Entropy is a powerful tool for nonlinear analysis of time series. We have recently introduced dispersion entropy (DispEn), which has widely drawn attention from researchers in a variety of settings in data analysis. However, DispEn is sensitive to its parameters, especially number of classes (quantization level). Additionally, in some cases, a small change in the signal amplitude due to noise changes the quantized series and so entropy value. To address these limitations, we develop fuzzy dispersion entropy (FuzzDispEn) on the basis of fuzzy membership functions for the quantization of a signal and Shannon entropy. In FuzzDispEn, the concept of fuzzification in Shannon entropy-based methods is introduced for the first time. Several straightforward concepts in signal processing using a set of time series were utilized to show the advantages of FuzzDispEn over DispEn in terms of the detection of the dynamical variability of signals. The results showed that FuzzDispEn has lower sensitivity to noise and signal length compared to DispEn. FuzzDispEn was also applied to five real world applications: mechanical fault detection in gearbox, fault diagnosis of rolling element bearings, damage level detection of bearings, detection of blood pressure in salt-sensitive rats, and detection of seizure and seizure-free EEGs. The results suggest that FuzzDispEn outperforms DispEn as well as popular sample entropy and permutation entropy in distinguishing different kinds of dynamical patterns in synthetic and real data.

Rotary machines often exhibit nonlinear behavior due to factors such as nonlinear stiffness, damping, friction, coupling effects, and defects. Consequently, their vibration signals display nonlinear characteristics. Entropy techniques prove to be effective in detecting these nonlinear dynamic characteristics. Recently, an approach called fuzzy dispersion entropy (DE–FDE) was introduced to quantify the uncertainty of time series. FDE, rooted in dispersion patterns and fuzzy set theory, addresses the sensitivity of DE to its parameters. However, FDE does not adequately account for the presence of multiple time scales inherent in signals. To address this limitation, the concept of multiscale fuzzy dispersion entropy (MFDE) was developed to capture the dynamical variability of time series across various scales of complexity. Compared to multiscale DE (MDE), MFDE exhibits reduced sensitivity to noise and higher stability. In order to enhance the stability of MFDE, we propose a refined composite MFDE (RCMFDE). In comparison with MFDE, MDE, and RCMDE, RCMFDE’s performance is assessed using synthetic signals and three real bearing datasets. The results consistently demonstrate the superiority of RCMFDE in detecting various patterns within synthetic and real bearing fault data. Importantly, classifiers built upon RCMFDE achieve notably high accuracy values for bearing fault diagnosis applications, outperforming classifiers based on refined composite multiscale dispersion and sample entropy methods.

Bearing vibration signals typically have nonlinear components due to their interaction and coupling effects, friction, damping, and nonlinear stiffness. Bearing faults affect the signal complexity at various scales. Hence, measuring signal complexity at different scales is helpful to diagnosis of bearing faults. Numerous studies have investigated multiscale algorithms; nevertheless, multiscale algorithms using the first moment lose important complexity data. Accordingly, generalized multiscale algorithms have been recently introduced. The present research examined the use of refined composite generalized multiscale dispersion entropy (RCGMDispEn) based on the second moment (variance) and third moment (skewness) along with refined composite multiscale dispersion entropy (RCMDispEn) in bearing fault diagnosis. Moreover, multiclass FCM-ANFIS, which is a combination of adaptive network-based fuzzy inference systems (ANFIS), was developed to improve the efficiency of rotating machinery fault classification. According to the results, it is recommended that generalized multiscale algorithms based on variance and skewness be examined for diagnosis, along with multiscale algorithms, and be used to achieve an improvement in the results. The simultaneous usage of the multiscale algorithm and generalized multiscale algorithms improved the results in all three real datasets used in this study.