# Early Warning Signals for Bearing Failure Using Detrended Fluctuation Analysis
## 1 Introduction  
Rolling element bearings (REBs) are one of the essential elements in rotary machinery, such as pumps, compressors, gearboxes, and electric motors. Usually, REBs are susceptible to catastrophic failures when they are operating under abnormal conditions, including heavier loading, corrosion, overheating, etc. As a result of frequent wear and friction, they will experience a gradual degradation and functionality loss during their operation. Evidently, REBs go through different stages (mild, moderate, and serious) of functionality loss before their failure. Prognostics and diagnostics have been active research areas in recent years [1, 2]. Diagnostic techniques evaluate the fault severity during operation to help in planning immediate reactive maintenance and avoiding any future failure. Usually, once a bearing fault has been detected, the time remaining before bearing failure is relatively very short. So, reactive maintenance often cannot be completed before the failure. On the other hand, prognostic techniques provide an early warning for bearing breakdown to plan the needed maintenance ahead of time. That permits a continuous and reliable operation of the mechanical system while enabling proactive maintenance. The prognostic techniques can be classified into three categories: physics-based models (PbMs), data-driven model (DDMs), and hybrid models. The PbM approach depends on the mechanical component models to determine its deterioration over time. Moreover, the hybrid model approach is a combination of the PbM and DDM approaches. The main challenge in these two approaches is the difficulty of deriving the mathematical models accurately. On the other hand, the DDM approach extracts the features in the data collected from the mechanical component to determine its proximity toa failure. It is considered a trusted approach in predicting future failures. In general, the DDM approach has two subcategories: the machine-learning approach [3; 4; 5] and the statistical-based approach. The statistical-based approach is based on several techniques in the time and frequency domains. In the time domain, the root mean square (RMS) is used to detect the starting point of bearing fault from its vibration data [6]. Furthermore, advanced statistical features, such as kurtosis [7], are applied to predict the remaining useful life (RUL) of bearings. On the other hand, novel frequency-domain statistical parameters, such as envelope spectrum and spectral kurtosis (SK) [8; 9], are proposed to detect bearing fault features. [PERSON] et al. have studied the correlation between nominal and degraded vibration signals to estimate the RUL of bearings [10]. [PERSON] et al. have used an improved exponential model and a particle filter (PF) to predict bearing failure [11]. The adoption of one statistical model does not explain the discrepancy between the different degradation rates in the vibration data. [PERSON] et al. have developed a multi-mode PF to track the different stages of bearing performance degradation [12]. [PERSON] et al. have proposed adaptive predictive models to understand the trend in the vibration data and predict bearing failure [13]. [PERSON] et al. have estimated the extended Kalman filter parameters to predict bearing failure under different operating conditions [14]. [PERSON] et al. have used Hilbert-Huang transform to monitor ball bearings and track their degradation [15]. The statistical techniques have different approaches to predict bearing failure up to a certain level. However, the used statistical features in these approaches have no clear trend of degradation or monotonic change over time before bearing failure. Moreover, despite signal denoising, the statistical parameters keep fluctuating over time and sometimes go through a sudden spike when bearings near their end of life. From the prognostics point of view, these drawbacks limit the usage of these statistical methods in practical applications.