## 1 Introduction  
In various large industrial equipment, the rolling bearing is a key equipment in the rotor system; moreover, it frequently works in the adverse environment of high velocity, high temperature, high voltage, etc., this is the very reason why it is particularly important to exercise fault diagnosis over it through its real-time monitoring data [1]. In the rail transit field, the potential fault of rolling bearing has seriously threatened the rail transit safety, especially nowadays the rapid popularization of high-speed railway puts up the urgent requirement of real-time monitoring and online fault analysis to the critical equipment of rolling bearing [2; 3; 4; 5]. By installing sensors at the special location of rolling bearing, vibration data are obtained that possess the characteristics of large amount, nonlinear, and non-stationary [6; 7]. A rapid and effective fault diagnosis algorithm is the key to the systematic implementation [8; 9]. The fault diagnosis algorithms of equipment mainly include statistical-based methods, signal processing-based methods, analytical model-based methods, and knowledge-based methods [10; 11]. The statistical-based method is to compare the statistical characteristics of faults and normal time-domain data. [PERSON] et al. proposed to compare the peak value of the vibration acceleration data of the bearing with the peak value of normal bearing to judge the running state of the bearing [12]. The signal processing-based methods include time-domain and frequency-domain analysis methods [13; 14; 15; 16]. Fourier transform is one of the classical frequency-domain analysis methods. The feature extraction methods for nonlinear, non-stationary signals are mainly wavelet transform and Hilbert transform [17; 18]. On this basis, researchers have proposed some new improved methods, including improved denoising method based on empirical mode decomposition and improved Hilbert Huang transform method (HHT) [19], bearing fault diagnosis method based on adaptive Fourier decomposition, fault diagnosis method based on new nonlinear dynamic model, and fusion of time-frequency-domain feature parameters. [PERSON] et al. [20] proposed a fault diagnosis method by combining local mean decomposition (LMD) and the ratio correction method to process the short-time signals. The vibration signal of rolling bearing is decomposed into a series of product functions (PFs) by LMD. The PF, which contains the richest fault information, is selected to perform envelope spectrum analysis by the Hilbert transform (HT). [PERSON] et al. [21] proposed a new method based on wavelet denoising and nonlinear independent component analysis (ICA) to tackle the nonlinear blind source separation problem with additive noise. The analytical model-based method includes state estimation method and parameter estimation method [22; 23]. State estimation is a method of estimating the internal state of a dynamic system based on available measurement data. The use of state estimation to design a distributed fault observer can solve the problem that the calculation is difficult when there are uncertain parameters in the system. Parameter estimation is the process of calculating system model parameters using system input and output data when the system model structure is known. Commonly used methods of parameter estimation include moment estimation, least squares estimation, maximum likelihood estimation, etc. With the increasing complexity of industrial equipment systems, the number and types of data recorded during the monitoring process are increasing. Knowledge-based fault diagnosis methods have gradually become the mainstream method in the field of monitoring [24]. Among them, artificial neural network [25], D-S evidence theory [26], Bayesian [27] are the most widely used. This kind of method, which can be applied in nonlinear, dynamic, and high-latitude systems can avoid the establishment of complex mechanism models. [PERSON] et al.