## 4 Results and Observations
### RUL Prediction of the Test Bearings under Condition C2  
To validate the prediction performance of the proposed method, the bearings under Condition 2 are analyzed, in which the shaft speed is 37.5 Hz with the loading at 11 kN. The EKF model is further applied to the ESI of Bearings 3 and 4 for RUL prediction at the early stage of bearing degradation. According to the envelope analysis results in Table 4, the presence of the inner-race fault is identified in Bearing 3 after 446 minutes. Then, the ESI data from the 436 th minute to the RUL prediction time \(t_{PT}\) are employed for the degradation state estimation using the proposed EKF models. The state process noise and the measurement noise are, respectively, set as \(v_{t}=0.02\), \(v_{t}=0.05\). After recursive filtering using the ESI observations (marked with a red circle in Figure 12), the estimated EOL time \(\overset{\sim}{t}_{EOL}\) at the prediction time can be extrapolated based on the updated state at \(t_{PT}\) according to (38). Four predictions within 20 minutes after the fault is detected (\(t_{FDT}\)) are shown in Figure 12.  
Figure 12: Bearing 3 degradation state prediction based on the EKF: (**a**) State prediction at the 458 th minute. The corresponding estimated RUL is \(\overset{\sim}{\mathit{rul}}_{458}=\overset{\sim}{t}_{EOL}-t_{PT}=28\) minutes. (**b**) State prediction at the 459 th minute, with \(\overset{\sim}{\mathit{rul}}_{459}=30\) minutes. (**c**) State prediction at the 460 th minute, with \(\overset{\sim}{\mathit{rul}}_{460}=32\) minutes. (**d**) State prediction at the 462 nd minute, with \(\overset{\sim}{\mathit{rul}}_{462}=23\) minutes. In the prediction result, an evident surge in the ESI of Bearing 3 is observed following the initiation of a fault. This surge results in the prediction of an early failure time at \(\overset{\sim}{t}_{EOL}=486\), which is extrapolated from the state estimation at \(t_{PT}=458\), as in Figure 12a. Meanwhile, the further predictions of the EOL time at \(t_{PT}=459\) and \(t_{PT}=460\) (Figure 12b,c) are well projected to the vicinity of the baseline EOL time at \(t_{EOL}=490\). Overall, the 95% credible intervals of the four predictions provided in the green dashed lines consistently cover the baseline end-of-life time. Analogously, Figure 13 displays the state estimation results by applying the EKF algorithm for the ESI of Bearing 4. The noise levels for the state process and the observations are, respectively, set at \(v_{t}=0.05\), \(w_{t}=0.08\). Notably, the state extrapolation at different prediction times effectively represents the overall pattern of the ESI (in the grey solid line). The predicted end-of-life times at \(t_{PT}=67\) minute and \(t_{PT}=68\) minute are estimated, respectively, as \(\overset{\sim}{t}_{EOL}=159\) minute and \(\overset{\sim}{t}_{EOL}=158\) minute, which closely correspond to the baseline end-of-life time at \(t_{EOL}=160\) minute. Figure 13: Bearing 4 degradation state prediction based on the EKF: (**a**) State prediction at the 67 th minute. The corresponding estimated RUL is \(\overset{\sim}{t}_{EOL}=\overset{\sim}{t}_{EOL}-t_{PT}=92\) minutes. (**b**) State prediction at the 68 th minute, with \(\overset{\sim}{t}_{68}=90\) minutes. (**c**) State prediction at the 69 th minute, with \(\overset{\sim}{t}_{69}=82\) minutes. (**d**) State prediction at the 70 th minute, with \(\overset{\sim}{t}_{70}=80\) minutes. For the assessment of the prediction accuracy, the state estimation results and the corresponding RUL are provided in Table 6, where the mean estimated RUL, the residuals between the estimated RUL and the baseline RUL, and the 95% credible intervals are summarized. It is seen that the overall prediction of RUL is fairly accurate and tends to be conservative given the negative residuals of the RUL estimate. Also, the absolute errors of the predicted RUL are no more than 5 minutes for Bearing 3, where the predicting times of the RUL are within 20 minutes after the fault detection.