Change ΔS of the entropy in natural time under time reversal: Complexity measures upon change of scale

Abstract

-The entropy S in natural time as well as the entropy in natural time under time reversal S− have already found useful applications in the physics of complex systems, e.g., in the analysis of electrocardiograms (ECGs). Here, we focus on the complexity measures Λ l which result upon considering how the statistics of the time series ∆S [≡ S − S−] changes upon varying the scale l. These scale specific measures are ratios of the standard deviations σ(∆S l ) and hence independent of the mean value and the standard deviation of the data. They focus on the different dynamics that appear on different scales. For this reason, they can be considered complementary to other standard measures of heart rate variability in ECG, like SDNN, as well as other complexity measures already defined in natural time. An application to the analysis of ECG -when solely using NN intervals-is presented: We show how Λ l can be used to separate ECG of healthy individuals from those suffering from congestive heart failure and sudden cardiac death.Introduction. -Natural time analysis [1][2][3] focuses in the sequential order of events occurring in a complex system. It extracts the maximum information possible from a given time series [4]. If we consider a series of N events in a complex system, the natural time attributed to the k-th event is given by χ k = k/N . For example, as shown in fig.1, in the case of an electrocardiogram (ECG) as events we may consider the heartbeats. In natural time analysis, χ k is complemented by a quantity Q k which is proportional to the energy emitted during the k-th event. Further analysis is made by studying the pair (χ k , Q k ) and introducing the normalized energy. . N ) sum up to unity, they can be considered as probabilities corresponding to χ k . The entropy S in natural time is defined [1,5,6] by S = ⟨χ ln χ⟩ − ⟨χ⟩ ln⟨χ⟩,

Entropy in Natural Time and the Associated Complexity Measures

“…Hence, the entropy S in natural time does satisfy the condition to be "causal". As the concept of entropy is equally applicable to deterministic, as well as stochastic processes, the natural time entropies S and S − may provide a useful tool [11,22,[26][27][28][29] for the analysis of physiological time series (cf. the latter in most cases is due to processes involving both stochastic and deterministic components).…”

“…Complexity measures quantifying the variability of S and ∆S upon changing either the (natural time window) length scale or shuffling the consecutive events randomly are classified into two categories depending on whether they make use of either S or the change ∆S of the entropy in natural time under time reversal [22,26,27,29,31].…”

“…Complexity measures Λ and N can be defined [27,29], in a similar fashion with the measures λ and ν defined above, by using the time series of ∆S l obtained for a sliding natural time window l and considering its standard deviation σ[∆S l ] instead of δS. For example, we can define [27] the measure:…”

Entropy in Natural Time and the Associated Complexity Measures

“…Hence, the entropy S in natural time does satisfy the condition to be 'causal'. As the concept of entropy is equally applicable to deterministic as well as stochastic processes, the natural time entropies S and S − may provide a useful tool [11,22,[26][27][28][29] for the analysis of physiological time series (cf. the latter in most cases are due to processes involving both stochastic and deterministic components).…”

“…Hence, for the study of time series coming from ECG recordings one should construct the corresponding time series of S or ∆S obtained for certain scales l of the length of the natural time sliding window. The ECG analyzed come from publicly available databases [123] (for more details see Section 5) and the following key results have been obtained [11,22,[26][27][28][29]]:…”