To simplify the quantification of time irreversibility, we employ order patterns instead of the raw multidimension vectors in time series, and considering the existence of forbidden permutation, we propose a subtraction-based parameter, Y S , to measure the probabilistic differences between symmetric permutations for time irreversibility. Two chaotic models, the logistic and Henon systems, and reversible Gaussian process and their surrogate data are used to validate the time-irreversible measure, and time irreversibility of epileptic EEGs from Nanjing General Hospital is detected by the parameter. Test results prove that it is promising to quantify time irreversibility by measuring the subtraction-based probabilistic differences between symmetric order patterns, and our findings highlight the manifestation of nonlinearity of whether healthy or diseased EEGs and suggest that the epilepsy leads to a decline in the nonlinearity of brain electrical activities during seize-free intervals.

No abstract

The association of equal heartbeat intervals with cardiac conditions and the effect of the equality on permutationbased time irreversibility are investigated in this paper. We measure the distributions of equal heartbeat intervals under three conditions, namely congestive heart failure (CHF), healthy young and elderly, whose time irreversibility of heartbeats is detected by measuring the probabilistic difference between permutations instead of raw vectors. We demonstrate that heartbeats contain high rates of equal states, particularly the CHF with around 20% equalities, and the distributions of equal values discriminate the heartbeats at very short data length. The CHF have more equal values than the healthy young (p<1.47 * 10 −15 ) and elderly (p<2.48 * 10 −11 ), and the healthy young have less equalities than the elderly (p<3.16 * 10 −4 ). Time irreversibility considering equal values is promising to extract nonlinear behaviors of heartbeats, confirming the decreased nonlinear complexity of the diseased and aging heart rates, while that involving no equality leads to erroneous nonlinearity detection. In our contribution, we highlight the pathological or physiological information contained by the distribution of equal heartbeat intervals that might contribute to develop relevant biomarkers in the area of heart analysis, and demonstrate the effectiveness of equality-based time irreversibility in the nonlinearity detection of heartbeats.Electrocardiography (ECG) is a common process of recording the cardiac electrical activities to obtain the heart structure and/or function. Derived from the ECG, different cardiac intervals, like the PR interval (from the beginning of P wave to QRS complex), QRS interval, ST interval (between the end of QRS complex and the start of T wave), etc, carry a lot of important information, among which the RR interval between successive R waves, generally used to represent heart rate, has more clinical and scientific applications and arguably manifests with nonlinear properties [1,2,3]. As a consequence, besides the time and frequency domains techniques, nonlinear dynamics is adopted to detect subtle nonlinear changes in heart rate to provide indices of cardiac autonomic regulation, such as Poincare plot [4], fractal measures [5,6], symbolic dynamics [7,8,9,10], entropy methods [11,12,13], and so on [14,15].

No abstract

The Shannon entropy and quantitative time irreversibility (qTIR) are statistical quantifiers that are widely used for characterizing complex processes. However, the differences and associations between them have not been subjected to detailed investigation. In this Letter, we report a comparative analysis of the Shannon entropy and qTIR using model series and real-world heartbeats. We find that the permutation-based Shannon entropy (PEn) and time irreversibility (PYs) detect nonlinearities in the model series differently according to the surrogate theory. Moreover, PEn and PYs, based on either the original or the equal-value permutation, give contradictory results for congestive heart failure cases and healthy young and elderly heartbeats. PEn quantifies the complexity by calculating the amount of mean information, whereas PYs measures the probabilistic differences among symmetric nonequilibrium distributions, and these yield different or even contradictory outcomes. Our findings demonstrate the statistical associations between the Shannon entropy and qTIR, contribute to more reliable elucidation of the nonlinear dynamics of heartbeats, and improve our understanding of the complexity and nonequilibrium nature of complex systems.

No abstract

Symbolizations, the base of symbolic dynamic analysis, are classified as global static and local dynamic approaches which are combined by joint entropy in our works for nonlinear dynamic complexity analysis. Two global static methods, symbolic transformations of Wessel N. symbolic entropy and base-scale entropy, and two local ones, namely symbolizations of permutation and differential entropy, constitute four double symbolic joint entropies that have accurate complexity detections in chaotic models, logistic and Henon map series. In nonlinear dynamical analysis of different kinds of heart rate variability, heartbeats of healthy young have higher complexity than those of the healthy elderly, and congestive heart failure (CHF) patients are lowest in heartbeats' joint entropy values. Each individual symbolic entropy is improved by double symbolic joint entropy among which the combination of base-scale and differential symbolizations have best complexity analysis. Test results prove that double symbolic joint entropy is feasible in nonlinear dynamic complexity analysis.

No abstract