## 4 Results and Observations
### RUL Prediction of Test Bearings under Condition C1  
Once the early fault is detected, the EKF is applied to estimate the latent degradation level of bearings for RUL prediction. As defined in Section 3.4.3, the degradation state of the test bearing is assumed to follow an accelerating increasing trend before reaching the average end-of-life threshold \(\overline{\gamma}\). In Figure 10a, the degradation-stage ESI of Bearing 1 is presented as a grey solid line. The first detection time of fault \(t_{PDT}\) is the 35 th minute as marked by the black dotted line. Observations reveal that the ESI after \(t_{PDT}\) exhibits a gradual upward trend with more fluctuation at the later time. The maximum value of ESI corresponds to the end-of-life time of Bearing 1 (\(t_{EOL}=126\)) as represented by the red dash-dot line, indicative of the need for replacement. Considering the uncertainty in the initial states, 10 additional data points preceding \(t_{PDT}\) are incorporated to update the state estimations in the EKF. Henceforth, the ESIs starting from the 25 th minute, specifically marked with red circles, are utilized for the EKF-based RUL prediction model. After initializing the model according to Section 3.4.4 and setting the noise levels empirically as \(v_{t}=0.1\), \(w_{t}=0.05\), the EKF algorithm is implemented for degradation state estimation until the prediction time \(t_{PT}\), at which the estimated failure time \(\overset{\sim}{t}_{EOL}\) is predicted based on the state extrapolation and the average EOL threshold \(\overline{\gamma}\). Figure 10a-d show the extrapolation results of the bearing degradation state at four prediction times: the 40 th, 45 th, 50 th and 55 th minutes, which are, respectively 5, 10, 15 and 20 minutes after the fault detection. It can be observed that the estimated EOL time, \(\overset{\sim}{t}_{EOL}\), in each prediction is closely in the vicinity of the baseline, \(t_{EOL}\). Figure 9: The envelope spectral indicator (ESI) of bearing degradation state: (**a–f**): Bearings 1 to 6. (Blue dotted scatter plot: the extracted ESI. Red solid line: the smoothed ESI using 4-point moving average). As for Bearing 2, Figure 11 demonstrates that the ESI of the bearing follows an accelerating increasing pattern, reaching the maximum at the 151 st minute. An outer-race fault is detected at the 59 th minute based on the spectral inspection results of envelope analysis (see Section 4.1). Similarly, the ESI data from the 49 th minute (marked with red circles) are employed for state estimation in the EKF algorithm. Using the noise levels at \(v_{t}=0.1\), \(w_{t}=0.05\), the estimated EOL time at various predicting times can be obtained. Figure 11a-d present the results of four \(t_{PT}\) at the 66 th, 69 th, 71 st and 74 th minutes. It is observed that despite the early prediction at the 66 th minute in Figure 11a, the prediction of the EOL time \(\overset{\sim}{t}_{EOL}\) in Figure 11b is exactly the same as the baseline EOL time, and that the extrapolations of \(\overset{\sim}{t}_{EOL}\) at the later times are also accurate, as in Figure 11c,d. To evaluate the performance of the proposed method for bearing RUL prediction, the state estimation results with uncertainty estimation for Bearings 1 and 2 are summarized in Table 5. The result shows that the established degradation model well expresses the initial upward trend in the ESI during the early stage of bearing degradation using one dynamic model parameter, \(b_{t}\). It also demonstrates that the EKF algorithm is capable of estimating the RUL of bearings with a few ESI data points since the first detection of faults. Figure 10: Bearing 1 degradation state prediction based on the EKF: (**a**) State prediction at the 40 th minute. The corresponding estimated RUL is \(87\overset{\sim}{\mathit{rul}}_{40}=\overset{\sim}{t}_{EOL}-t_{PT}=87\) minutes. (**b**) State prediction at the 45 th minute, with \(\overset{\sim}{\mathit{rul}}_{45}=81\) minutes. (**c**) State prediction at the 50 th minute, with \(\overset{\sim}{\mathit{rul}}_{50}=84\) minutes.