## 1 Introduction  
Rotating machinery is commonly used in modern industrial fields such as aerospace, power generation systems, and transportation [1]. Among the parts, rolling bearings and gearbox always work under high speed and heavy load environments; fatigue and damage are very susceptible to occur. These faults may cause serious economic losses and even lead to significant safety accidents [2]. Therefore, the incipient fault diagnosis and health condition monitoring of rotating machinery can improve the reliability of engineering machinery, which is of great importance to industrial production efficiency and safety. As the occurrence of bearing and gear failure will produce a series of bandwidth shock and shock decay responses, vibration signal analysis has become a mainstream fault diagnosis method [3]. The features extracted from the vibration signal can characterize the health status of the rotating machinery. However, due to the complexity of the mechanical system structure, the fault characteristics are often disturbed by strong background noise, especially for weaker early faults, which are difficult to identify directly by conventional time-frequency analysis methods alone [4]. In recent years, entropy-based fault characterization methods have brought new approaches to feature extraction [5]. Entropy can effectively describe the degree of chaos of a nonlinear system. For example, when the bearing is under normal conditions, the corresponding entropy value is larger due to the non-periodic and non-modulated nature of the vibration signal. When a fault occurs, the vibration signal complexity is lower corresponding to a smaller entropy value due to the generation of periodic pulses [6]. Classical entropy methods include approximate entropy (AE), sample entropy (SE), fuzzy entropy (FE), and permutation entropy (PE). AE was proposed by [PERSON] [7], which calculates the complexity of a time series by the probability of the appearance of a new pattern. However, the accuracy of AE can be plagued by the length of the time series. To overcome the shortcomings of AE, [PERSON] et al. proposed SE [8], which eliminates the bias caused by its own similarity statistics. Unfortunately, the presentation of the Heaviside function in SE still brings the problem of inaccurate estimates [9]. [PERSON] et al. proposed FE based on SE by replacing the Heaviside function with an exponential function [10]. FE can effectively improve the accuracy of track similarity estimation and increase the robustness of entropy estimation. In addition, [PERSON] and [PERSON] proposed PE [11], which is centered on computing the state probability of the track alignment and has high computational efficiency. Moreover, these four entropy methods have been applied maturely in the field of rolling bearing fault diagnosis. To provide a more comprehensive description of the fault features, [PERSON] et al. combined entropy theory with the multiscale analysis method and proposed multiscale sample entropy (MSE) [12]. MSE segments the time series at different scales, making the extracted fault features more informative. Inspired by MSE, many scholars have extended fuzzy entropy and fuzzy entropy to multi-scale space and applied them to rolling bearing fault detection. Meanwhile, more and more new entropy measurement methods have been proposed. [PERSON] et al. introduced a calculation of cosine similarity from nonlinear system dynamics and proposed a novel multiscale diversity entropy (MDE) [13], which has the advantage of high consistency. For the problem of weak fault feature extraction, [PERSON] et al. propose symbolic fuzzy entropy (SFE) based on symbolic dynamic filtering and fuzzy entropy [14], which led to the improvement of both computational efficiency and anti-interference performance. [PERSON] et al. extended the multiscale entropy analysis method to multichannel signals and proposed multivariate multiscale dispersion entropy (mv-MDE) [15], which was successfully used for feature extraction in biomedical engineering. [PERSON] et al.