## 3 Uncertainty-Controlled Remaining Useful Life Prediction with a Data Augmentation Strategy
### Data Augmentation Based on Degradation Modeling and Sobol Sampling  
This study proposes a data generation method for producing enough data for training based on degradation modeling and Sobol sampling. The flowchart of the data generation method can be divided into four steps. Detailed descriptions of each step are given as follows. 1. Degradation modeling is the first step. The RMS is a commonly used time-domain feature, which can reflect the degradation process. Thus, the RMS of the monitored bearing is chosen as the health indicator. The RMS of the whole life bearing signal is shown in Figure 2a, which can be divided into two different stages [36]. The bearing is in the health stage at an early time. RUL prediction is not necessary for this stage because RMS value in the health stage shows a smooth trend. At the Fault Occurrence Time (FOT), the bearing gets into the degradation stage. The degradation can be modeled as \[X_{t}=\left\{ \begin{array}{ll}\exp(m+\sigma_{1}\varepsilon(t))&t\leqslant \gamma\\ a\exp(-\frac{\sigma^{2}}{2}+b(t-\gamma)+e\varepsilon(t-\gamma))&t>\gamma\\ \end{array} \right.,\] (6) where the \(\gamma\) is the FOT, it is a random variable because different bearing has different FOT. When \(t\leqslant\gamma\), the bearing is in the healthy stage. The \(m\) is a normal random variable representing a stable level of normal bearing health. The \(\sigma_{1}\) is a noise variance in this healthy stage. \(\varepsilon(1)\), \(\varepsilon(2)......\) in \(\varepsilon(t)\) is independent at different time. RUL prediction often starts from FOT to the failure time, which means the stage \(t>\gamma\) is always used for RUL prediction. At this stage, \(a\) is a log-normal random variable representing the magnitude of the exponential degradation trend. \(b\) is a normal random variable representing the slope of the exponentially degenerate trend. \(e\) is the  
Figure 1: The flowchart of the proposed method. Three stages can be divided in the method. The generated data in the data augmentation process are sent to the training process of the Wiener–LSTM model. What is more, the hyperparameters of the Wiener–LSTM model are optimized by PSO algorithm. At the end, the testing data are sent to the trained Wiener–LSTM model to verify the effectiveness of the proposed method. noise variance in the degradation process. If there are N number of vibration signal data at each sampling point as \(D=[d_{1},d_{2},d_{3},\ldots,d_{N}]\). In [37], commonly used health indicators in residual useful life prediction are root mean square (RMS), kurtosis, peak value, and so on. The RMS is a commonly used indicator to describe the degradation trends of bearings, due to its ability to reflect the vibration energy characteristics with robustness. In [38], it is also stated that the RMS indicator is a commonly used indicator in the machinery fault diagnosis and remaining useful life prediction. The RMS is chosen as the indicator in the proposed method, and the unit of RMS is 'g'. The RMS of the vibration signal is calculated firstly as the health indicator as  
\[x_{i}^{rms}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(d_{i}-d_{m})^{2}}, \tag{7}\]  
where \(d_{i}\) is the signal in the sampling point. \(d_{m}\) is the mean of the signal recorded from 0-\(N\). The FOT of the RMS is then calculated using the laida criterion, also known as the 3\(\sigma\) rules. The RMS health stage mean \(\mu\) and standard deviation \(\sigma\) are calculated. The first value, which exceeds 3 \(\sigma\) is assumed as the FOT. Then the data from FOT to failure are collected and prepared for the next step. 2. Since the RUL prediction is not necessary for the stage \(t\leqslant\gamma\). The model fitting is implemented only in the stage \(t>\gamma\) to acquire the parameters \(a\), \(b\), and \(e\) in Equation (6). The \(\varepsilon(t)\) follows a normal distribution, which can not be fitted in this process. All the training samples are fitted by the function of the degradation model. Different parameters are acquired after fitting different samples.