The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 4) HIERARCHICAL 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> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 4) HIERARCHICAL SAMPLE ENTROPY <s> We develop a hierarchical entropy (HE) method to quantify the complexity of a time series based on hierarchical decomposition and entropy analysis. The proposed method is applied to the Gaussian white noise and the 1/f noise. We prove that the difference frequency components of the Gaussian white noise with the same scale factor have the same value of entropies, and the values decline as the scale factor increases. We also apply the HE method to the 1/f noise, and prove mathematically that a lower frequency component of a 1/f noise is also a 1/f noise and verify numerically that a higher frequency component of a 1/f random vector is approximately equal to a Gaussian random vector. The theoretical results are confirmed by numerical results. Moreover, we show that the HE method is an efficient method to analyze heartbeat signals by applying it to the cardiac interbeat interval time series of healthy young and elderly subjects, congestive heart failure (CHF) subjects and atrial fibrillation (AF) subjects. <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 4) HIERARCHICAL SAMPLE ENTROPY <s> Entropy is a powerful tool for the analysis of time series, as it allows describing the probability distributions of the possible state of a system, and therefore the information encoded in it. Nevertheless, important information may be codified also in the temporal dynamics, an aspect which is not usually taken into account. The idea of calculating entropy based on permutation patterns (that is, permutations defined by the order relations among values of a time series) has received a lot of attention in the last years, especially for the understanding of complex and chaotic systems. Permutation entropy directly accounts for the temporal information contained in the time series; furthermore, it has the quality of simplicity, robustness and very low computational cost. To celebrate the tenth anniversary of the original work, here we analyze the theoretical foundations of the permutation entropy, as well as the main recent applications to the analysis of economical markets and to the understanding of biomedical systems. <s> BIB003 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> 4) HIERARCHICAL SAMPLE ENTROPY <s> Abstract The fault diagnosis of planetary gearboxes is crucial to reduce the maintenance costs and economic losses. This paper proposes a novel fault diagnosis method based on adaptive multi-scale morphological filter (AMMF) and modified hierarchical permutation entropy (MHPE) to identify the different health conditions of planetary gearboxes. In this method, AMMF is firstly adopted to remove the fault-unrelated components and enhance the fault characteristics. Second, MHPE is utilized to extract the fault features from the denoised vibration signals. Third, Laplacian score (LS) approach is employed to refine the fault features. In the end, the obtained features are fed into the binary tree support vector machine (BT-SVM) to accomplish the fault pattern identification. The proposed method is numerically and experimentally demonstrated to be able to recognize the different fault categories of planetary gearboxes. <s> BIB004
MSE can provide a comprehensive analysis of vibration signals, however, it may discard the fault information hidden in the high frequency components because the multiscale analysis in MSE only considers the fault information in low frequency components BIB004 , BIB002 . To address this problem, Zanin et al. BIB003 developed the hierarchical decomposition and proposed the hierarchical sample entropy (HSE). Hierarchical decomposition has been demonstrated to be more effective than multiscale analysis BIB001 . There are two main steps in HSE method in Eq. BIB003 . (1) Construct the hierarchical tree. Figure 8 illustrates the hierarchical tree with 3 hierarchical layers. For a given time series X {x(i), i = 1, 2, · · · , N }, the hierarchical component Z e k can be obtained by the averaging operator Q 0 and difference operator Q 1 as follows: (2) Calculate the SampEn of each layers of the hierarchical component and then plotted as the function of hierarchical lay k. The definition of HSE is given as follows.