## 5 Conclusions  
This paper proposes a novel prognosis approach for the degradation trend of rotating machinery based on a long-range dependence and particle filter algorithm. The main objective is to address the problem of prognostics in FOC models while health indicators time series possess a weak LRD property and the STPs are contained in the predicted range. The two main contributions of this work can be summarized as:  
1. For engineering applications, a new condition prognostic method is introduced as an effective tool for the long lifetime prediction and assessment of mechanical equipment in the PHM filed, the mechanical running degradation pattern can be predicted well in real-time and the most vital components of the mechanical equipment can be repaired and replaced prior to actual catastrophic failure.
2. For theoretical analysis, the particle filter frame (PFF) is embedded in the FOC model, an update scheme for online model parameters has been introduced to adapt the initial state model based on the PFF algorithm. Due to the adaptability of the parameters, the performance of our proposed model is remarkably efficient than classical time-series ARMA model and the FOC models, even the initial health status is unknown and the failure degradation behavior is time-varying.  
Figure 8: The kurtosis prediction results by the proposed method and the other state-of-the-art methods. (**a**) Prediction result using the ARMA model; (**b**) Prediction result using the _f_-ARIMA model; (**c**) Prediction result using the FBM model; (**d**) Prediction result using the proposed model; (**e**) Prognostic errors.  
That is to say, the main advantage of the proposed approach is that the prognostic model has an ability to generate the reliable probabilistic results despite the uncertainties of the initial model parameters.  
As we all known, in the case of offline, the vibration data has been collected for further analysis (e.g., trend prediction or remaining useful life, i.e., RUL), and many adaptive prediction methods (such as Wavelet neural network (WNN) and ARMA-recursive least squares, i.e., ARMA-RLS, etc.) with parameters adjustment strategy can be performed only under the known data. Thus, the forecast errors are adjusted based on the predicted data and original data. If the original data is known before the prediction the adaptive prediction methods can work well, but in most cases, the original data is unknown, especially when the equipment is working under normal conditions, so the parameters adjustment strategy does not make sense. Based on this principle, if the original data is unknown, the proposed model is actually able to correctly identify when the bearing lifetime will enter the incipient fault region based on the data from the normal operational region. If not, the proposed model may lose its function.  
Future research will investigate the suitability of the proposed model under harsh working conditions or different operational conditions, and the prediction for the remaining useful life (RUL) of the bearing will be investigated. As well, the proposed model might be expanded to other complex industrial applications such as diesel engine failure degradation and aero-engine fault monitor, etc.  
Algorithm improvement, programming and analysis, paper writing was done by Q.L. Review and suggestions were provided by S.Y.L. All authors have read and approved the final manuscript.  
This research is funded by the Fundamental Research Funds for the Central Universities (Grant No. CUSF-DH-D-2017059 and BCZD2018013) and the Research Funds of Worldtech Transmission Technology (Grant No. 12966 EM).  
The authors declare no conflict of interest.