# Remaining Useful Life Prediction of Broken Rotor Bar Based on Data-Driven and Degradation Model
## 5 Synthetic Data Manipulation
### Feature Selection  
In this study, the three most popular metrics defined in the literature, including monotonicity, trendability, and prognosability, evaluate the significance of the extracted features and select the correlated ones with the fault propagation. The first metric for the prognosis is monotonicity, which characterizes the increasing or decreasing trends of the CIs as the SCIM evolves toward failure [43]. The more the monotonicity score is close to 1, the more the feature has a better monotonic trend. It is the absolute difference between the numbers of positive and negative derivatives for each feature. The expression of the monotonicity is defined as  
\[M_{i}(f_{i})=\left|\frac{No.\ of\ dx/dt>0-No.\ of\ dx/dt<0}{n-1}\right| \tag{7}\]  
where \(M_{i}\) is the monotonicity value for the \(i\)th feature \(f_{i}\) with length of \(n\). The second metric is the trendability, which measures similarity between the trajectories of the extracted CIs and represents the correlation between them. The constant CIs  
Figure 5: Individual smoothed feature trending, (**a**) torque signals (blue: before smoothing, red: after smoothing), (**b**) current signals, and (**c**) speed signals. have zero correlation with time and, therefore, zero trendability, and the CIs with linear functions have a strong correlation with time, showing large trendability. The trendability for this objective is calculated as [43]. \[T_{i}(f_{i})=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][ n\sum y^{2}-(\sum y)^{2}]}} \tag{8}\]  
where \(x\) and \(y\) represent the vector of measurements and time index of the feature, respectively, and \(n\) is the number of measurements. The third metric is prognosability, which measures the variance of the critical value of failure in a population of systems. It mainly consists of the exponential ratio of the standard deviation of CIs to its mean value, and it is given by [23]  
\[P_{i}(f_{i})=exp\left(-\frac{Std_{j}(x_{j}(N_{j}))}{mean_{j}(x_{j}(1)-x_{j}(N _{j}))}\right),j=1,...,M \tag{9}\]  
where \(x_{j}\) is the observations of a feature on the \(j\)th system, \(M\) represents the number of the monitored system, and \(N_{j}\) represents the number of observations on the \(j\)th system. Higher values of the three metrics give better performance in the prognosis. Therefore, the suitability criterion (Equation (10)) for feature selection is given by the sum of the three metrics in this study. \[Suitability=M_{i}(f_{i})+T_{i}(f_{i})+P_{i}(f_{i}) \tag{10}\]  
The sum of the metrics is calculated for each of the 45 time and frequency-domain features. The results are given in Figure 6, in which the importance rank of the extracted features from the torque, stator current, and speed signals show different suitability scores. It can be seen that the skewness, kurtosis, Peak-to-Peak, crest factor, and impulse factor have higher suitability scores compared with the other CIs for the torque signals. As given in Figure 6b, the crest factor, impulse factor, and Peak-to-Peak are the best features with a suitability score higher than 0.5 for the current signals. Figure 6c gives the performance metrics for each CI extracted from the speed signals. Std and Peak-to-Peak are the most sensitive features to the degradation phenomena, with a suitability score equal to 0.6. As a result, the rank of features shows that the time-domain features have a higher suitability score than the frequency-domain features. Therefore, the features with a suitability score higher than 0.5 are selected for the prediction task. It is possible to choose the skewness, crest factor, and std features to be a HI machine due to their higher suitability score, but there are other features with the same score that can provide useful information about the degradation behavior and EOL of the system. Therefore, to determine unique and proper HIs that can represent relevant information about the system performance degradation, a compression of the selected features is required.