The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VI. FUZZY ENTROPY A. DESCRIPTION OF FUZZY ENTROPY <s> Abstract A new nonprobabilistic entropy measure is introduced in the context of fuzzy sets or messages. Fuzzy units, or fits , replace bits in a new framework of fuzzy information theory. An appropriate measure of entropy or fuzziness of messages is shown to be a simple ratio of distances: the distances between the fuzzy message and its nearest and farthest nonfuzzy neighbors. Fuzzy conditioning is examined as the degree of subsethood (submessagehood) of one fuzzy set or message in another. This quantity is shown to behave as a conditional probability in many contexts. It is also shown that the entropy of A is the degree to which A ∪ A c is a subset of A ∩ A c , an intuitive relationship that cannot occur in probability theory. This theory of subsethood is then shown to solve one of the major problems with Bayes-theorem learning and its variants—the problem of requiring that the space of alternatives be partitioned into disjoint exhaustive hypotheses. Any fuzzy subsets will do. However, a rough inverse relationship holds between number and fuzziness of partitions and the information gained from experience. All results reduce to fuzzy cardinality. <s> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VI. FUZZY ENTROPY A. DESCRIPTION OF FUZZY ENTROPY <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> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VI. FUZZY ENTROPY A. DESCRIPTION OF FUZZY ENTROPY <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> BIB003 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VI. FUZZY ENTROPY A. DESCRIPTION OF FUZZY ENTROPY <s> Abstract To timely detect the incipient failure of rolling bearing and find out the accurate fault location, a novel rolling bearing fault diagnosis method is proposed based on the composite multiscale fuzzy entropy (CMFE) and ensemble support vector machines (ESVMs). Fuzzy entropy (FuzzyEn), as an improvement of sample entropy (SampEn), is a new nonlinear method for measuring the complexity of time series. Since FuzzyEn (or SampEn) in single scale can not reflect the complexity effectively, multiscale fuzzy entropy (MFE) is developed by defining the FuzzyEns of coarse-grained time series, which represents the system dynamics in different scales. However, the MFE values will be affected by the data length, especially when the data are not long enough. By combining information of multiple coarse-grained time series in the same scale, the CMFE algorithm is proposed in this paper to enhance MFE, as well as FuzzyEn. Compared with MFE, with the increasing of scale factor, CMFE obtains much more stable and consistent values for a short-term time series. In this paper CMFE is employed to measure the complexity of vibration signals of rolling bearings and is applied to extract the nonlinear features hidden in the vibration signals. Also the physically meanings of CMFE being suitable for rolling bearing fault diagnosis are explored. Based on these, to fulfill an automatic fault diagnosis, the ensemble SVMs based multi-classifier is constructed for the intelligent classification of fault features. Finally, the proposed fault diagnosis method of rolling bearing is applied to experimental data analysis and the results indicate that the proposed method could effectively distinguish different fault categories and severities of rolling bearings. <s> BIB004
Because SampEn measures the similarity between the two vectors using the Heaviside function, which is jumping. However, the boundaries of the two classes are mostly ambiguous in real applications, thereby, the Heaviside function is unsuitable to measure the similarity of two vectors BIB003 , BIB001 . To avoid this shortcoming, fuzzy entropy (FE) was developed by Chen et al. BIB002 , which replaces the Heaviside function with a Gaussian function. As the continuity of the exponential function, FE can overcome the drawbacks of SampEn effectively.  Then, FE can be expressed as: A flowchart of the FE method is shown in Figure 10 . Compared with SampEn method, FE has a better performance in robustness to noise because FE evaluates ambiguous uncertainties from the highly irregular signals. Therefore, the advantage of FE is that FE is insensitive to background noises and highly sensitive to the dynamical change BIB004 , BIB003 . In addition, the main shortcoming of FE lies in its low calculation efficiency.