The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> C. IMPROVED SAMPLE ENTROPY METHODS <s> Multiscale entropy (MSE) was proposed to characterize complexity as a function of the time-scale factor tau. Despite its broad use, this technique suffers from two limitations: 1) the artificial MSE reduction due to the coarse graining procedure and 2) the introduction of spurious MSE oscillations due to the suboptimal procedure for the elimination of the fast temporal scales. We propose a refined MSE (RMSE), and we apply it to simulations and to 24-h Holter recordings of heart rate variability (HRV) obtained from healthy and aortic stenosis (AS) groups. The study showed that the refinement relevant to the elimination of the fast temporal scales was more helpful at short scales (spanning the range of short-term HRV oscillations), while that relevant to the procedure of coarse graining was more useful at large scales. In healthy subjects, during daytime, RMSE was smaller at short scales (i.e., tau = 1-2) and larger at longer scales (i.e., tau = 4-20) than during nighttime. In AS population, RMSE was smaller during daytime both at short and long time scales (i.e., tau = 1 -11) than during nighttime. RMSE was larger in healthy group than in AS population during both daytime (i.e., tau = 2 -9) and nighttime (i.e., tau = 2). RMSE overcomes two limitations of MSE and confirms the complementary information that can be derived by observing complexity as a function of the temporal scale. <s> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> C. IMPROVED SAMPLE ENTROPY METHODS <s> Abstract Multiscale entropy (MSE) is an effective algorithm for measuring the complexity of a time series that has been applied in many fields successfully. However, MSE may yield an inaccurate estimation of entropy or induce undefined entropy because the coarse-graining procedure reduces the length of a time series considerably at large scales. Composite multiscale entropy (CMSE) was recently proposed to improve the accuracy of MSE, but it does not resolve undefined entropy. Here we propose a refined composite multiscale entropy (RCMSE) to improve CMSE. For short time series analyses, we demonstrate that RCMSE increases the accuracy of entropy estimation and reduces the probability of inducing undefined entropy. <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> C. IMPROVED SAMPLE ENTROPY METHODS <s> We introduce a generalization of multiscale entropy (MSE) analysis. The method is termed MSEn, where the subscript denotes the moment used to coarse-grain a time series. MSEμ, described previously, uses the mean value (first moment). Here, we focus on MSEσ2 , which uses the second moment, i.e., the variance. MSEσ2 quantifies the dynamics of the volatility (variance) of a signal over multiple time scales. We use the method to analyze the structure of heartbeat time series. We find that the dynamics of the volatility of heartbeat time series obtained from healthy young subjects is highly complex. Furthermore, we find that the multiscale complexity of the volatility, not only the multiscale complexity of the mean heart rate, degrades with aging and pathology. The “bursty” behavior of the dynamics may be related to intermittency in energy and information flows, as part of multiscale cycles of activation and recovery. Generalized MSE may also be useful in quantifying the dynamical properties of other physiologic and of non-physiologic time series. <s> BIB003 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> C. IMPROVED SAMPLE ENTROPY METHODS <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> BIB004
This section aims to describe the improved sample entropy methods: multiscale sample entropy, modified multiscale sample entropy, composite multiscale sample entropy and hierarchical sample entropy. In addition, there are some other improved SampEn methods, such as refined composite multiscale entropy BIB002 , multivariate sample entropy BIB004 , generalized multiscale entropy BIB003 refined multiscale entropy BIB001 . Such improved will not be described in this paper due to they are not applied in the fault diagnosis of rotating machinery until now.