## 5 Discussion  
As mentioned in Section 1, the pure physics-based models for RUL prediction are beneficial when a clear understanding of the failure mechanism is available, while the deterioration dynamics of the bearing is not yet fully unveiled and often in practice, not all the prior physics information (e.g., defect dimension) is available. Therefore, the prediction accuracy of the pure physics-based methods may be limited in practical applications. In general, the pure data-driven approaches and statistical approaches are not strictly separated. Although pure statistics methods (e.g., survival analysis) may require a large amount of historical data that provide consistent statistical meaningful results, such as  
Figure 15: Bearing 6 degradation state prediction based on the EKF: (**a**) State prediction at the 1428 th minute. The corresponding estimated RUL is \(\widetilde{\mathit{rul}}_{1428}=\widetilde{t}_{\mathit{EOL}}-t_{PT}=44\) minutes. (**b**) State prediction at the 1429 th minute, with \(\widetilde{\mathit{rul}}_{1429}=47\) minutes. (**c**) State prediction at the 1430 th minute, with \(\widetilde{\mathit{rul}}_{1430}=47\) minutes. (**d**) State prediction at the 1441 st minute, with \(\widetilde{\mathit{rul}}_{1441}=37\) minutes. the data requirement in pure data-driven approaches (e.g., neural networks) to ensure generalization performance, both have this advantage: no detailed knowledge of the failure mechanism is needed. In this study, the recursive Bayesian filter is applied combining the advantages of both physics-based and data-driven approaches. The envelope spectral indicator is crafted relating to bearing fault characteristics and is incorporated to the EKF for sequential estimation of the bearing degradation state with uncertainty estimation. The experimental results on the test bearings under various operating conditions have demonstrated promising performance of the ESI in representing the bearing degradation process and the EKF in RUL prediction. In the above cases, the EKF-based RUL predictions at the discrete time steps shortly after the fault detection exhibit commendable accuracy and precision. The prediction results suggest that the EKF could effectively integrate the bearing ESI data and update the state estimation, which underscores the capability of the proposed model to capture short-term trends and variations in the ESI during bearing deterioration, providing valuable insights into the impending failure and the RUL of bearings. On the other hand, it is also important to acknowledge that the prediction based on more ESI observations may exhibit a more intricate pattern. In the cases of continuous RUL prediction, it is justifiable to claim that there is a convergence of prediction accuracy towards the end of the bearing life because of the health indicators approaching the pre-specified end-of-life threshold. Yet, the significance of the late-stage RUL predictions might be diminished due to their proximity to the bearing failure, thereby limiting their practical value for maintenance decision making. In contrast, the significance of the presented method is highlighted for the ability to provide meaningful RUL predictions during the early phase of the bearing degradation. Regarding the computation cost of the proposed methods, the proposed averaged envelope spectrum can be efficiently computed based on the Hilbert transform and the fast Fourier transform. As for the EKF, the state process model for the bearing degradation has been further optimized and only one-dimensional measurement is leveraged. Also, the study targets the early phase RUL prediction by using only a few observations of the health indicators at the fault detection phase, which results in low computation requirements. In summary, the effectiveness of the EKF-based RUL prediction methodology has been demonstrated on the experimental bearing vibration data of various operating conditions. The bearing RUL can be predicted shortly after the fault detection based on the ESI at the early phase of bearing degradation.