## 1 Introduction  
The prognostics and health management (PHM) in mechanical equipment and systems has become extremely important in many industrial applications including aero-engines, metallurgy machinery, wind turbine, etc. ; owing to the fact that the PHM is able to minimize downtime and maintenance costs, maximize machinery utilization, and ensure system availability [1; 2]. Prognostics is a key issue in PHM, which aims at utilizing real monitoring data to facilitate relevant degradation indicators and trends (DITs) that depict the current system health status and evaluate of remaining useful life (RUL) of a device. Therefore, reasonable prediction models with high accuracy are very crucial in the fields of industrial applications, because they should overcome different challenges and obstacles from the sampled data with external noise, variation trend, model perturbations and other underlying uncertainties, etc. [3; 4]. Over the past few decades, various prognostic efforts have been proposed for mechanical condition maintenance. Current existing techniques can roughly be classified into two categories:(i) physics-based prognostics and (ii) data driven prognostics models. For a physics-based approach, the degradation model requires the knowledge of material properties, failure physical mechanisms (FPM), and the geometric structure of the mechanical equipment to estimate its failure degradation trend or remaining useful life. Generally, physics-based models with sufficient physical knowledge in a specific system may obtain an accurate prediction assessment, for example, fatigue crack growth (FCG) model [5] and fatigue spall progression life (FSPL) model [6], etc. However, it is difficult to obtain adequate modeling information regarding specific knowledge in most actual applications, especially the failure degradation process, which is too complicated. On the other hand, usually, once a physics-based model is established, the relevant model's parameters cannot be adjusted in real-time, which means that the actual degeneration in running states as reflected in the observed vibration signals will not be used. In contrast, data-driven prediction models such as the Bayesian method [7; 8], machine learning techniques [9; 10], support vector machine [11; 12], etc., which utilize the measured historical data to establish the failure degradation model and avoid merging all kinds of physical knowledge of the system allow for an adjustment of the model parameters to capture the degradation trend of vibration signals. However, the common drawback of data-driven approaches is that they do not establish a theoretical mechanism between the physical degradation state (e.g., defect size) and the parameters obtained from the prediction models. Today, fractional order characteristic (FOC) methods have aroused great interest for time series prediction among scholars. The FOC method emphasize long-range time correlation between the historical degradation and the predicted data and it has been demonstrated that this characteristic is difficult to restrain and control in practical applications [13; 14]. In this regard, the PHM community has chosen FOC models such as long-range dependence (LRD) and fractional Brownian motion (FBM) models as the state-of-the-art approaches to address this scenario. For instance, in ref. [15], [PERSON] et al., utilized a fractionally autoregressive integrated moving average (\(f\)-ARIMA) model to investigate the degradation tendency of a rolling bearing based on the equivalent vibration intensity (EVI) health indicator, and also proved that the prediction accuracy outperforms the classical statistical time series and computational intelligence models. In ref. [16], [PERSON] et al., proposed an improved rescaled range (R/S) statistic and hybrid FBM model for the rolling bearing degradation process. In ref. [17], [PERSON] et al., developed a FBM degradation model to predict the bearing running tendency during a serious fault condition.