# An Ensemble Learning and RUL Prediction Method Based on Bearings Degradation Indicator Construction
## 3 Methodology
### LSTM-AdaBoost Ensemble Learning and Prediction Model  
After constructing the LSTM neural network model mentioned in Section 2.1, the prediction ability of the model did not meet the requirements for robust prediction. AdaBoost is an iterative algorithm that was originally mainly used in classification problems, and it is sensitive to abnormal features. We considered using the AdaBoost algorithm to enhance the LSTM network model and achieve robust prediction. Suppose we want to make the m-step ahead prediction for a time-series. The iterative prediction strategy is implemented in this paper, which can be expressed as: \(\hat{x}_{t+m}=f(x_{t},x_{t-1},\cdots,x_{t-(p-1)})\), where \(\hat{x}\) is the predicted value, \(x_{t}\) is the actual value in period \(t\), and \(p\) denotes the lag orders. In this study, the AdaBoost algorithm was introduced to integrate a set of LSTM predictors. The proposed LSTM-AdaBoost ensemble learning approach consists of seven steps as shown in Algorithm 3. ```
0: Training data set: \(S=\left\{\left(x_{t_{1}},\hat{x}_{t_{1}}\right),\left(x_{t_{2}},\hat{x}_{t_{2} }\right),\cdots,\left(x_{t_{N}},\hat{x}_{t_{N}}\right)\right\}\), LSTM weak predictor. 0: Strong predictor \(P(x)\)
1: Initialize the weight vector. The weight distribution of the training data is initialized to: \(W=\left(\frac{1}{N},\frac{1}{N},\cdots,\frac{1}{N}\right)\), \(k=1,2,\cdots,K\),
2: Suppose the weight distribution is \(W_{k}\), the prediction error of the predictor \(P_{k}\) on the training data set is calculated by \(\varepsilon_{k}^{i}=\left|P_{k}\left(x_{i}\right)-y_{i}\right|/E_{k}\), where \(E_{k}=\underset{i}{\text{sup}}\ \left(\left|P_{k}\left(x_{i}\right)-y_{i}\right|\right)\), the output interval of the function is \(\left[0,1\right]\),
3: Calculate the total error of training sample sets: \(\varepsilon_{k}=\sum\limits_{i=1}^{n}W_{k}^{i}\varepsilon_{k}^{i}\),
4: Calculate the weights of the current predictor \(a_{k}=\frac{1}{2}\ln\left(\frac{1}{\beta_{i}}\right)\), \(\beta_{i}=\frac{\varepsilon_{k}}{1-\varepsilon_{k}}\),
5: Update the distribution of weights of training datasets as follows: \(W_{k+1}^{i}=\frac{W_{k}^{i}\cdot\beta_{k}^{-\varepsilon_{i}}}{Z(k)}\), where \(Z(k)=\sum\limits_{i=1}^{N}W_{k}^{i}\),
6: Repeat steps 1-5 until all the LSTM predictors are obtained. Record the connection weight of the LSTM predictors \(W=\left(w_{1},w_{2},\cdots w_{K}\right)\), where \(w_{i}=\frac{a_{i}}{\sum_{i=1}^{K}a_{i}}\),
7: Build the final predictor and integrate the above trained predictors according to the connection weights to obtain the final strong predictor. \[P\left(x\right)=w_{1}P_{1}\left(x\right)+w_{2}P_{2}\left(x\right)+\cdots w_{K }P_{K}\left(x\right).\]

Through the LSTM-AdaBoost ensemble learning approach, multiple weak predictors are integrated into a strong predictor and the features of the degradation indicator set are predicted by the strong predictor. Finally, the prediction results are ensembled to obtain the remaining useful life of the bearings in the next moment. The main steps are as follows:

First, the feature extraction method proposed above is used to construct the indicator set of bearings degradation \(F_{opt}\). Next, for each feature of the bearing degradation indicator set \(F_{opt}\), the LSTM-AdaBoost ensemble learning approach is adopted to obtain the predicted remaining useful life value \(\hat{f}_{i,\left(t+1\right)}\) corresponding to each feature at moment \(t+1\). Finally, the above prediction results are ensembled to obtain the predicted value \(R\hat{U}L_{t+1}\) of useful bearing life at \(t+1\). That is \(R\hat{U}L_{t,t+1}=\sum_{i=1}^{n}\hat{f}_{i,t+1}\). ## 4 Experiment and Analysis

The run-to-failure data acquired from accelerated degradation tests of rolling element bearings were used to demonstrate the effectiveness of the proposed prediction approach. The proposed approach was compared with other two features selected methods. ### Data Description

The bearings testbed is shown in Figure 4. These faults occurred accidentally in accelerated degradation experiments.