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Multiscale entropy (MSE) was recently developed to evaluate the complexity of time series over different time scales. Although the MSE algorithm has been successfully applied in a number of different fields, it encounters a problem in that the statistical reliability of the sample entropy (SampEn) of a coarse-grained series is reduced as a time scale factor is increased. Therefore, in this paper, the concept of a composite multiscale entropy (CMSE) is introduced to overcome this difficulty. Simulation results on both white noise and 1/f noise show that the CMSE provides higher entropy reliablity than the MSE approach for large time scale factors. On real data analysis, both the MSE and CMSE are applied to extract features from fault bearing vibration signals. Experimental results demonstrate that the proposed CMSE-based feature extractor provides higher separability than the MSE-based feature extractor.

Atrial fibrillation (AF) is nowadays the most common cardiac arrhythmia, being associated with an increase in cardiovascular mortality and morbidity. When AF lasts for more than seven days, it is classified as persistent AF and external interventions are required for its termination. A well-established alternative for that purpose is electrical cardioversion (ECV). While ECV is able to initially restore sinus rhythm (SR) in more than 90% of patients, rates of AF recurrence as high as 20–30% have been found after only a few weeks of follow-up. Hence, new methods for evaluating the proarrhythmic condition of a patient before the intervention can serve as efficient predictors about the high risk of early failure of ECV, thus facilitating optimal management of AF patients. Among the wide variety of predictors that have been proposed to date, those based on estimating organization of the fibrillatory (f-) waves from the surface electrocardiogram (ECG) have reported very promising results. However, the existing methods are based on traditional entropy measures, which only assess a single time scale and often are unable to fully characterize the dynamics generated by highly complex systems, such as the heart during AF. The present work then explores whether a multi-scale entropy (MSE) analysis of the f-waves may provide early prediction of AF recurrence after ECV. In addition to the common MSE, two improved versions have also been analyzed, composite MSE (CMSE) and refined MSE (RMSE). When analyzing 70 patients under ECV, of which 31 maintained SR and 39 relapsed to AF after a four week follow-up, the three methods provided similar performance. However, RMSE reported a slightly better discriminant ability of 86%, thus improving the other multi-scale-based outcomes by 3–9% and other previously proposed predictors of ECV by 15–30%. This outcome suggests that investigation of dynamics at large time scales yields novel insights about the underlying complex processes generating f-waves, which could provide individual proarrhythmic condition estimation, thus improving preoperative predictions of ECV early failure.

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.

The aim of this study was to assess the ability of multiscale sample entropy (MSE), refined composite multiscale entropy (RCMSE), and complexity index (CI) to characterize gait complexity through trunk acceleration patterns in subjects with Parkinson’s disease (swPD) and healthy subjects, regardless of age or gait speed. The trunk acceleration patterns of 51 swPD and 50 healthy subjects (HS) were acquired using a lumbar-mounted magneto-inertial measurement unit during their walking. MSE, RCMSE, and CI were calculated on 2000 data points, using scale factors (τ) 1–6. Differences between swPD and HS were calculated at each τ, and the area under the receiver operating characteristics, optimal cutoff points, post-test probabilities, and diagnostic odds ratios were calculated. MSE, RCMSE, and CIs showed to differentiate swPD from HS. MSE in the anteroposterior direction at τ4 and τ5, and MSE in the ML direction at τ4 showed to characterize the gait disorders of swPD with the best trade-off between positive and negative posttest probabilities and correlated with the motor disability, pelvic kinematics, and stance phase. Using a time series of 2000 data points, a scale factor of 4 or 5 in the MSE procedure can yield the best trade-off in terms of post-test probabilities when compared to other scale factors for detecting gait variability and complexity in swPD.

The health and financial cost of falls has spurred research to differentiate the characteristics of fallers and non-fallers. Postural stability has received much of the attention with recent studies exploring various measures of entropy. This study compared the discriminatory ability of several entropy methods at differentiating two paradigms in the center-of-pressure of elderly individuals: (1) eyes open (EO) vs. eyes closed (EC) and (2) fallers (F) vs. non-fallers (NF). Methods were compared using the area under the curve (AUC) of the receiver-operating characteristic curves developed from logistic regression models. Overall, multiscale entropy (MSE) and composite multiscale entropy (CompMSE) performed the best with AUCs of 0.71 for EO/EC and 0.77 for F/NF. When methods were combined together to maximize the AUC, the entropy classifier had an AUC of for 0.91 the F/NF comparison. These results suggest researchers and clinicians attempting to create clinical tests to identify fallers should consider a combination of every entropy method when creating a classifying test. Additionally, MSE and CompMSE classifiers using polar coordinate data outperformed rectangular coordinate data, encouraging more research into the most appropriate time series for postural stability entropy analysis.

The mild cognitive impairment (MCI) stage plays an essential role in preventing the progression of older adults to Alzheimer's disease. In this study, neurofeedback training (NFT) is applied to improve MCI brain cognitive function. To assess the improvement effect, a novel algorithm called Weighted Multiple Multiscale Entropy (WMMSE) is proposed to extract and analyze the electroencephalogram (EEG) features of patients with MCI. To overcome the information loss problem of traditional multiscale entropy (MSE), WMMSE fully considered the correlation of the sequence and the contribution of each sequence to the total entropy. The experimental group composed of 39 patients with MCI was subjected to NFT for 10 days during two sessions. The control group included 21 patients with MCI without any intervention. The Lempel-Ziv complexity (LZC) was used for primary assessment, and WMMSE was used to accurately analyze the effect of NFT. The results show that the WMMSE values of F4, C3, C4, O1, and T5 channels post-NFT are higher compared with pre-NFT and significant differences ( P < 0.05). Moreover, the cognitive subscale of the Montreal Cognitive Assessment (MoCA) results shows that the post-NFT score is higher than the pre-NFT in the vast majority of the patients with MCI and significant differences ( P < 0.05). When compared with the control group, the WMMSE values of the experimental group increased in each channel. Therefore, the NFT intervention method contributes to brain cognitive functional recovery, and WMMSE can be used as a biomarker to evaluate the state of MCI brain cognitive function.

Multiscale entropy (MSE) and refined multiscale entropy (RMSE) techniques are being widely used to evaluate the complexity of a time series across multiple time scales ‘ t ’. Both these techniques, at certain time scales (sometimes for the entire time scales, in the case of RMSE), assign higher entropy to the HRV time series of certain pathologies than that of healthy subjects, and to their corresponding randomized surrogate time series. This incorrect assessment of signal complexity may be due to the fact that these techniques suffer from the following limitations: (1) threshold value ‘ r ’ is updated as a function of long-term standard deviation and hence unable to explore the short-term variability as well as substantial variability inherited in beat-to-beat fluctuations of long-term HRV time series. (2) In RMSE, entropy values assigned to different filtered scaled time series are the result of changes in variance, but do not completely reflect the real structural organization inherited in original time series. In the present work, we propose an improved RMSE (I-RMSE) technique by introducing a new procedure to set the threshold value by taking into account the period-to-period variability inherited in a signal and evaluated it on simulated and real HRV database. The proposed I-RMSE assigns higher entropy to the age-matched healthy subjects than that of patients suffering from atrial fibrillation, congestive heart failure, sudden cardiac death and diabetes mellitus, for the entire time scales. The results strongly support the reduction in complexity of HRV time series in female group, old-aged, patients suffering from severe cardiovascular and non-cardiovascular diseases, and in their corresponding surrogate time series.

Multiscale fuzzy entropy (MFE), as an enhanced multiscale sample entropy (MSE) method, is an effective nonlinear method for measuring the complexity of time series. In this paper, an improved MFE algorithm termed composite interpolation-based multiscale fuzzy entropy (CIMFE) is proposed by using cubic spline interpolation of the time series over different scales to overcome the drawbacks of the coarse-grained MFE process. The proposed CIMFE method is compared with MSE and MFE by analyzing simulation signals and the result indicates that CIMFE is more robust than MSE and MFE in analyzing short time series. Taking this into account, a new fault diagnosis method for rolling bearing is presented by combining CIMFE for feature extraction with Laplacian support vector machine for fault feature classification. Finally, the proposed fault diagnosis method is applied to the experiment data of rolling bearing by comparing with the MSE, MFE and other existing methods, and the recognition rate of the proposed method is 98.71%, 98.71%, 98.71%, 98.71% and 100% under different training samples (5, 10, 15, 20 and 25), which is higher than that of the existing methods.

A typical symptom of vibration signals collected from rolling bearings with local faults is the existence of periodic transients, which makes the intrinsic structure of vibration signals become more and more regular. Generally, the vibration always contains multiple intrinsic oscillatory modes on different scales, which generally is caused by the interaction and coupling of machine components. Therefore, it is necessary to detect the behavior change of complexity of vibration signals in the view of multiple scales for fault information representation. The complexity and nonlinear failure symptom of rolling bearing can be evaluated by the recently proposed nonlinear dynamic tools, such as multiscale entropy (MSE) and its variants. Recently, the improved MSE method, multiscale dispersion entropy (MDE) and its improvement refined composite MDE (RCMDE) are developed to measure the complexity of time domain data. However, the intrinsic shortages of coarse graining approach that used in MDE and RCMDE have limited their application to fault feature representation. In this paper, an improved RCMDE approach named generalized refined composite multiscale fluctuation-based fractional dispersion entropy (GRCMFDE) is proposed to enhance MDE and RCMDE in complexity measurement of time series. GRCMFDE was compared with MPE, MDE, RCMDE by analyzing synthetic simulation signals to verify its advantages. After that, an intelligent fault diagnosis method was proposed by combining GRCMFDE with supervised multi-clustering feature selection and gray wolf optimized SVM for fault classification of rolling bearing. Lastly, the proposed fault diagnostic method was applied to two experimental data set analysis by comparing with multiscale permutation entropy, MDE- and RCMDE-based fault diagnostic methods and the comparison results indicate that the proposed method can effectively diagnose the fault locations and severities of rolling bearing and get a higher fault identifying rate than the comparative methods.

The vibration signals of rolling bearings are often nonlinear and non-stationary. Multiscale entropy (MSE) has been widely applied to measure the complexity of nonlinear mechanical vibration signals, however, at present only the single channel vibration signals are used for fault diagnosis by many scholars. In this paper multiscale entropy in multivariate framework, i.e., multivariate multiscale entropy (MMSE) is introduced to machinery fault diagnosis to improve the efficiency of fault identification as much as possible through using multi-channel vibration information. MMSE evaluates the multivariate complexity of synchronous multi-channel data and is an effective method for measuring complexity and mutual nonlinear dynamic relationship, but its statistical stability is poor. Refined composite multivariate multiscale fuzzy entropy (RCMMFE) was developed to overcome the problems existing in MMSE and was compared with MSE, multiscale fuzzy entropy, MMSE and multivariate multiscale fuzzy entropy by analyzing simulation data. Finally, a new fault diagnosis method for rolling bearing was proposed based on RCMMFE for fault feature extraction, Laplacian score and particle swarm optimization support vector machine (PSO-SVM) for automatic fault mode identification. The proposed method was compared with the existing methods by analyzing experimental data analysis and the results indicate its effectiveness and superiority.