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Fractional refined composite multiscale fuzzy entropy (FRCMFE), which aims to relieve the large fluctuation of fuzzy entropy (FuzzyEn) measure and significantly discriminate different short-term financial time series with noise, is proposed to quantify the complexity dynamics of the international stock indices in the paper. To comprehend the FRCMFE, the complexity analyses of Gaussian white noise with different signal lengths, the random logarithmic returns and volatility series of the international stock indices are comparatively performed with multiscale fuzzy entropy (MFE), composite multiscale fuzzy entropy (CMFE) and refined composite multiscale fuzzy entropy (RCMFE). The empirical results show that the FRCMFE measure outperforms the traditional methods to some extent.

Since existing selection methods of surgical treatment schemes of renal cancer patients mainly depend on physicians’ clinical experience and judgments, the surgical treatment options of renal cancer patients lack their scientifical and reasonable information expression and group decision-making model for renal cancer patients. Fuzzy multi-sets (FMSs) have a number of properties, which make them suitable for expressing the uncertain information of medical diagnoses and treatments in group decision-making (GDM) problems. To choose the most appropriate surgical treatment scheme for a patient with localized renal cell carcinoma (RCC) (T1 stage kidney tumor), this article needs to develop an effective GDM model based on the fuzzy multivalued evaluation information of the renal cancer patients. First, we propose a conversion method of transforming FMSs into entropy fuzzy sets (EFSs) based on the mean and Shannon entropy of a fuzzy sequence in FMS to reasonably simplify the information expression and operations of FMSs and define the score function of an entropy fuzzy element (EFE) for ranking EFEs. Second, we present the Aczel-Alsina t-norm and t-conorm operations of EFEs and the EFE Aczel-Alsina weighted arithmetic averaging (EFEAAWAA) and EFE Aczel-Alsina weighted geometric averaging (EFEAAWGA) operators. Third, we develop a multicriteria GDM model of renal cancer surgery options in the setting of FMSs. Finally, the proposed GDM model is applied to two clinical cases of renal cancer patients to choose the best surgical treatment scheme for a renal cancer patient in the setting of FMSs. The selected results of two clinical cases verify the efficiency and rationality of the proposed GDM model in the setting of FMSs.

The study focuses on the fault signals of rolling bearings, which are characterized by nonlinearity, periodic impact, and low signal-to-noise ratio. The advantages of entropy calculation in analyzing time series data were combined with the high calculation accuracy of Multiscale Fuzzy Entropy (MFE) and the strong noise resistance of Multiscale Permutation Entropy (MPE), a multivariate coarse-grained form was introduced, and the coarse-grained process was improved. The Composite Multivariate Multiscale Permutation Fuzzy Entropy (CMvMPFE) method was proposed to solve the problems of low accuracy, large entropy perturbation, and information loss in the calculation process of fault feature parameters. This method extracts the fault characteristics of rolling bearings more comprehensively and accurately. The CMvMPFE method was used to calculate the entropy value of the rolling bearing experimental fault data, and Support Vector Machine (SVM) was used for fault diagnosis analysis. By comparing with MPFE, the Composite Multiscale Permutation Fuzzy Entropy (CMPFE) and the Multivariate Multiscale Permutation Fuzzy Entropy (MvMPFE) methods, the results of the calculations show that the CMvMPFE method can extract rolling bearing fault characteristics more comprehensively and accurately, and it also has good robustness.

This paper presents anomaly detection in activities of daily living based on entropy measures. It is shown that the proposed approach will identify anomalies when there are visitors representing a multi-occupant environment. Residents often receive visits from family members or health care workers. Therefore, the residents’ activity is expected to be different when there is a visitor, which could be considered as an abnormal activity pattern. Identifying anomalies is essential for healthcare management, as this will enable action to avoid prospective problems early and to improve and support residents’ ability to live safely and independently in their own homes. Entropy measure analysis is an established method to detect disorder or irregularities in many applications: however, this has rarely been applied in the context of activities of daily living. An experimental evaluation is conducted to detect anomalies obtained from a real home environment. Experimental results are presented to demonstrate the effectiveness of the entropy measures employed in detecting anomalies in the resident’s activity and identifying visiting times in the same environment.

Arc fault diagnosis is necessary for the safety and efficiency of PV stations. This study proposed an arc fault diagnosis algorithm formed by combining variational mode decomposition (VMD), improved multi-scale fuzzy entropy (IMFE), and support vector machine (SVM). This method first uses VMD to decompose the current into intrinsic mode functions (IMFs) in the time-frequency domain, then calculates the IMFE according to the IMFs associated with the arc fault. Finally, it uses SVM to detect arc faults according to IMFEs. Arc fault data gathered from a PV arc generation experiment platform are used to validate the proposed method. The results indicated the proposed method can classify arc fault data and normal data effectively.

Fuzzy divergence measure provides an information-theoretic distance between two fuzzy sets. It has been used successfully in different application areas. In the present work, we study fuzzy information measures based on the exponential function. First, the work defines a new exponential fuzzy divergence and proves its basic and advance mathematical properties. We also show that the developed exponential fuzzy divergence is a valid fuzzy distance measure. Second, we propose a new exponential fuzzy entropy to measure the fuzziness associated with a fuzzy set. A comparative study between different fuzzy entropy measures is also presented. Third, based on exponential fuzzy entropy and divergence measures, a fuzzy version of a multi-attributive border approximation area comparison (MABAC) method is developed to solve multiple attribute group decision-making problems with completely unknown/partially known attribute weights. A numerical example is considered to illustrate the decision-making steps, and a comparative study with some existing methods is conducted to show the effectiveness of the developed approach.

Bearing fault diagnosis is essential for reducing equipment operating and maintenance costs. We propose a bearing fault diagnosis method that combines hierarchical symbolic fuzzy entropy (HSFE) and sparse Bayesian extreme learning machine (SBELM). Multiscale symbolic fuzzy entropy (MSFE) is a recently proposed fault diagnosis method. Compared with multiscale sample entropy (MSE), multiscale permutation entropy (MPE), and multiscale fuzzy entropy (MFE), MSFE has high noise resistance and computational efficiency. The multiscale analysis method using the average operator can only extract information in the low frequency component, but cannot use the feature information in the high frequency component. Aiming at this defect, symbolic fuzzy entropy is formed by combining the hierarchical decomposition with the symbolic fuzzy entropy. Hierarchical decomposition uses average and difference operators to decompose the sequence, which can extract the fault information of high-frequency and low-frequency components at the same time. Then, the extracted fault information is efficiently identified and classified using the SBELM. The effectiveness and superiority of the HSFE method are verified by simulation signals and experimental vibration signals. At the same time, experimental comparisons were carried out using MPE, HPE, MSE, HSE, MSFE and HSFE. The experimental results indicate that the HSFE method has the best effect on the identification of rotating machinery fault types.

Multiscale entropy (MSE), as a complexity measurement method of time series, has been widely used to extract the fault information hidden in machinery vibration signals. However, the insufficient coarse graining in MSE will result in fault pattern information missing and the sample entropy used in MSE at larger factors will fluctuate heavily. Combining fractal theory and fuzzy entropy, the time shift multiscale fuzzy entropy (TSMFE) is put forward and applied to the complexity analysis of time series for enhancing the performance of MSE. Then TSMFE is used to extract the nonlinear fault features from vibration signals of rolling bearing. By combining TSMFE with the Laplacian support vector machine (LapSVM), which only needs very few marked samples for classification training, a new intelligent fault diagnosis method for rolling bearing is proposed. Also the proposed method is applied to the experiment data analysis of rolling bearing by comparing with the existing methods and the analysis results show that the proposed fault diagnosis method can effectively identify different states of rolling bearing and get the highest recognition rate among the existing methods.

One of the main drawbacks of the well-known Fuzzy C-means clustering algorithm (FCM) is the random initialization of the centers of the clusters as it can significantly affect the performance of the algorithm, thus not guaranteeing an optimal solution and increasing execution times. In this paper we propose a variation of FCM in which the initial optimal cluster centers are obtained by implementing a weighted FCM algorithm in which the weights are assigned by calculating a Shannon Fuzzy Entropy function. The results of the comparison tests applied on various classification datasets of the UCI Machine Learning Repository show that our algorithm improved in all cases relating to the performances of FCM.

A fuzzy relation is a basic concept of fuzzy set theory. Fuzzy entropy as information entropy for a fuzzy relation is to measure the uncertainty of a fuzzy relation. Some measures on the uncertainty of a fuzzy relation have been presented in recent years by generalizing information entropy. However, a fuzzy relation can induce to a family of upper-fuzzy sets and a family of lower-fuzzy sets, respectively; the existing methods consider only the upper-fuzzy sets in a fuzzy relation and do not address the lower-fuzzy sets. Moreover, these methods sometimes need to use the equality between fuzzy sets, but the equality between fuzzy sets is actually very difficult to achieve. To solve the above problem, this paper proposes new fuzzy entropy based on class-consistent technology and considers its application in attribute reduction for heterogeneous data. Class-consistent technology is a technique for dealing with approximate equality of values. It replaces equality with approximate equality between two values in unit close interval. First of all, based on this technology, two equivalence relations are put forward by means of upper-fuzzy and the lower-fuzzy sets in a fuzzy relation. Then, upper-fuzzy entropy and lower-fuzzy entropy are presented by using these two equivalence relations. Next, new fuzzy entropy is constructed to measure the uncertainty of a fuzzy relation, and the constructed fuzzy entropy overcomes the weakness of the existing methods. Moreover, new fuzzy conditional entropy is defined. Finally, new fuzzy conditional entropy is applied to carry out attribute reduction for heterogeneous data. The experiment results confirm that the given reduction method is more effective than other methods.