# An Ensemble Learning and RUL Prediction Method Based on Bearings Degradation Indicator Construction
## 3 Methodology
### LSTM-AdaBoost Ensemble Learning and Prediction Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Operating** & **Rotating Speed** & **Radial Force** & **Bearings Dataset** \\
**Condition** & **(rpm)** & **(KN)** & **Bearings Dataset** \\ \hline Condition A & 2100 & 12 & Bearing1\_1 Bearing1\_2 Bearing1\_3 Bearing1\_4 Bearing1\_5 \\ Condition B & 2250 & 11 & Bearing2\_1 Bearing2\_2 Bearing2\_3 Bearing2\_4 Bearing2\_5 \\ Condition C & 2400 & 10 & Bearing3\_1 Bearing3\_2 Bearing3\_3 Bearing3\_4 Bearing3\_5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operating Conditions of the Tested Bearings. Figure 6: Typical vertical vibration signals. #### 4.2.2 Construction of Bearing Degradation Indicator Set

The features mentioned above represent bearings degradation from the different perspective. However, if all these features have been taken as input parameters to the model, then it may result into model over-fitting. Thus, before using these features as input parameters to the model, it is desirable to select the most sensitive features from the feature set and remove the less indicative features to improve the model accuracy [44]. In this paper, the three-stage feature selection method is used to select the sensitive features that can characterize bearings degradation state, which is used to construct the bearings degradation indicator set. Taking operating condition A as an example, the construction process of degradation indicator set is described in detail as follows. Figure 7 shows the sensitive features extracted by the first-stage FCBF feature extraction method. The reference value symmetric uncertainty \(SU\) is sorted in descending order, the threshold \(\lambda\) given in this paper is 0.1, i.e., the feature with the SU value greater than 0.1 is selected to be placed in the feature subset \(F^{\prime}\), then, an AMB de-redundancy method is used to de-redundant the features in feature subset \(F^{\prime}\), construct the feature subset in Figure 8 as the initial bearings degradation indicator subset \(F^{*}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature** & **Time-Domain Feature Parameters** & **Feature** & **Frequency-Domain Feature Parameters** \\ \hline \(F0\) & \(f_{0}=\frac{\sum_{i=1}^{N}x(n)}{N-1}\) & \(F12\) & \(f_{12}=\frac{\sum\limits_{k=1}^{K}s(k)}{K-1}\) \\ \(F1\) & \(f_{1}=\sqrt{\frac{\sum_{i=1}^{N}



In the second stage, the \(MIC\) method mentioned above is used to measure the correlation between features and failure modes, features and features, and construct a strong subset \(F_{F-R}\) with strong correlations between features and failure modes, and a strongly uncorrelated subset \(F_{F-F}\) consisting of less redundant features. The two subsets of bearing degradation indicators are shown in Figure 9a,b. In the third stage, the above three bearing degradation indicator subsets, \(F^{*}\), \(F_{F-R}\), and \(F_{F-F_{r}}\) are cross-operated based on the fusion method to obtain a strong correlation and low redundancy optimal degradation indicator set \(F_{opt}\). The final bearings degradation indicator set consists of eight features shown in Figure 10, which will be applied to the bearing remaining useful life prediction as the degradation indicators of the bearing. Figure 8: Initial bearing degradation indicator subset \(F^{*}\). Figure 10: Optimal degradation indicator set \(F_{opt}\) of operating condition A.

Figure 7: The sensitive features were extracted and sorted in descending order in the feature subset \(F^{\prime}\). According to the proposed feature selection method, features selected by the proposed method are shown in Figure 11.

#### 4.2.3 Train Prediction Model

After obtaining the optimal degradation indicator set revealing the state of bearing degradation, the prediction model is trained. The model is trained by LSTM network, and the AdaBoost algorithm is used to optimize the LSTM prediction model to form a strong predictor. The input of the model is the degradation indicators of the optimal set, the output is the RUL of the bearings.