## 1 Introduction
expanded the application of mv-MDE to the fault diagnosis of rotating machinery, and the simulation and experimental results showed that mv-MDE has high noise immunity and low dependence on serial length [16]. [PERSON] et al. combined hierarchical entropy with fuzzy entropy and proposed the concept of hierarchical fuzzy entropy (HFE) [17], which realizes the extraction of fault features from the high-frequency part and low-frequency part of vibration signals simultaneously. The essence of the rolling bearing fault diagnosis problem is a process of pattern recognition. The entropy-based feature extraction method combined with machine learning can complete the intelligent diagnosis of faults. [PERSON] et al. used multi-scale fuzzy entropy (MFE) to extract fault features and then used a support vector machine (SVM) to identify different bearing faults [18]. [PERSON] et al. took the multi-scale permutation entropy (MPE) of vibration signals as the input of convolutional neural networks (CNN) and obtained a high fault identification accuracy [19]. [PERSON] et al. used linear discriminant analysis (LDA) to reduce the MPE dimension, and afterward, low-dimensional features were imported into a new fuzzy logic classifier to achieve the classification of bearing faults [20]. [PERSON] et al. used multi-scale sample entropy (MSE) to extract fault features and subsequently used adaptive neuro-fuzzy inference (ANFI) classifier to diagnose different bearing faults [21]. Although many studies have laid a great foundation for the extraction of entropy features of rolling bearings, there are still some problems. For certain deterministic systems, the entropy estimation results should be consistent with their dynamical complexity. However, this consistency is often difficult to guarantee due to the interferences and interactions between mechanical vibration system components. Even worse, it is possible to generate negative and undefined values that contradict the definition of entropy value. In addition, since the early fault features of rolling bearings are usually very weak, this difficulty can lead to inaccurate quantitative estimation of entropy methods, and the tracking of dynamic changes is hard to achieve. In order to further improve the extraction effect of the entropy method on weak fault features of rotating machinery, in this paper, a refined multiscale symbolic diversity entropy (RMSDE) is proposed. This method integrates information from multiple coarse-grained sequences in the vibration signal for comprehensive feature description, and the feature extraction process is more stable and reliable. At the same time, this method simplifies the time series into a symbolic sequence consisting of a finite number of symbols. The computational efficiency is improved while the noise immunity is optimized. Moreover, the proposed RMSDE uses dynamical complexity to describe the system state, which can better reflect changes in internal patterns. Apart from this, based on RMSDE and a self-organizing fuzzy logic classifier (SOF), the intelligent diagnosis of rolling bearing faults is also achieved in this paper. The rest of this paper is organized as follows. Section 2 introduces the underlying theory of the RMSD. Section 3 designs three sets of simulation tests to evaluate the performance of the proposed method. Section 4 introduces the process of the overall RMSDE-SOF method. In Section 5, the proposed method is applied to the fault diagnosis of bearings and gears, respectively. Finally, conclusions are drawn in Section 6.