# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 4 Health Indicator Construction
### Feature Extraction  
Various signal processing techniques, including time domain, frequency domain, and time-frequency domain analysis, are used to construct the features. We consider not only basic statistical measures such as mean, standard deviation, and root mean square (RMS) but also the higher-order statistics such as skewness and kurtosis. All these statistics can be expected to change as a deteriorating fault signature intrudes the nominal signal [4,7]. Table 1 shows all statistical features used to build the health indicator, including 10 time domain features, 5 frequency domain features, and 12 time-frequency domain features.  
Figure 1: Remaining useful life (RUL) prediction framework for bearings. We add the section number corresponding to the explanation of each component.  
* Time domain feature: Simple statistical features of time domain signals can serve as health indicators for predicting RUL. For instance, the average value or variance of a specific signal increases as the system performance degrades [7]. Furthermore, the higher-order statistics provide insight into system behavior through the third moment (skewness) and fourth moment (kurtosis) of the signal [4]. We use various statistical metrics of the time domain analysis including mean, standard deviation, RMS, skewness, kurtosis, maximum-to-minimum difference, sum of the square called energy, signal median absolute deviation, peak value divided by the RMS called crest factor, and RMS divided by the mean of the absolute value called shape factor.
* Frequency domain feature: Spectral analysis extracts the useful features for predicting RUL, such as bearings, gears, and engines [7; 11]. The frequency domain features include power bandwidth, mean frequency, signal-to-noise ratio, and local maxima of the power spectrum of the signal. For example, the peak value of a signal spectrum or the frequency at which the peak magnitude occurs is changed as the machine degrades. The mean frequency, kurtosis, skewness of the power spectrum, and mean and standard deviation of the local maxima of the power spectrum are used as the statistical features of the frequency domain.
* Time-frequency domain feature: Another way to quantify the chaotic behavior is the time-frequency spectral properties such as spectral kurtosis and spectral entropy. Spectral kurtosis, for example, in the frequency domain, is considered a powerful method for the RUL prediction of the wind turbine [42]. Furthermore, the time-frequency moment effectively characterizes the frequency changes in time of non-stationary signals [43]. The short-time Fourier transform technique is used to capture the time-varying frequency behavior because the classical Fourier analysis fails to analyze the time-varying behavior. The conditional spectral moment of the time-frequency distribution of a signal is computed for a given sampling rate and order between 2 and 4. We then use the statistical metrics such as mean, standard deviation, skewness, and kurtosis of the conditional spectral moment with different orders.  
The noise of various features substantially degrades the accuracy of the RUL prediction. Furthermore, the feature importance rank based on the monotonicity, as described below, is vulnerable to noise. We apply a moving average filter to reduce the noise effect of the features where no future feature value is used.