## 2 Description of the Datasets and Data Preprocessing
### HI Construction  
HI is very important in RUL prediction of bearings, and RMS can reflect the degradation status of bearings and is commonly used as an indicator for dividing bearing status as HI. However, the generator bearing speed in the run-to-failure data in the actual wind farm is not fixed, and there must be random fluctuations in the RMS calculated under composite working conditions, which is not a typical phenomenon of bearing degradation and will seriously affect the prediction effect. To reduce the random fluctuations in RMS, this paper uses the Savitzky-Golay filter (SG) to smooth RMS and obtains SRMS as the HI and the other 11 features as model inputs [31; 32].  
\begin{table}
\begin{tabular}{c c c c} \hline
**Features** & **Equations** & **Features** & **Equations** \\ \hline Peak & \(TF_{1}=\max\lvert x_{t}\rvert\) & Kurtosis & \(TF_{7}=\frac{1}{T:TF_{8}^{3}}\sum\limits_{i=1}^{T}\left(x_{t}-TF_{2}\right)^{4}\) \\ Mean absolute value & \(TF_{2}=\frac{1}{T}\sum\limits_{t=1}^{T}\left\lvert x_{t}\right\rvert\) & Square root amplitude & \(TF_{8}=\left(\frac{1}{T}\sum\limits_{t=1}^{T}\sqrt{\lvert x_{t}\rvert}\right)^ {2}\) \\ Peak to peak & \(TF_{3}=\max(x_{t})-\min(x_{t})\) & Crest factor & \(TF_{9}=\frac{TF_{6}}{TF_{4}}\) \\ Root mean square(RMS) & \(TF_{4}=\sqrt{\frac{1}{T}\sum\limits_{t=1}^{T}x_{t}^{2}}\) & Clearance factor & \(TF_{10}=\frac{TF_{1}}{TF_{8}}\) \\ Standard deviation & \(TF_{5}=\sqrt{\frac{1}{T-1}\sum\limits_{t=1}^{T}\left(x_{t}-TF_{2}\right)^{2}}\) & Impulse factor & \(TF_{11}=\frac{TF_{6}}{TF_{2}}\) \\ Skewness & \(TF_{6}=\frac{1}{T:TF_{8}^{3}}\sum\limits_{t=1}^{T}\left(x_{t}-TF_{2}\right)^{3}\) & Shape factor & \(TF_{12}=\frac{TF_{6}}{TF_{2}}\) \\ \hline \end{tabular}
\end{table}
Table 1: Time domain degradation feature.  
SG is a smoothing technique that relies on interval polynomial fitting, with the ability to filter out random fluctuations in the data without changing the shape and trend of the original data. The key to SG smoothing is to use a \((k-1)\)-order polynomial to fit points centered on the selected data point within a window length of \(2m+1\), and the polynomial formula is:  
\[y=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{k-1}x^{k-1} \tag{1}\]  
Then, for \(2m+1\) points in the window, there are \(2m+1\) equations:  
\[\left(\begin{array}{c}y_{-m}\\ y_{1-m}\\ \vdots\\ y_{m}\end{array}\right)=\left(\begin{array}{cccc}1&x_{-m}&\cdots&(x_{-m})^{ k-1}\\ 1&x_{1-m}&\cdots&(x_{1-m})^{k-1}\\ \vdots&\vdots&\ddots&\vdots\\ 1&x_{m}&\cdots&(x_{m})^{k-1}\end{array}\right)+\left(\begin{array}{c}e_{-m} \\ e_{1-m}\\ \vdots\\ e_{m}\end{array}\right) \tag{2}\]  
Using matrix form, it can be represented as:  
\[\mathbf{Y}=\mathbf{A}\mathbf{X}+\mathbf{E} \tag{3}\]  
where \(\mathbf{Y}\) is the fitting vector, \(\mathbf{A}\) is the coefficient matrix, \(\mathbf{X}\) is the independent variable matrix, and \(\mathbf{E}\) is the residual. The above equation can usually be solved by using the least square method.  
\[\overset{-}{\mathbf{A}}=\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\mathbf{Y} \tag{4}\]  
Figure 4: TFs and CTFs of 10# WT generator bearings.  
\[\begin{split}\begin{array}{c}-\\ Y=X\mathbf{A}=X\Big{(}\mathbf{X}^{T}\mathbf{X}\Big{)}^{-1}\mathbf{X}^{T}\mathbf{Y}\end{array}\end{split} \tag{5}\]  
The window slides in chronological order until all data points are fitted, resulting in a smoothed-curve SRMS.