# New Approaches of Stochastic Models to Examine the Vibration Features in Roller Bearings
## 1 Introduction  
Bearing rolling is an advanced forming manufacturing technology for high quality bearings, and China has become an important country in bearing rolling research and production [1]. Applications for rolling bearings include the automobile, aerospace, and machine tool industries [2]. Rolling element bearings play a pivotal role in the functionality of rotating machinery, providing essential support to the machine structure and facilitating rotational motion. The repercussions of bearing failure directly impact the overall health of the machinery, making it a critical component. Various factors contribute to bearing failure, including contamination, deterioration, poor lubrication, and improper mounting [3]. Recent research data highlights that a significant majority, approximately 80%, of rolling bearing failures in rotating machines result from inadequate lubrication (refer to Figure 1) [4]. Radial cracking and fracture may lead large rolling element bearings to fail; hoop stress, roller diameter, and Hertz pressure are important contributors to this process [5]. The peculiar circumstances surrounding a set of roller bearings' failure exposed intriguing aspects of the rollers' metallurgy, the failure mechanism, and the assessment of oils used for lubrication [6]. Consequently, it is imperative to monitor the condition of these bearings and implement intelligent, robust, and reliable diagnostic and prognostic methods for optimal system design and machinery maintenance. Substantial research efforts have been dedicated to exploring these crucial aspects over the past few decades [7]. The monitoring of bearing degradation stands as a crucial aspect of prognostic preservation. Traditionally, researchers rely on frequency-based vibration analysis to assess the progression of mechanical damages [8]. Predictive maintenance becomes imperative when the operational characteristics of bearings deviate significantly from their norm, aiming to prevent actual failures and subsequent machinery breakdowns. In pursuit of this objective, researchers have employed various machine learning (ML) algorithms to construct models capable of forecasting the remaining useful life (RUL) of bearings [9; 10]. While these algorithms have demonstrated effective results and substantial predictive performance, there tends to be a tendency to overlook support for censored observations in the field. Only a limited number of studies have made efforts to construct models that accommodate this crucial aspect [11; 12]. Given the infrequent occurrence of ball bearing failures, encountering censored observations is common, and outright dismissal may lead to reduced efficiency and the introduction of estimation bias [13]. Survival analysis, as a form of regression, has the capacity to harness subdued information [14]. However, recent investigations in the realm of bearing prognostics, utilizing survival analysis, have shown deficiencies in predictive accuracy or ranking performance [15]. Fault diagnosis for rotating equipment can be viewed as the identification of defect patterns within machinery health data. This process typically comprises two primary steps: extracting relevant features from the data, and classifying the faults. In the initial step, various vibration-based signal processing techniques are commonly employed. These techniques involve analyzing the acquired health data in the time domain, frequency domain, or time-frequency domain [16]. Time domain indicators are extensively utilized in bearing fault detection, prized for their intuitive nature and high efficiency. These indicators fall into two main categories: dimensional and non-dimensional. Dimensional signatures hinge on factors such as loading situations, rotating speed, and machinery operating conditions. On the other hand, non-dimensional indicators are independent of loads and speeds, making them preferable for monitoring bearing health.