Entropy in Natural Time and the Associated Complexity Measures

Abstract

Natural time is a new time domain introduced in 2001. The analysis of time series associated with a complex system in natural time may provide useful information and may reveal properties that are usually hidden when studying the system in conventional time. In this new time domain, an entropy has been defined, and complexity measures based on this entropy, as well as its value under time-reversal have been introduced and found applications in various complex systems. Here, we review these applications in the electric signals that precede rupture, e.g., earthquakes, in the analysis of electrocardiograms, as well as in global atmospheric phenomena, like the El Niño/La Niña Southern Oscillation.

Natural Time Analysis: Results Related to Two Earthquakes in Greece during 2019

“…It was preceded by an anomalous geolectric signal that was recorded on 2 October 2018 at a measuring station 70km away from the epicenter [2]. Upon analyzing this signal in natural time, it was found [2] that it conforms to the conditions suggested (e.g., see [3][4][5])) for its clarification as precursory Seismic Electric Signal (SES) activity [4,6,7]. Notably, the observed lead time of 23 days lies within the range of values that has been very recently identified [8] as being statistically significant for the precursory variations of the electric field of the Earth.…”

“…For example in the case of SES, Q k is proportional to the duration of each SES pulse [40,41], while for EQs it may be considered proportional to the seismic moment [40,42,43]. How the time series coming from a variety of complex systems are read in natural time can be seen in Figure 1 of [5].…”

“…The pair (χ k , Q k ) is studied by considering the normalized energy for the k-th event p k = Q k /∑ N n=1 Q n , where p k can be also considered as a probability distribution [5,44]. In view of the latter, the function [4,[40][41][42]44]…”

“…for a variety of systems approaching criticality. Thus, κ 1 reaches the value of 0.070 for a critical system or 0.083 for a system exhibiting stationary or quasi-periodic behavior [5]. Apart from κ 1 , another useful quantity in NTA [4,5] is the entropy S given by [40,46,48]…”

A Prototype Photoplethysmography Electronic Device that Distinguishes Congestive Heart Failure from Healthy Individuals by Applying Natural Time Analysis

“…gives us useful information in the study of complex systems that operate far away from the equilibrium state, as is the case of the heart [30] and enables the distinction of healthy individuals from those who suffer from heart disease [24,31].We clarify that living systems are examples of systems operating far from equilibrium which could be understood as follows (e.g., see [32,33] and references therein): living systems are subject to mass, energy, entropy, and information fluxes across their boundaries. Time irreversibility, which is fundamental property of far from equilibrium systems, is related to the unidirectionality of the energy flow across the boundaries of the system.…”