The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract ::: Techniques to determine changing system complexity from data are evaluated. Convergence of a frequently used correlation dimension algorithm to a finite value does not necessarily imply an underlying deterministic model or chaos. Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes. The capability to discern changing complexity from such a relatively small amount of data holds promise for applications of ApEn in a variety of contexts. <s> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Approximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity, which appears to have potential application to a wide variety of relatively short (greater than 100 points) and noisy time‐series data. The development of ApEn was motivated by data length constraints commonly encountered, e.g., in heart rate, EEG, and endocrine hormone secretion data sets. We describe ApEn implementation and interpretation, indicating its utility to distinguish correlated stochastic processes, and composite deterministic/ stochastic models. We discuss the key technical idea that motivates ApEn, that one need not fully reconstruct an attractor to discriminate in a statistically valid manner—marginal probability distributions often suffice for this purpose. Finally, we discuss why algorithms to compute, e.g., correlation dimension and the Kolmogorov–Sinai (KS) entropy, often work well for true dynamical systems, yet sometimes operationally confound for general models, with the aid of visual ... <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings is presented in this paper. Detection of both localized and distributed categories of defect has been considered. An explanation for the vibration and noise generation in bearings is given. Vibration measurement in both time and frequency domains along with signal processing techniques such as the high-frequency resonance technique have been covered. Other acoustic measurement techniques such as sound pressure, sound intensity and acoustic emission have been reviewed. Recent trends in research on the detection of defects in bearings, such as the wavelet transform method and automated data processing, have also been included. <s> BIB003 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series. <s> BIB004 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations. <s> BIB005 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <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> BIB006 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract This paper compares three related measures of complexity, ApEn, SampEn, and FuzzyEn. Since vectors’ similarity is defined on the basis of the hard and sensitive boundary of Heaviside function in ApEn and SampEn, the two families of statistics show high sensitivity to the parameter selection and may be invalid in case of small parameter. Importing the concept of fuzzy sets, we developed a new measure FuzzyEn, where vectors’ similarity is defined by fuzzy similarity degree based on fuzzy membership functions and vectors’ shapes. The soft and continuous boundaries of fuzzy functions ensure the continuity as well as the validity of FuzzyEn at small parameters. The more details obtained by fuzzy functions also make FuzzyEn a more accurate entropy definition than ApEn and SampEn. In addition, similarity definition based on vectors’ shapes, together with the exclusion of self-matches, earns FuzzyEn stronger relative consistency and less dependence on data length. Both theoretical analysis and experimental results show that FuzzyEn provides an improved evaluation of signal complexity and can be more conveniently and powerfully applied to short time series contaminated by noise. <s> BIB007 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> PURPOSE ::: The cardiovascular system (CVS) regulation can be studied from a central viewpoint, through heart rate variability (HRV) data, and from a peripheral viewpoint, through laser Doppler flowmetry (LDF) signals. Both the central and peripheral CVSs are regulated by several interacting mechanisms, each having its own temporal scale. The central CVS has been the subject of many multiscale studies. By contrast, these studies at the level of the peripheral CVS are very recent. Among the multiscale studies performed on the central CVS data, multiscale entropy has been proven to give interesting physiological information for diagnostic purposes. However, no multiscale entropy analysis has been performed on LDF signals. The authors' goal is therefore to propose a first multiscale entropy study of LDF data recorded in healthy subjects. ::: ::: ::: METHODS ::: The LDF signals recorded in the forearm of seven healthy subjects are processed. Their period sampling is T=50 ms, and coarse-graining scales from T to 23T are studied. Also, for validation, the algorithm is first tested on synthetic signals of known theoretical multiscale entropy. ::: ::: ::: RESULTS ::: The results reveal nonmonotonic evolution of the multiscale entropy of LDF signals, with a maximum at small scales around 7T and a minimum at longer scales around 18T, singling out in this way two distinctive scales where the LDF signals undergo specific changes from high to low complexity. This also marks a strong contrast with the HRV signals that usually display a monotonic increase in the evolution of the multiscale entropy. ::: ::: ::: CONCLUSIONS ::: Multiscale entropy of LDF signals in healthy subjects shows variation with scales. Moreover, as the variation pattern observed appears similar for all the tested signals, multiscale entropy could potentially be a useful stationary signature for LDF signals, which otherwise are probe-position and subject dependent. Further work could now be conducted to evaluate possible diagnostic purposes of the multiscale entropy of LDF signals. <s> BIB008 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> This paper presents a novel framework to determine the number of resolution levels in the application of a wavelet transformation to a rainfall time series. The rainfall time series are decomposed using the a trous wavelet transform. Then, multi-scale entropy (MSE) analysis that helps to elucidate some hidden characteristics of the original rainfall time series is applied to the decomposed rainfall time series. The analysis shows that the Mann-Kendall (MK) rank correlation test of MSE curves of residuals at various resolution levels could determine the number of resolution levels in the wavelet decomposition. The complexity of rainfall time series at four stations on a multi-scale is compared. The results reveal that the suggested number of resolution levels can be obtained using MSE analysis and MK test. The complexity of rainfall time series at various locations can also be analyzed to provide a reference for water resource planning and application. <s> BIB009 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Bearing fault diagnosis has attracted significant attention over the past few decades. It consists of two major parts: vibration signal feature extraction and condition classification for the extracted features. In this paper, multiscale permutation entropy (MPE) was introduced for feature extraction from faulty bearing vibration signals. After extracting feature vectors by MPE, the support vector machine (SVM) was applied to automate the fault diagnosis procedure. Simulation results demonstrated that the proposed method is a very powerful algorithm for bearing fault diagnosis and has much better performance than the methods based on single scale permutation entropy (PE) and multiscale entropy (MSE). <s> BIB010 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> This paper investigates the usage of permutation entropy for working status characterization of rotary machines. As a statistical measure, the permutation entropy describes complexity of a time series or signal measured on a physical system through phase space reconstruction, and takes into account non-linear behavior of the time series, as often seen in vibration signals of rotary machines. Thus it can be served as a viable tool for detecting dynamic changes of the machine working status. The effect of embedded dimension and time delay on calculation of the permutation entropy value has been studied, and the validity of the permutation entropy for detecting dynamic change of a physical system is studied through a well known non-linear system, the Logistic Map. Comparison with other complexity measures using a numerically formulated signal has also been investigated. Experimental results on bearing vibration analysis have then confirmed that the permutation entropy provides an effective measure for monitoring the working status of rolling bearings. <s> BIB011 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract Nonstationary signal analysis is one of the main topics in the field of machinery fault diagnosis. Time–frequency analysis can identify the signal frequency components, reveals their time variant features, and is an effective tool to extract machinery health information contained in nonstationary signals. Various time–frequency analysis methods have been proposed and applied to machinery fault diagnosis. These include linear and bilinear time–frequency representations (e.g., wavelet transform, Cohen and affine class distributions), adaptive parametric time–frequency analysis (based on atomic decomposition and time–frequency auto-regressive moving average models), adaptive non-parametric time–frequency analysis (e.g., Hilbert–Huang transform, local mean decomposition, and energy separation), and time varying higher order spectra. This paper presents a systematic review of over 20 major such methods reported in more than 100 representative articles published since 1990. Their fundamental principles, advantages and disadvantages, and applications to fault diagnosis of machinery have been examined. Some examples have also been provided to illustrate their performance. <s> BIB012 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract When the fault happens, the complexity of the intrinsic oscillation from the mechanical system will change. Fuzzy entropy (FuzzyEn), which is defined to measure the complexity and self-similarity of the time series, can be utilized to measure the complexity of vibration signal and reflect changes of complexity of the intrinsic oscillation. Since the changes distribute in different scales, a new non-stationary signal analysis method, local characteristic-scale decomposition (LCD), is proposed and used to decompose the vibration signal adaptively into series of intrinsic scale components (ISC) in different scales. And then a new rolling bearing fault diagnosis approach based on LCD and FuzzyEn is proposed. Namely, firstly, by using the LCD rolling bearing vibration signal is decomposed into numbers of ISCs; then the FuzzyEns of the first few ISCs that contain main failure information are extracted. Thirdly, the FuzzyEns obtained are taken as the inputs to the adaptive neuro-fuzzy inference systems (ANFIS) classifier. Finally, the proposed method is applied to experimental data, and the analysis results show that the proposed method performs effectively for the rolling bearing fault diagnosis. <s> BIB013 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract Planetary gearboxes significantly differ from fixed-axis gearboxes and exhibit unique behaviors, which invalidate fault diagnosis methods working well for fixed-axis gearboxes. Much work has been done for condition monitoring and fault diagnosis of fixed-axis gearboxes, while studies on planetary gearboxes are not that many. However, we still notice that a number of publications on condition monitoring and fault diagnosis of planetary gearboxes have appeared in academic journals, conference proceedings and technical reports. This paper aims to review and summarize these publications and provide comprehensive references for researchers interested in this topic. The structures of a planetary gearbox as well as a fixed-axis one are briefly introduced and contrasted. The unique behaviors and fault characteristics of planetary gearboxes are identified and analyzed. Investigations on condition monitoring and fault diagnosis of planetary gearboxes are summarized based on the adopted methodologies. Finally, open problems are discussed and potential research topics are pointed out. <s> BIB014 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract A new rolling bearing fault diagnosis method based on multi-scale fuzzy entropy (MFE), Laplacian Score (LS) and variable predictive model-based class discrimination (VPMCD) is proposed in this paper. Compared with previous approximate entropy (ApEn) and sample entropy (SampEn), MFE has taken into account the dynamic nonlinearity, interaction and coupling effects among mechanical components and thus it provides much more hidden information in different scales of vibration signal. Hence, MFE is employed to characterize the complexity and irregularity of rolling bearing vibration signals. Besides, to fulfill an automatical fault diagnosis, the VPMCD, as a new classification approach, is employed to construct a multi-fault classifier for making decision. Also, Laplacian Score (LS) for feature selection is utilized to refine the feature vector by sorting the features according to their importance and correlations with the fault information to eschew a high dimension of feature vector. Finally, the proposed method is implemented to rolling bearing experimental data and the results indicate that the proposed method is able to discriminate the different fault categories and degrees effectively. <s> BIB015 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Epilepsy can be detected using EEG signals.The entropy indicates the complexity of the EEG signal.Various entropies are used to diagnose epilepsy.Unique ranges for various entropies are proposed. Epilepsy is the neurological disorder of the brain which is difficult to diagnose visually using Electroencephalogram (EEG) signals. Hence, an automated detection of epilepsy using EEG signals will be a useful tool in medical field. The automation of epilepsy detection using signal processing techniques such as wavelet transform and entropies may optimise the performance of the system. Many algorithms have been developed to diagnose the presence of seizure in the EEG signals. The entropy is a nonlinear parameter that reflects the complexity of the EEG signal. Many entropies have been used to differentiate normal, interictal and ictal EEG signals. This paper discusses various entropies used for an automated diagnosis of epilepsy using EEG signals. We have presented unique ranges for various entropies used to differentiate normal, interictal, and ictal EEG signals and also ranked them depending on the ability to discrimination ability of three classes. These entropies can be used to classify the different stages of epilepsy and can also be used for other biomedical applications. <s> BIB016 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract This paper presents a new rolling bearing fault diagnosis method based on local mean decomposition (LMD), improved multiscale fuzzy entropy (IMFE), Laplacian score (LS) and improved support vector machine based binary tree (ISVM-BT). When the fault occurs in rolling bearings, the measured vibration signal is a multi-component amplitude-modulated and frequency-modulated (AM–FM) signal. LMD, a new self-adaptive time-frequency analysis method can decompose any complicated signal into a series of product functions (PFs), each of which is exactly a mono-component AM–FM signal. Hence, LMD is introduced to preprocess the vibration signal. Furthermore, IMFE that is designed to avoid the inaccurate estimation of fuzzy entropy can be utilized to quantify the complexity and self-similarity of time series for a range of scales based on fuzzy entropy. Besides, the LS approach is introduced to refine the fault features by sorting the scale factors. Subsequently, the obtained features are fed into the multi-fault classifier ISVM-BT to automatically fulfill the fault pattern identifications. The experimental results validate the effectiveness of the methodology and demonstrate that proposed algorithm can be applied to recognize the different categories and severities of rolling bearings. <s> BIB017 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Fault diagnosis of rotary machine is becoming extremely important because of the increasing complexity of modern industrial systems and the rising demands for quality, cost efficiency, reliability, and safety. In this paper, a fault diagnosis method based on complementary ensemble empirical mode decomposition (CEEMD)-sample entropy (SampEn) combined with random forest (RF) is presented and applied to practical fault diagnosis for centrifugal pumps. CEEMD, a novel noise assisted method based on empirical mode decomposition (EMD), is applied to decompose the nonlinear and non-stationary vibration signals into a series of intrinsic mode functions (IMFs). Then, the SampEn is utilized to characterize the complexity of the IMF components in different time scales. After the feature extraction, an RF classifier is introduced for identification and classification of fault modes of the centrifugal pumps. The analysis results demonstrate that the proposed method has desirable diagnostic performance for different fault types of centrifugal pumps. <s> BIB018 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> One of the most powerful tools to assess the dynamical characteristics of time series is entropy. Sample entropy (SE), though powerful, is not fast enough, especially for long signals. Permutation entropy (PE), as a broadly used irregularity indicator, considers only the order of the amplitude values and hence some information regarding the amplitudes may be discarded. To tackle these problems, we introduce a new method, termed dispersion entropy (DE), to quantify the regularity of time series. We gain insight into the dependency of DE on several straightforward signal-processing concepts via a set of synthetic time series. The results show that DE, unlike PE, can detect the noise bandwidth and simultaneous frequency and amplitude change. We also employ DE to three publicly available real datasets. The simulations on real-valued signals show that the DE method considerably outperforms PE to discriminate different groups of each dataset. In addition, the computation time of DE is significantly less than that of SE and PE. <s> BIB019 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract Diagnosis of bearings generally plays an important role in fault diagnosis of mechanical system, and machine learning has been a promising tool in this field. In many real applications of bearings fault diagnosis, the data tend to be online imbalanced, which means, the number of fault data is much less than the normal data while they are all collected in online sequential way. Suffering from this problem, many traditional diagnosis methods will get low accuracy of fault data which acts as the minority class in the collected bearing data. To address this problem, an online sequential prediction method for imbalanced fault diagnosis problem is proposed based on extreme learning machine. This method introduces the principal curve and granulation division to simulate the flow distribution and overall distribution characteristics of fault data, respectively. Then a confident over-sampling and under-sampling process is proposed to establish the initial offline diagnosis model. In online stage, the obtained granules and principal curves are rebuilt on the bearing data which are arrived in sequence, and after the over-sampling and under-sampling process, the balanced sample set is formed to update the diagnosis model dynamically. A theoretical analysis is provided and proves that, even existing information loss, the proposed method has lower bound of the model reliability. Simulation experiments are conducted on IMS bearing data and CWRU bearing data. The comparative results demonstrate that the proposed method can improve the fault diagnosis accuracy with better effectiveness and robustness than other algorithms. <s> BIB020 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> This paper presents a novel signal processing scheme, bandwidth empirical mode decomposition, and adaptive multiscale morphological analysis (BEMD-AMMA) for early fault diagnosis of rolling bearings. In this scheme, we propose a bandwidth based method to select the best envelope interpolation method. First, multiple envelope algorithms are defined and separately subtracted from the original data to obtain the preintrinsic mode functions (PIMFs). Second, an IMF with the smallest frequency bandwidth is selected to be the optimal IMF (OIMF). Third, this OIMF is subtracted from the original signal, and then repeat the sifting process until the residual is a constant or monotonic. Since the OIMF has the smallest frequency bandwidth, the mode mixing phenomenon can be significantly weakened. After that the OIMFs with clear fault information are used to construct the main component of the original signal. Then, the AMMA is introduced to demodulate the constructed main component. Simulation and experimental vibration signals are employed to evaluate the effectiveness of the proposed method. Results show that the proposed method outperforms EMD-AMMA, ensemble empirical mode decomposition-AMMA, and generalized empirical mode decomposition-empirical envelope demodulation in detecting early inner race fault. <s> BIB021 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract Health condition identification of planetary gearboxes is crucial to reduce the downtime and maximize productivity. This paper aims to develop a novel fault diagnosis method based on modified multi-scale symbolic dynamic entropy (MMSDE) and minimum redundancy maximum relevance (mRMR) to identify the different health conditions of planetary gearbox. MMSDE is proposed to quantify the regularity of time series, which can assess the dynamical characteristics over a range of scales. MMSDE has obvious advantages in the detection of dynamical changes and computation efficiency. Then, the mRMR approach is introduced to refine the fault features. Lastly, the obtained new features are fed into the least square support vector machine (LSSVM) to complete the fault pattern identification. The proposed method is numerically and experimentally demonstrated to be able to recognize the different fault types of planetary gearboxes. <s> BIB022 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Alzheimer’s disease (AD) is a degenerative brain disorder leading to memory loss and changes in other cognitive abilities. The complexity of electroencephalogram (EEG) signals may help to characterise AD. To this end, we propose an extension of multiscale entropy based on variance (MSEσ2) to multichannel signals, termed multivariate MSEσ2 (mvMSEσ2), to take into account both the spatial and time domains of time series. Then, we investigate the mvMSEσ2 of EEGs at different frequency bands, including the broadband signals filtered between 1 and 40 Hz, θ, α, and β bands, and compare it with the previously-proposed multiscale entropy based on mean (MSEµ), multivariate MSEµ (mvMSEµ), and MSEσ2, to distinguish different kinds of dynamical properties of the spread and the mean in the signals. Results from 11 AD patients and 11 age-matched controls suggest that the presence of broadband activity of EEGs is required for a proper evaluation of complexity. MSEσ2 and mvMSEσ2 results, showing a loss of complexity in AD signals, led to smaller p-values in comparison with MSEµ and mvMSEµ ones, suggesting that the variance-based MSE and mvMSE can characterise changes in EEGs as a result of AD in a more detailed way. The p-values for the slope values of the mvMSE curves were smaller than for MSE at large scale factors, also showing the possible usefulness of multivariate techniques. <s> BIB023 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract Variational mode decomposition (VMD), an effective signal decomposing technique, has attracted considerable attention in recent years. The successful applicability of the VMD method mainly depends on the selection of decomposition parameters. Most of the existing VMD-like fault diagnostic methods only aim to optimize the number of either decomposed modes or balance parameters (or both) by adopting qualitative analysis, priori criteria, or genetic-like algorithms. However, the types of initial center frequencies (ICFs) in VMD are seldom discussed alongside the fault diagnosis of rotating machines. In actual scenarios, the selected ICFs have significant effects on the analytical results. Thus, in this study, the variation features of the center frequency (CF) of extracted modes are investigated with different ICFs, in which the converging U-shape phenomenon is found. Motived by this phenomenon, a novel ICF-guided VMD method is proposed to extract accurately the weak damage features of rotating machines. In particular, the proposed method is composed of two steps. First, an energy fluctuation spectrum is presented to rapidly highlight the ICF of the latent mode. Second, a CF-guided VMD optimization strategy is constructed to extract the optimal mode by adaptively refining the balance parameter. The simulations and experimental verifications confirm the effectiveness of the proposed method to enhance the fault diagnosis of rotating machines. A comparison with existing methods demonstrates the superiority of the proposed method to detect weak faults. <s> BIB024 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract The effective fault diagnosis of rotating machinery is critical to ensure the continuous operation of equipment and is more economical than scheduled maintenance. Traditional signal processing-based and artificial intelligence-based methods, such as wavelet packet transform and support vector machine, have been proved effective in fault diagnosis of rotating machinery, which prevents unexpected machine breakdowns due to the failure of significant components. However, these methods have several disadvantages that make them unable to automatically and effectively extract valid fault features for the effective fault diagnosis of rotating machinery. A novel adaptive learning rate deep belief network combined with Nesterov momentum is developed in this study for rotating machinery fault diagnosis. Nesterov momentum is adopted to replace traditional momentum to enable declining in advance and to improve training performance. Then, an individual adaptive learning rate method is used to select a suitable step length for accelerating descent. To confirm the utility of the proposed deep learning network architecture, two examinations are implemented on datasets from gearbox and locomotive bearing test rigs. Results indicate that the method achieves impressive performance in fault pattern recognition. Comparisons with existing methods are also conducted to demonstrate that the proposed method is more accurate and robust. <s> BIB025 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract Multiscale permutation entropy (MPE) is a recently proposed nonlinear dynamic method for measuring the randomness and detecting the nonlinear dynamic change of time series and can be used effectively to extract the nonlinear dynamic fault feature from vibration signals of rolling bearing. To solve the drawback of coarse graining process in MPE, an improved MPE method called generalized composite multiscale permutation entropy (GCMPE) was proposed in this paper. Also the influence of parameters on GCMPE and its comparison with the MPE are studied by analyzing simulation data. GCMPE was applied to the fault feature extraction from vibration signal of rolling bearing and then based on the GCMPE, Laplacian score for feature selection and the Particle swarm optimization based support vector machine, a new fault diagnosis method for rolling bearing was put forward in this paper. Finally, the proposed method was applied to analyze the experimental data of rolling bearing. The analysis results show that the proposed method can effectively realize the fault diagnosis of rolling bearing and has a higher fault recognition rate than the existing methods. <s> BIB026 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract This paper proposes a novel fault diagnosis method based on variational mode decomposition (VMD) and generalized composite multi-scale symbol dynamic entropy (GCMSDE) to identify the different health conditions of planetary gearboxes. First, VMD is adopted to remove the noises and highlight the fault symptoms. Second, GCMSDE is utilized to extract the fault features from the denoised vibration signals. Third, the Laplacian score (LS) approach is employed to refine the fault features. Finally, the new features are fed into Softmax regression to identify the health conditions of planetary gearboxes. The proposed method is numerically and experimentally demonstrated to be able to differentiate seven localized fault types on the sun gear, planet gear and ring gear of planetary gearboxes. <s> BIB027 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> FIGURE 1. <s> Abstract Early fault diagnosis of rolling bearings is crucial to operating and maintenance cost reduction of the equipment with bearings. This paper aims to propose a novel early fault feature extraction method based on the proposed hierarchical symbol dynamic entropy (HSDE) and the binary tree support vector machine (BT-SVM). Multiscale symbolic dynamic entropy (MSDE) has been recently proposed to characterize the dynamical behavior of time series. MSDE has several merits comparing with multiscale sample entropy (MSE) and multiscale permutation entropy (MPE), such as high computational efficiency and robustness to noise. However, MSDE only utilizes the fault information in the low frequency components and consequently the fault information hidden in the high frequency components is discarded. To address this shortcoming, a new method, namely HSDE, is proposed to extract the fault information in the high frequency components. Then, the BT-SVM is utilized to automatically complete the fault type identification. The effectiveness of the proposed method is validated using simulated and experimental vibration signals. Meanwhile, a comparison is conducted between MPE, hierarchical permutation entropy (HPE), MSE, hierarchical sample entropy (HSE), MSDE and HSDE. Results show that the proposed method performs best to recognize the early fault types of rolling bearings. <s> BIB028
Some applications of rotating machinery  . to various damages BIB006 - BIB024 . In order to guarantee the production security and minimize the unexpected breakdowns and economic loss, it is important to detect these damages as early as possible BIB020 - BIB025 . Due to the rich fault-related information embedded in contaminated vibration signals, the vibrationbased fault diagnosis method has become a mainstream in the field of health condition monitoring (HCM) BIB014 , BIB003 . The collected vibration signals of rotating machinery often represent non-linear and non-stationary characteristics and the fault features are usually weakened and disturbed by the strong environment noises and other neighboring components. However, most of the existing methods are based on a stationary assumption, which is inefficient to analyze these complex vibration signals BIB012 . Therefore, how to extract these periodic features is the crucial issue in HCM of rotating machinery BIB021 . As a statistical measure, entropy can quantify the complexity and detect dynamic change through taking into account non-linear behavior of time series BIB010 , BIB015 - BIB026 . Recently, entropy-based method has been widely applied in the fault diagnosis of rotating machinery BIB026 - BIB017 . Known that the vibration signal collected from a healthy machine has a larger entropy value due to its high irregularity, while that collected from a faulty rotating machine has a smaller entropy value due to its low irregularity caused by the localized damage BIB018 , BIB022 . Compared with traditional methods, entropy-based method has several advantages, such as good cluster ability, high classification accuracy, robust to noise, independent on prior knowledge, etc BIB013 . Therefore, entropy can be served as a promising tool to extract the dynamic characteristics of rotating machines, which shows considerable potential for HCM of rotating machinery. Entropy, as a measure of uncertainty or irregularity of time series, was first proposed by Shannon in 1948  . Shannon entropy estimates the complexity using the probability distribution of its states. For a given time series, if the probability values of different states are similar, it is difficult to determine the future status, thus the time series has its maximum entropy value. In contrast, if there is only one state, the time series has its minimum entropy BIB019 . After that, other forms of Shannon entropy are conducted and the most representative one is rényi entropy  . Inspired by Shannon entropy, Pincus introduced approximate entropy (ApEn) to quantify the irregularity and self-similarity of time series BIB001 , BIB002 . However, a very long data is required in ApEn, and if the data length is short, the obtained value is often smaller than the real one BIB004 . To address this problem, Richman and Moorman BIB004 proposed sample entropy (SampEn). SampEn, though powerful, has two main shortcomings. First, SampEn utilizes a jumping self-similarity function (Heaviside function) to measure the complexity of time series, resulting in inaccurate estimated value in real applications BIB015 , BIB017 . Second, SampEn has lower calculation efficiency, especially for long time series BIB022 . To address the first problem, fuzzy entropy (FE) was developed by Chen et al. BIB007 , which replaced the Heaviside function with a kind of fuzzy membership function. To address the second problem, a new irregularity indicator, permutation entropy (PE) was proposed by Bandt and Pompe BIB005 . PE assesses the dynamical characteristics by considering the order of the amplitude value, which has a higher calculation efficiency compared with SampEn. Entropy-based method has been widely used in many fields such as biology BIB004 , BIB016 , BIB023 , medicine BIB008 , geography BIB009 , image-processing BIB004 , and engineering BIB027 , BIB011 , BIB028 . In this paper, we will give a comprehensive review on entropy-based methods and their applications in fault diagnosis of rotating machinery. For each entropy method, we will review the mathematical theorem, improved entropy methods, and applications in the fault diagnosis of rotating machinery, respectively. The organization of the rest of this paper is as follows: Section II to VII presents the fundamentals of six main entropy methods, their variants and engineering applications. Section VIII reviews the other four entropy methods and their applications in the fault diagnosis of rotating machinery. Section IX gives a summary of the entropy-based methods and their applications. Section X describes some prospects of entropy in fault diagnosis of rotating machinery.