## 3 Results
### Experimental Study of a Benchmark Bearing
#### 3.1.3 Discussion and Comparison  
To show the superiority of the developed bearing degradation model in this paper, it is compared with the fixed degradation model established by the traditional empirical formula. The degradation characteristic index RRMS is used as the input of the model to calculate the state prediction values of five bearings, respectively. In order to quantitatively evaluate the performance of the proposed method, the prediction error is designed and calculated, which can be evaluated by the root-mean-square error (RMSE) using the following equation:  
\[RMSE=\sqrt{\frac{1}{M}\sum_{i=1}^{M}\left(x_{p}(i)-x_{r}(i)\right)^{2}} \tag{23}\]  
where \(x_{p}\) is the predicted data point, \(x_{r}\) is the real data point, and \(M\) is the number of predicted data points. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Outer race diameter & 39.80 mm & Inner race diameter & 29.30 mm \\ Bearing mean diameter & 34.55 mm & Ball diameter & 7.92 \\ Number of balls & 8 & Contact angle & 0 rad \\ Load rating (static) & 6.65 kN & Load rating (dynamic) & 12.82 kN \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the tested bearings. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Operating** & **Bearing Dataset** & \begin{tabular}{c} **Number** \\ **of Files** \\ \end{tabular} & \
\begin{tabular}{c} **Bearing** \\ **Lifespan** \\ \end{tabular} &
\begin{tabular}{c} **Fault** \\ **Element** \\ \end{tabular} \\ \hline \multirow{3}{*}{Condition1} & Bearing1\_1 & 123 & 2 h 3 min & Outer race \\  & Bearing1\_2 & 161 & 2 h 41 min & Outer race \\  & Bearing1\_3 & 158 & 2 h 38 min & Outer race \\  & Bearing1\_4 & 122 & 2 h 2 min & Cage \\  & Bearing1\_5 & 52 & 52 min & Inner and outer race \\ \hline \hline \end{tabular}
\end{table} 
Table 2: XJTU-SY bearing dataset. Figure 4: (**a**) HI of bearing1_1; (**b**) HI of bearing1_2; (**c**) HI of bearing1_3; (**d**) HI of bearing1_4; (**e**) HI of bearing1_5. Figure 5: (**a**) RRMS prediction result of bearing1_1; (**b**) RRMS prediction result of bearing1_2; (**c**) RRMS prediction result of bearing1_3; (**d**) RRMS prediction result of bearing1_4; (**e**) RRMS prediction result of bearing1_5. The standard deviation of the state estimation error (RMSE) of the five bearingss is calculated as shown in Figure 6. Taking the result of bearing1_1 as an example, the RMSE of the degradation model based on time-varying parameters is 0.7123, while that of the fixed degradation model based on empirical formula is 0.8323. This shows that the performance of the degradation model based on time-varying parameters is better. The state prediction results of the two models are mapped to the RUL of the bearing as shown in Figure 7. The RUL prediction result of bearing1_1 is also taken as an example to illustrate. The RUL prediction error of the degradation model based on time-varying parameters is 10 min, while that of the fixed degradation model based on empirical formula is 16 min. Therefore, the prediction results of the proposed model are more convergent than those of the traditional empirical model. The accuracy is higher and is closer to the actual RUL. To illustrate the advantages of the proposed time-varying model compared with the data-driven method and the fixed-model based method, the same dataset was compared with four advanced methods published in the literature, including the mixed prediction  
Figure 6: Comparison of RMSE predicted by two models. Figure 7: (**a**) Comparison of RUL prediction of two models for bearing1_1; (**b**) Comparison of RUL prediction of two models for bearing1_2; (**c**) Comparison of RUL prediction of two models for bearing1_3; (**d**) Comparison of RUL prediction of two models for bearing1_4; (**e**) Comparison of RUL prediction of two models for bearing1_5. method based on RVM and the random degradation model [35], the method based on RVM [36], the method based on PF [37], and the method based on EKF [38].