# New Approaches of Stochastic Models to Examine the Vibration Features in Roller Bearings
## 1 Introduction
Time-domain dimensionless indicators not only overcome the limitations associated with dimensional counterparts, but also prove effective in early fault detection in rotating machinery [17; 18; 19]. In contrast, dimensionless parameters derived from raw vibration signals offer insights into the overall changes in bearing health. Among these parameters, Kurtosis stands out for its sensitivity to impact signals, making it widely employed in rolling bearing fault detection. However, these indicators predominantly capture impulsiveness measures. To address the cyclic behavior in rotating machine vibration signals, an indicator based on the negentropy of the square envelope spectrum is introduced [20]. Additionally, statistical point-of-view indicators and bandpass filter techniques are proposed to identify cyclo-stationary behavior [21]. Spectral kurtosis (SK) is then presented as a statistical tool for pinpointing transients in the frequency domain [22]. Investigators further introduce a novel spectral extraction method called \"in fogram\", which considers both transient characteristics using negentropy of the square envelope and its spectrum [23]. Stochastic approaches play a crucial role in analyzing bearing inner race and outer race faults, employing probabilistic techniques to account for uncertainties in the bearings' operation and fault occurrence [24]. One commonly employed stochastic model is the Markov chain model, assuming that the bearing's health status can be represented by discrete states such as healthy, incipient fault, and severe fault. The transition probabilities  
Figure 1: Distribution of the causes of bearing failure in terms of percentages. between these states are determined based on the bearing's operating conditions and fault characteristics [25; 26]. Another stochastic model in bearing fault analysis is the hidden Markov model (HMM), which posits that the bearing's health status is not directly observable, but can only be inferred from measurable data like vibration signals or acoustic emissions. The HMM assumes that the bearing's health status follows a hidden state process, modelled by a Markov chain, with observable data generated based on the current hidden state [27; 28; 29; 30; 31]. For machine RUL prediction, stochastic process models reduce risks and repair expenditures while improving productivity, accessibility, sustainability, and accuracy [32]. Stochastic model roll bearing testing performs well most of the time, despite perhaps minimizing signals from other components such as gears [33]. For the literature, see references [34; 35; 36; 37]. In our study, we employ the stochastic resonance approach to extract features of faulty bearings under various loading conditions. The extracted features identify the presence of defects such as spalls, cracks, wear, and other faults. These features are then used to formulate stochastic equations describing uncertainties in the system's working conditions. These models offer a well-defined mathematical structure for classifying non-stationary periodicities in time series, providing valuable constraints suitable for integrated auto-diagnosis. This approach proves effective in fault detection, classification, and prognosis, all within a unified methodology.