**5. Conclusions**

Accurate prognosis of the degradation trend of rotating machinery plays an important role in industrial applications. Currently, most developments in the mechanical fault prognostics area have been targeted towards directly utilizing degradation-based data to trace the degradation trajectories, very few studies have used the idea of sparse decomposition. In this work, a novel intelligent prediction approach based on asymmetric penalty sparse decomposition (APSD) combined with WNN and ARMA-RLS models for health indicator degradation trajectories of four rolling bearings is proposed. The original health indicators od degradation trajectory are rearranged as two components, i.e., LFC and HFC. In particular, the HFC corresponds to the stable change around the zero line of health indicators, whereas the LFC is essentially related to the evolutionary trend of health indicators which rarely occurs in practice. The LFC and HFC are, respectively, predicted using the WNN and ARMA-RLS models. The final degradation regions (e.g., last 100 points) is correspondingly obtained by combining the predicted LFC and predicted HFC. Experimental results of four rolling bearings have demonstrated the superiority of the proposed method in terms of quantitative and qualitative evaluation compared to three commonly-used parametric-based and nonparametric-based methods, i.e., ARMA, FARIMA, WNN, L-Lyap methods.

This observation motivates the study of an integrated method for bearing prediction, which combines the strength of both parametric-based and nonparametric-based techniques. This paper focuses on the development of an intelligent degradation prognosis model that the error reverse transmission is employed to optimize the initial parameters of WNN, and the initial modelling parameters of ARMA are also optimized by the RLS algorithm step by step. The proposed method is more robust to different operating conditions and outperforms other one-step prediction methods by taking the LFC and HFC properties of health indicators into account. Therefore, the proposed method has huge potential in the field of PHM of mechanical equipment.

For future research, it would be interesting to investigate more advanced a sparse low-rank matrix decomposition (SLMD) algorithm to robustly separate the health indicator time series (HITS). Both LFC and HFC could be predicted using more sophisticated adaptive algorithms under more complex and harsh environments.

**Author Contributions:** Algorithms improvement, programming, experimental analysis and paper writing were done by Q.L. Review and suggestions were provided by S.Y.L. All authors have read and approved the final manuscript.

**Acknowledgments:** This research is supported the Fundamental Research Funds for the Central Universities (Grant Nos. CUSF-DH-D-2017059 and BCZD2018013) and the Research Funds of Worldtech Transmission Technology (Grant No. 12966EM).

**Conflicts of Interest:** The authors declare no conflict of interest.
