# Remaining Useful Life Prediction of Broken Rotor Bar Based on Data-Driven and Degradation Model
## 5 Synthetic Data Manipulation
### Feature Extraction  
Tracking the broken bar fault pattern of the SCIM is a challenging task in our prognostic approach since the accuracy of the estimated RUL depends on the extracted CIs. In this work, the recorded signals are processed using time- and frequency-domains techniques. Signal processing in the time domain mainly includes time-domain statistical feature extraction. Therefore, it is the most used approach to reflect the statistical properties from its time-domain waveform for detection, diagnostic, and prognostic purposes. The extracted time-domain features from the torque, stator current, and speed signals are summarized in Table 1, where \(x\) is the signal series for i = 1, 2,..., N and N is the number of data points of each signal. Among the features shown in Table 1, the statistical indicators, such as mean, root mean square (RMS), kurtosis, standard deviation (std), and peak-to-peak are widely used thanks to their ability to capture changes in the signal pattern. On the other hand, the indicators representing the overall shape of the signals, such as the crest factor, impulse factor, shape factor, margin factor, and energy, are decisive for incipient fault. Each feature can contain fault information with different levels; even many cannot be sensitive to the fault. Therefore, more CIs should be extracted to track the broken rotor bar fault of the SCIM. Signal processing in the frequency domain is to disclose the inherent nature by providing a frequency spectrum for each time sample. It can reflect the frequency of rotor faults and the distribution of a signal. In this work, the spectral kurtosis \((SK)\) analysis technique is applied to the torque, stator current, and speed signals, which is defined \begin{table}
\begin{tabular}{c c} \hline \hline
**Feature** & **Expression** \\ \hline Peak-To-Peak & \(x_{max}-x_{min}\) \\ Root Mean Square (RMS) & \((\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2})^{\frac{1}{2}}\) \\ Mean & \(\frac{1}{N}\sum_{i=1}^{N}x_{i}\) \\ Kurtosis & \(\frac{1}{N}\sum_{i=1}^{N}\frac{(x_{i}-\mu)^{4}}{(N-1)\sigma^{4}}\) \\ Skewness & \(\frac{\sum_{i=1}^{N}(x_{i}-\mu)^{3}}{(N-1)\sigma^{3}}\) \\ Std & \((\frac{1}{N}\sum_{i=1}^{N}(x_{i}-mean))^{\frac{1}{2}}\) \\ Energy & \(\sum_{i=1}^{N}x_{i}^{2}\) \\ Crest factor & \(\frac{x_{max}}{RMS}\) \\ Shape factor & \(\frac{RMS}{N}\sum_{i=1}^{N}|x_{i}|\) \\ Margin factor & \((\frac{x_{max}}{N}\sum_{i=1}^{N}|x_{i}|)^{2}\) \\ Impulse factor & \(\frac{x_{max}}{N}\sum_{i=1}^{N}|x_{i}|\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Time-domain features. as the kurtosis of the signal spectral components. \(SK\) of the signal \(x(t)\) is defined as the normalized fourth-order spectral moment as given in the following [42]. \[SK(f)=\frac{\left\langle\left|X^{4}(t,f)\right|\right\rangle}{\left\langle\left| X^{2}(t,f)\right|\right\rangle^{2}}-2 \tag{6}\]  
where \(\left\langle\cdot\right\rangle\) represents the time-frequency averaging operator, and \(X^{4}(t,f)\) and \(X^{2}(t,f)\) are the fourth-order and the second-order cumulants, respectively, of a band-pass filtered signal of \(x(t)\) around \(f\). The four extracted features in the frequency domain include the std, mean, skewness, and kurtosis of the \(SK\). In this work, 45 CIs, including 33 time-domain features and 12 frequency-domain features, were extracted from the torque, stator current, and speed signals, respectively. Those features are associated with noise, which can be harmful to the RUL prediction. Therefore, the smoothing process of the 45 CIs is performed using a moving average filter. As shown in Figure 5, the rotor degradation behavior is correlated with statistical features of raw torque, stator current, and speed signals, respectively. As can be seen, most of the extracted CIs reflect a clear degradation trend over time, such as mean, std, rms, kurtosis, crest factor, etc., compared with the others. Kurtosis-SK and skewness-SK fluctuate clearly and do not reflect any monotonicity for torque signals.