The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 1) MULTISCALE SAMPLE ENTROPY <s> There has been considerable interest in quantifying the complexity of physiologic time series, such as heart rate. However, traditional algorithms indicate higher complexity for certain pathologic processes associated with random outputs than for healthy dynamics exhibiting long-range correlations. This paradox may be due to the fact that conventional algorithms fail to account for the multiple time scales inherent in healthy physiologic dynamics. We introduce a method to calculate multiscale entropy (MSE) for complex time series. We find that MSE robustly separates healthy and pathologic groups and consistently yields higher values for simulated long-range correlated noise compared to uncorrelated noise. <s> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 1) MULTISCALE SAMPLE ENTROPY <s> Traditional approaches to measuring the complexity of biological signals fail to account for the multiple time scales inherent in such time series. These algorithms have yielded contradictory findings when applied to real-world datasets obtained in health and disease states. We describe in detail the basis and implementation of the multiscale entropy (MSE) method. We extend and elaborate previous findings showing its applicability to the fluctuations of the human heartbeat under physiologic and pathologic conditions. The method consistently indicates a loss of complexity with aging, with an erratic cardiac arrhythmia (atrial fibrillation), and with a life-threatening syndrome (congestive heart failure). Further, these different conditions have distinct MSE curve profiles, suggesting diagnostic uses. The results support a general "complexity-loss" theory of aging and disease. We also apply the method to the analysis of coding and noncoding DNA sequences and find that the latter have higher multiscale entropy, consistent with the emerging view that so-called "junk DNA" sequences contain important biological information. <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 1) MULTISCALE SAMPLE ENTROPY <s> Condition-based maintenance (CBM) is a maintenance program that recommends maintenance decisions based on the information collected through condition monitoring. It consists of three main steps: data acquisition, data processing and maintenance decision-making. Diagnostics and prognostics are two important aspects of a CBM program. Research in the CBM area grows rapidly. Hundreds of papers in this area, including theory and practical applications, appear every year in academic journals, conference proceedings and technical reports. This paper attempts to summarise and review the recent research and developments in diagnostics and prognostics of mechanical systems implementing CBM with emphasis on models, algorithms and technologies for data processing and maintenance decision-making. Realising the increasing trend of using multiple sensors in condition monitoring, the authors also discuss different techniques for multiple sensor data fusion. The paper concludes with a brief discussion on current practices and possible future trends of CBM. <s> BIB003 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 1) MULTISCALE SAMPLE ENTROPY <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> BIB004 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 1) MULTISCALE SAMPLE ENTROPY <s> A bearing fault diagnosis method has been proposed based on multi-scale entropy (MSE) and adaptive neuro-fuzzy inference system (ANFIS), in order to tackle the nonlinearity existing in bearing vibration as well as the uncertainty inherent in the diagnostic information. MSE refers to the calculation of entropies (e.g. appropriate entropy, sample entropy) across a sequence of scales, which takes into account not only the dynamic nonlinearity but also the interaction and coupling effects between mechanical components, thus providing much more information regarding machinery operating condition in comparison with traditional single scale-based entropy. ANFIS can benefit from the decision-making under uncertainty enabled by fuzzy logic as well as from learning and adaptation that neural networks provide. In this study, MSE and ANFIS are employed for feature extraction and fault recognition, respectively. Experiments were conducted on electrical motor bearings with three different fault categories and several levels of fault severity. The experimental results indicate that the proposed approach cannot only reliably discriminate among different fault categories, but identify the level of fault severity. Thus, the proposed approach has possibility for bearing incipient fault diagnosis. <s> BIB005
The measured vibration signals from the rotating machinery and the fault information often embedded in multiple scale structures. However, SampEn only analyzes the vibration signal from single scale and much useful information will be ignored. This limits its performance in extracting the embedded fault features BIB001 . In order to avoid such disadvantage, Costa et al. put forward a multiscale procedure and combined it with SampEn, called multiscale sample entropy (MSE), to estimate the complexity of the original time series over a range of scales BIB005 . MSE can enhance the physical meanings and statistical sense of SampEn. There are two steps in MSE method BIB002 . (1) For a given time sereis {x 1 , . . . , x i , . . . , x N } , the coarsegraining procedure for scale i is obtained by averaging the samples of the time series inside consecutive but non overlapping windows of length i. Therefore, it can be divided into several coarse-grained time series y (τ ) using Eq. (12) as follows. Figure 4 gives an example of the coarse-grained procedure. where τ is the scale factor and τ should be a positive integer. When τ = 1, the time series y BIB003 is the original time series. (2) Compute the SampEn for each coarse-grained time series and then plotted as the function of the scale factor τ . The definition of MSE is given as follows. A flowchart of the MSE method is shown in Figure 5 . For convenience, the applications of MSE in fault diagnosis of rotating machinery are summarized in Table 8 . MSE can estimate the dynamical characteristics of time series over different scales. However, MSE has three main shortcomings as follows. First, the coarse-graining procedure reduces the data length with the scale factor increasing, which may lead an inaccurate estimation; Second, MSE can be regarded as a low-pass filter, which cannot prevent aliasing when the downsampling procedure is applied BIB004 ; Third, the standard deviation of time series may become lower with the scale factor increasing. This would cause the patterns becoming closer, resulting a decreasing entropy value.