# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 7 Performance Evaluation
### Feature Extraction and Postprocessing  
Figure 3 shows the power spectrum as a function of time and frequency of BearingB_2. The colorbar indicates the fault severity normalized to 1, dependent on the RUL value of the experiment. As the machine progressively approaches to failure, the power spectrum around 13 kHz gradually increases. Hence, the statistical features of the power spectrum are potential indicators of degraded bearings.  
Various statistical features of time domain, frequency domain, and time-frequency domain signals, as listed in Table 1, are used to construct the health indicator. We only select main features with monotonicity greater than 0.4 as the input to PCA for feature fusion by considering all training sets. Note that we only use the training data to select the main features for the health indicator. These features are standard deviation, RMS, energy, median absolute deviation of the time domain, and mean and standard deviation of the conditional spectral moment with orders 2 and 3 in the time-frequency domain. Besides, the mean, skewness, kurtosis, and standard deviation of the local maxima of the power spectrum are selected in the frequency domain. One interesting finding is that the operating condition is not a critical factor in the feature importance ranking using the monotonicity.  
Once the main features are selected using the monotonicity rank, PCA is applied to construct the health indicator by reducing the dimension of feature spaces. Before performing PCA, the mean and standard deviation of training data normalize the whole data. Figure 4 shows the projected data onto the first two principal components obtained by PCA of Bearing A_2. We clearly observe that the first principal component increases as the machine gets closer to failure. We use the first principal component as the health indicator for the RUL prediction.  
Figure 5 presents the health indicators of various training cases Bearing A_1, Bearing A_2, Bearing B_1, Bearing B_2, Bearing C_1, Bearing C_2 as a function of experimental time. We observe that the health indicator monotonically increases under various operating conditions with different cases. While each case shows significantly different health indicator at the initial state, all cases converge to a similar health indicator value as the machine approaches failure. Based on these observations, we set the threshold as 9, around 90% of the maximum value of the health indicators of the training data, to reduce the delay effect of smoothing.  
Another interesting observation is that the operating condition is not a dominant factor to characterize the health indicator. For instance, two different cases Bearing A_1 and Bearing A_2 of same operating condition are considerably different. On the other hand, the health indicators of both Bearing B_2 and Bearing C_2 are similar and rise rapidly when the bearing is closer to the end, even if they have different operating conditions. Hence, it clearly shows that the health indicator significantly varies due to the complex interactions of machine components and high uncertainty.  
Figure 4: Two principal components of Bearing A_2.  
Figure 3: Power spectrum over time and frequency of Bearing B_2.