature Extraction and Sensitive Feature Selection
#### 2.2.2 Sensitive Feature Selection  
Different features have varying sensitivities to bearing degradation. In this paper, the monotonicity, trendability, predictability, and robustness [16] are used to comprehensively evaluate the sensitivity of the signal features. The weighted sum is calculated to determine  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Serial Number** & **Name** & **Feature Expression** \\ \hline
5 & Mean Square Root Value & \(p_{5}=\sqrt{\frac{\sum\limits_{i=1}^{N}{x_{i}^{2}}}{N}}\)\(2^{2}\) \\
6 & Root Mean Square Value & \(p_{6}=\sqrt{\frac{\sum\limits_{i=1}^{N}{x_{i}^{2}}}{N}}\) \\
7 & Kurtosis & \(p_{7}=\frac{\sum\limits_{i=1}^{N}{x_{i}^{4}}}{N}\) \\
8 & Waveform & \(p_{8}=\frac{p_{8}}{|p_{1}|}\) \\
9 & Kurtosis & \(p_{9}=\frac{p_{7}}{|p_{6}|}\) \\
10 & Skewness & \(p_{10}=\frac{N}{(N-1)(N-2)}\sum\limits_{i=1}^{N}{(\frac{x_{i}-p_{1}}{p_{2}})}\)\(3^{3}\) \\
11 & Peak & \(p_{11}=\max(x(t_{i}))\) \\
12 & Mean Amplitude in & \(p_{12}=\frac{\sum\limits_{i=1}^{N}{s_{k}}}{N}\) \\
13 & Root Mean Square Frequency & \(p_{13}=\frac{\sum\limits_{k=1}^{N}{(f_{k})}^{2}s(k)}{\sum\limits_{k=1}^{K}{s( k)}}\) \\ \hline \hline \end{tabular} Where \(x_{i}\) is the time domain signal sequence such that \(i=1\), \(2\), \(\cdots\), \(N\) is the number of sample points, \(s(k)\) is the spectrum of signal \(x(n)\), \(k=1\), \(2\), \(\cdots\), \(k\) is the number of spectral lines, and \(f_{k}\) is the frequency of the \(k\)th spectral line.  
\end{table}
Table 1: _Cont._a comprehensive score. Note that a higher score indicates higher sensitivity and greater contribution to the prediction of bearing life.  
The Monotonicity indicator (\(M\)) determines the trend of a signal feature; it is expressed as [44]:  
\[M=\frac{1}{m}|\frac{positivediff(x_{i_{k}}^{n})-negativediff(x_{i_{k}}^{n})}{k_{a}-1}| \tag{5}\]  
where \(x_{i}\) represents the \(i_{a}\)-th feature, \(k_{a}\) represents the number of measurements for each feature, and m is the number of monitored systems.  
The Trendability indicator (\(Q\)) determines the correlation between the bearing running time and degradation features. It is computed as [44]:  
\[Q=\frac{|\sum_{i_{b}=1}^{N_{Q}}{(x_{i_{b}}-x_{m})(y_{i_{b}}-y_{m})}|}{\sqrt{ \sum_{i_{b}=1}^{N_{Q}}{(x_{i_{b}}-x_{m})^{2}\sum_{i_{b}=1}^{N_{Q}}{(y_{i_{b}}- y_{m})}^{2}}}} \tag{6}\]  
where \(x_{i_{b}}\) is the rank of features, \(y_{i_{b}}\) is the rank of time, and \(N\) is the number of feature measurements. Note that, for \(Q=1\), the feature is strictly monotonic, while for \(Q=0\), it is non-monotonic.  
The Predictability indicator (\(C\)) represents the ratio of the deviation (\(std\)) of the final failure value to the mean range of change for each feature. It is given by [44]:  
\[C=\exp(-\frac{std(f_{i_{c}}(N))}{mean[f_{i_{c}}(1)-f_{N_{c}}(N_{c})]}) \tag{7}\]  
where \(f_{i_{c}}(1)\) and \(f_{i_{c}}(N_{c})\) are the initial and failure values of the \(i_{c}\)-th feature, respectively.  
The Robustness indicator (\(R\)) represents the tolerance of specific features to existing outliers. Before calculation \(R\), the underlying features should be smoothed using the local weighted scatter plot smoothing technique. It is expressed as [44]:  
\[R=\frac{1}{N_{R}}\sum_{i_{R}}\exp(-\frac{f_{i_{R}}-\overset{\sim}{f}}{f_{i_{R} }}) \tag{8}\]  
where \(f_{i_{R}}\) is the \(i_{R}\)-th feature, \(N_{R}\) represents its measurement value, and \(\overset{\sim}{f}\) is the average trend value.  
The final score is calculated by the weighted sum of each indicator [44]:  
\[ScoreJ_{i_{0}}=w_{1}^{\prime}M_{i_{0}}+w_{2}^{\prime}Q_{i_{0}}+w_{3}^{\prime} F_{i_{0}}+w_{4}^{\prime}R_{i_{0}} \tag{9}\]  
where \(j_{0}\) represents the \(j_{0}\)-th indicator, \(w_{1}^{\prime}\), \(w_{2}^{\prime}\), \(w_{3}^{\prime}\), \(w_{4}^{\prime}\) represents the weight, and \(w_{1}^{\prime}+w_{2}^{\prime}+w_{3}^{\prime}+w_{4}^{\prime}=1\).