# A Tutorial for Feature Engineering in the Prognostics and Health Management of Gears and Bearings
## 5 Feature Evaluation and Selection for Prognosis  
This section addresses how to evaluate the predictability performance of the features and select useful ones for the purpose of prognosis. Note that the principle for this is very different from those of diagnosis as the words suggest. The procedure is described via the IMS bearing datasets mentioned in Section 3. Bear in mind that the IMS bearing datasets contain the vibration signals for four bearings, in which dataset 1 contains those of the both (horizontal and vertical) directions for each bearing, whereas datasets 2 and 3 contain only one direction. Among these, the horizontal signal for the bearing #3 of dataset 1 is used for the demonstration, where the inner race is damaged during the experiment. As opposed to the diagnosis where the data are acquired for each discrete class such as the normal, inner race fault and so on, the data in the prognosis are acquired with a certain interval over time for the purpose of monitoring the degradation of the fault. In the IMS bearing, the duration for a single measurement is one second, which are stored in an individual file. They are done by every 5 min for the first 43 numbers, which is followed by every 10 min until failure. In the prognosis, construction of a suitable health index (HI) is necessary, which represents the current health state and enables its monitoring against the failure. The HI can be constructed using the diverse features extracted in the previous section. According to the literature, a good HI is characterized by three metrics: correlation, monotonicity, and robustness in terms of time (or cycles). Higher values of the three metrics give better performance in the prognosis. Therefore, the criterion for feature selection is given by the average of the three metrics in this study. The first metric for the prognosis is the correlation, which represents the linearity between the features and time:  
\[Corr(T,X)=\frac{cov(T,X)}{\sigma_{T}\sigma_{X}}\, \tag{20}\]  
where \(T\) is time, \(X\) is the feature, and \(\sigma\) is the standard deviation over the period. The value toward 1 or \(-\)1 means that it has near the perfect linear relationship. The Matlab built-in function for this is _Rho = corr(X,Y)_. Note that this is also called the Pearson correlation. The second metric is the monotonicity, which evaluates the degree of continuous increase or decrease of the feature over time. It is also called Spearman correlation, which is obtained by replacing the variables of the Pearson correlation by its rank variable that represents the standing of the variable in the increasing order:  
Figure 12: Scatter plot of two features with (**a**) max \(J_{3}\) and (**b**) min \(J_{3}\) for CWRU bearing. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Top 5** & **Feature Combination** & \(J_{3}\)** **Value** & **Bottom 5** & **Feature Combination** & \(J_{3}\)** **Value** \\ \hline
1 & SF \& P2P & 151.38 & 1 & SK \& IF & 2.5375 \\
2 & SF \& BPFI & 133.34 & 2 & SK \& CR & 2.5435 \\
3 & PEAK \& SF & 110.48 & 3 & SK \& CL & 2.6373 \\
4 & KUR \& SF & 108.00 & 4 & KUR \& CL & 2.8185 \\
5 & BPFO \& BPFI & 100.29 & 5 & KUR \& IF & 2.9564 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Rank of \(J_{3}\) value for CWRU. \[Mon(T,X)=\frac{cov(rgT,rgX)}{\sigma_{rgT}\sigma_{rgX}}, \tag{21}\]  
where, \(rg\) is the rank of the variables, and \(\sigma\) is its standard deviation. The Matlab built-in function is \(Rho=corr(X,Y,'Spearman')\). Monotonicity also has a value between \(-1\) and \(1\), of which the absolute value near \(1\) means that the feature is good for the prognosis. The third metric is the robustness, which has to do with the measurement noise arising in the data acquisition. Because the larger noise can cause poorer performance in the prognosis, selecting features robust to this noise is important.