Objective: We propose a novel complexity measure to overcome the deficiencies of the widespread and powerful multiscale entropy (MSE), including, MSE values may be undefined for short signals, and MSE is slow for real-time applications. Methods: We introduce multiscale dispersion entropy (DisEn-MDE) as a very fast and powerful method to quantify the complexity of signals. MDE is based on our recently developed DisEn, which has a computation cost of O(N), compared with O(N²) for sample entropy used in MSE. We also propose the refined composite MDE (RCMDE) to improve the stability of MDE. Results: We evaluate MDE, RCMDE, and refined composite MSE (RCMSE) on synthetic signals and three biomedical datasets. The MDE, RCMDE, and RCMSE methods show similar results, although the MDE and RCMDE are faster, lead to more stable results, and discriminate different types of physiological signals better than MSE and RCMSE. Conclusion: For noisy short and long time series, MDE and RCMDE are noticeably more stable than MSE and RCMSE, respectively. For short signals, MDE and RCMDE, unlike MSE and RCMSE, do not lead to undefined values. The proposed MDE and RCMDE are significantly faster than MSE and RCMSE, especially for long signals, and lead to larger differences between physiological conditions known to alter the complexity of the physiological recordings. Significance : MDE and RCMDE are expected to be useful for the analysis of physiological signals thanks to their ability to distinguish different types of dynamics. The MATLAB codes used in this paper are freely available at http://dx.doi.org/10.7488/ds/1982.

Dispersion entropy (DispEn) is a recently introduced entropy metric to quantify the uncertainty of time series. It is fast and, so far, it has demonstrated very good performance in the characterisation of time series. It includes a mapping step, but the effect of different mappings has not been studied yet. Here, we investigate the effect of linear and nonlinear mapping approaches in DispEn. We also inspect the sensitivity of different parameters of DispEn to noise. Moreover, we develop fluctuation-based DispEn (FDispEn) as a measure to deal with only the fluctuations of time series. Furthermore, the original and fluctuation-based forbidden dispersion patterns are introduced to discriminate deterministic from stochastic time series. Finally, we compare the performance of DispEn, FDispEn, permutation entropy, sample entropy, and Lempel–Ziv complexity on two physiological datasets. The results show that DispEn is the most consistent technique to distinguish various dynamics of the biomedical signals. Due to their advantages over existing entropy methods, DispEn and FDispEn are expected to be broadly used for the characterization of a wide variety of real-world time series. The MATLAB codes used in this paper are freely available at http://dx.doi.org/10.7488/ds/2326.

Multiscale entropy (MSE) has been a prevalent algorithm to quantify the complexity of biomedical time series. Recent developments in the field have tried to alleviate the problem of undefined MSE values for short signals. Moreover, there has been a recent interest in using other statistical moments than the mean, i.e., variance, in the coarse-graining step of the MSE. Building on these trends, here we introduce the so-called refined composite multiscale fuzzy entropy based on the standard deviation (RCMFE_σ) and mean (RCMFE_μ) to quantify the dynamical properties of spread and mean, respectively, over multiple time scales. We demonstrate the dependency of the RCMFE_σ and RCMFE_μ, in comparison with other multiscale approaches, on several straightforward signal processing concepts using a set of synthetic signals. The results evidenced that the RCMFE_σ and RCMFE_μ values are more stable and reliable than the classical multiscale entropy ones. We also inspect the ability of using the standard deviation as well as the mean in the coarse-graining process using magnetoencephalograms in Alzheimer’s disease and publicly available electroencephalograms recorded from focal and non-focal areas in epilepsy. Our results indicated that when the RCMFE_μ cannot distinguish different types of dynamics of a particular time series at some scale factors, the RCMFE_σ may do so, and vice versa. The results showed that RCMFE_σ-based features lead to higher classification accuracies in comparison with the RCMFE_μ-based ones. We also made freely available all the Matlab codes used in this study at http://dx.doi.org/10.7488/ds/1477.

Fluctuation-based dispersion entropy (FDispEn) is a new approach to estimate the dynamical variability of the fluctuations of signals. It is based on Shannon entropy and fluctuation-based dispersion patterns. To quantify the physiological dynamics over multiple time scales, multiscale FDispEn (MFDE) is developed in this article. MFDE is robust to the presence of baseline wanders or trends in the data. We evaluate MFDE, compared with popular multiscale sample entropy (MSE), multiscale fuzzy entropy (MFE), and the recently introduced multiscale dispersion entropy (MDE), on selected synthetic data and five neurological diseases’ datasets: 1) focal and non-focal electroencephalograms (EEGs); 2) walking stride interval signals for young, elderly, and Parkinson’s subjects; 3) stride interval fluctuations for Huntington’s disease and amyotrophic lateral sclerosis; 4) EEGs for controls and Alzheimer’s disease patients; and 5) eye movement data for Parkinson’s disease and ataxia. MFDE avoids the problem of undefined MSE values and, compared with MFE and MSE, leads to more stable entropy values over the scale factors for white and pink noises. Overall, MFDE is the fastest and most consistent method for the discrimination of different states of neurological data, especially where the mean value of a time series considerably changes along the signal (e.g., eye movement data). This study shows that MFDE is a relevant new metric to gain further insights into the dynamics of neurological diseases recordings. The Matlab codes for MFDE and its refined composite form are available in Xplore.

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.