## 2 Signal Division Method of Bearing Health State
### Evaluation of Bearing HI  
To select a reasonable health indicator, the trendability, monotonicity, and robustness are introduced to quantitatively evaluate the health indicator. (1) Trendability  
Over time, the evolution process of bearing degradation becomes more and more serious, so the HI curve representing the development process of bearing degradation \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dimensional Feature** & **Equation** &
\begin{tabular}{c} **Dimensionless Feature** \\ \end{tabular} & **Equation** \\ \hline mean & \(\rm\bar{\kappa}=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}(t)\) & pulse factor & \(\rm C_{if}=\frac{F}{|u|}\) \\ Peak values & \(\rm\bar{F}=max|x_{i}(t)|\) & clearance factor & \(\rm C_{mf}=\frac{F}{\left(\frac{i}{k}|x_{i}(t)|^{2}\right)^{2}}\) \\ root mean square & \(\rm RMS=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}(t)}\) & Peak factor & \(\rm C_{f}=\frac{F}{RMS}\) \\ Population variance & \(\rm s^{2}=\frac{1}{N}\sum\limits_{i=1}^{N}(x_{i}-\rm\bar{\kappa})^{2}\) & kurtosis factor & \(\rm C_{kf}=\frac{k}{K\sum_{i=1}^{N}x_{i}^{2}(t)}{RMS}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Time-domain characteristics of bearing vibration signal. Figure 2: Signal analysis of bearing HI. should show a certain time correlation. The characteristics of this curve are defined as a trend, and the equation is as follows:  
\[\text{Tre}(\text{H},\text{T})=\frac{\left|\sum_{\text{k}=1}^{\text{K}}\left(\text{ h}_{\text{k}}-\text{ \leavevmode\
obreak\ }\overline{\text{H}}\right)\left(\text{t}_{\text{k}}-\overline{\text{T}} \right)\right|}{\sqrt{\sum_{\text{k}=1}^{\text{K}}\left(\text{h}_{\text{k}}- \overline{\text{H}}\right)^{2}\sum_{\text{k}=1}^{\text{K}}\left(\text{t}_{ \text{k}}-\overline{\text{T}}\right)^{2}}} \tag{1}\]  
In Equation (1), \(\text{h}_{\text{k}}\) is the sampling period HI value at \(\text{t}_{\text{k}}\) time; \(\overline{\text{H}}=(1/\text{K})\sum_{\text{k}=1}^{\text{K}}\text{h}_{\text{k}}\) is the average value of health indicators in the whole life cycle; \(\overline{\text{T}}=(1/\text{K})\sum_{\text{k}=1}^{\text{K}}\text{t}_{\text{k}}\) is the mean value of each sampling period. \(0\leq\text{Tre}(\text{H},\text{T})\leq 1\); the closer the value to 1, the better the trend. (2) Monotonicity  
The monotonicity of the HI curve reflects the degree of degeneration of the HI curve. Although the bearing operation is in a stable stage, the bearing has also degraded over time, but the degree of degradation is weak. The monotonicity is calculated as follows:  
\[\text{Mon}=\left|\frac{\text{num of dF}>0}{\text{T}-1}-\frac{\text{num of dF}<0}{\text{T}-1}\right| \tag{2}\]  
T is the number of sampling points in the bearing cycle; dF is the differential between the front and rear values in the HI curve. (3) Robustness [41]  
For the characteristic signal sequence \(\text{F}=[\text{f}(1),\text{f}(2),\ldots\text{f}(\text{K})]\), the time sequence \(\text{T}=[\text{t}(1),\text{t}(2),\ldots\text{t}(\text{K})]\), \(\text{f}(\text{t}_{\text{k}})\) represents the characteristic value obtained at the time \(\text{t}_{\text{k}}\), where \(\text{k}=1,2,\ldots\text{K}\), K represents the length of time. First, the Exponential Weighted Moving Average (EWMA) is used to divide the feature sequence into two parts, namely, the stationary trend term \(\text{f}_{\text{T}}(\text{t}_{\text{k}})\) and the random complementary vector \(\text{f}_{\text{R}}(\text{t}_{\text{k}})\). \[\text{f}(\text{t}_{\text{k}})=\text{f}_{\text{T}}(\text{t}_{\text{k}})+\text{ f}_{\text{R}}(\text{t}_{\text{k}}) \tag{3}\]  
EWMA is calculated as follows:  
\[\text{f}_{\text{T}}(\text{t}_{\text{k}})=\beta\text{f}_{\text{T}}(\text{t}_{ \text{k}-1})+(1-\beta)\text{f}(\text{t}_{\text{k}}) \tag{4}\]  
Equation (4) generally takes \(\beta\geq 0.9\) and \(\text{f}_{\text{T}}(\text{t}_{1})\) can be calculated by averaging the previous values. Robustness is the tolerance to outliers and measures the effect of possible random fluctuations in the bearing degradation process due to random changes in sensor noise.