The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 3) REFINED COMPOSITE MULTIVARIATE MULTISCALE FUZZY ENTROPY <s> Multiscale entropy (MSE) is an appealing tool to characterize the complexity of time series over multiple temporal scales. Recent developments in the field have tried to extend the MSE technique in different ways. Building on these trends, we propose the so-called refined composite multivariate multiscale fuzzy entropy (RCmvMFE) whose coarse-graining step uses variance (RCmvMFEσ2) or mean (RCmvMFEμ). We investigate the behavior of these multivariate methods on multichannel white Gaussian and 1/f noise signals, and two publicly available biomedical recordings. Our simulations demonstrate that RCmvMFEσ2 and RCmvMFEμ lead to more stable results and are less sensitive to the signals’ length in comparison with the other existing multivariate multiscale entropy-based methods. The classification results also show that using both the variance and mean in the coarse-graining step offers complexity profiles with complementary information for biomedical signal analysis. We also made freely available all the Matlab codes used in this paper. <s> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 3) REFINED COMPOSITE MULTIVARIATE MULTISCALE FUZZY ENTROPY <s> The vibration signals of rolling bearings are often nonlinear and non-stationary. Multiscale entropy (MSE) has been widely applied to measure the complexity of nonlinear mechanical vibration signals, however, at present only the single channel vibration signals are used for fault diagnosis by many scholars. In this paper multiscale entropy in multivariate framework, i.e., multivariate multiscale entropy (MMSE) is introduced to machinery fault diagnosis to improve the efficiency of fault identification as much as possible through using multi-channel vibration information. MMSE evaluates the multivariate complexity of synchronous multi-channel data and is an effective method for measuring complexity and mutual nonlinear dynamic relationship, but its statistical stability is poor. Refined composite multivariate multiscale fuzzy entropy (RCMMFE) was developed to overcome the problems existing in MMSE and was compared with MSE, multiscale fuzzy entropy, MMSE and multivariate multiscale fuzzy entropy by analyzing simulation data. Finally, a new fault diagnosis method for rolling bearing was proposed based on RCMMFE for fault feature extraction, Laplacian score and particle swarm optimization support vector machine (PSO-SVM) for automatic fault mode identification. The proposed method was compared with the existing methods by analyzing experimental data analysis and the results indicate its effectiveness and superiority. <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 3) REFINED COMPOSITE MULTIVARIATE MULTISCALE FUZZY ENTROPY <s> Rolling bearings’ operation under variable speed conditions exhibits complex time-varying modulations and spectral structures, resulting in difficulty in the fault diagnosis. In order to effectively remove the influence of the rotational speed and extract the fault characteristics, this paper develops a fault diagnosis scheme based on the Vold–Kalman filter (VKF), refined composite multi-scale fuzzy entropy (RCMFE), and Laplacian score (LS). In the proposed method, the VKF is first adopted to remove the fault-unrelated components and give a clear representation of the fault symptoms. Second, the RCMFE is applied to extract fault features from the denoised vibration signal. Third, the LS approach is introduced to refine the fault features by sorting the scale factors. In the end, the selected features are fed into the logistic regression to automatically complete the fault pattern identifications. The proposed method is experimentally demonstrated to be able to recognize the localized defect on the inner race, outer race, and rolling element under variable speed conditions. <s> BIB003
Recently, Azami and Escudero extended the MFE to multivariate signals (mvMFE) BIB001 . mvMFE measures each sequence in multi-channel data by taking into account their mutual predictability. However, mvMFE will have some fluctuations at larger scale. To address this issue, refined composite multivariate multiscale fuzzy entropy (RCMMFE) was proposed by Zheng et al. BIB002 . Because the multivariate multiscale analysis has a better fault feature extraction ability than mono channel analysis, RCMMFE has a better performance for fault feature extraction. In BIB002 , the CWRU bearing data is applied to verify the effectiveness of the RCMMFE method and the final classification accuracy is 100% BIB002 . Li et al. BIB003 combined Vold-Kalman filter and RCMFE to conduct the fault diagnosis of rolling bearing under speed fluctuation condition. Results demonstrated that their method is able to recognize the localized damage on the inner race, outer race, and rolling element under variable speed conditions. The definition of RCMMFE is expressed as follows. (1) For a given normalized p variate multi-channel time series X = {x k,i } N i=1 , k = 1, 2, · · · , p, the coarse-graining time series is similar as MSE: (2) Compute the averageφ m τ (r) of all ϕ m τ q (r) and the averageφ m+1 (3) The definition of RCMMFE is expressed as