Distinguishing noise from chaos: objective versus subjective criteria using horizontal visibility graph.

Martín Gómez Ravetti,Bruna Amin Gonçalves,Osvaldo A Rosso,Laura C Carpi,Alejandro C Frery,Satoru Hayasaka

doi:10.1371/journal.pone.0108004

Abstract

A recently proposed methodology called the Horizontal Visibility Graph (HVG) [Luque et al., Phys. Rev. E., 80, 046103 (2009)] that constitutes a geometrical simplification of the well known Visibility Graph algorithm [Lacasa et al., Proc. Natl. Sci. U.S.A. 105, 4972 (2008)], has been used to study the distinction between deterministic and stochastic components in time series [L. Lacasa and R. Toral, Phys. Rev. E., 82, 036120 (2010)]. Specifically, the authors propose that the node degree distribution of these processes follows an exponential functional of the form , in which is the node degree and is a positive parameter able to distinguish between deterministic (chaotic) and stochastic (uncorrelated and correlated) dynamics. In this work, we investigate the characteristics of the node degree distributions constructed by using HVG, for time series corresponding to chaotic maps, 2 chaotic flows and different stochastic processes. We thoroughly study the methodology proposed by Lacasa and Toral finding several cases for which their hypothesis is not valid. We propose a methodology that uses the HVG together with Information Theory quantifiers. An extensive and careful analysis of the node degree distributions obtained by applying HVG allow us to conclude that the Fisher-Shannon information plane is a remarkable tool able to graphically represent the different nature, deterministic or stochastic, of the systems under study.

Highlights

Time series, temporal sequences of measurements or observations, are one of the basic tools for investigating natural phenomena
Its horizontal and vertical axis are functionals of the pertinent probability distribution, namely, the normalized Shannon entropy (S) and the normalized Fisher Information measure (F ). We evaluate these quantifiers for the time series using as probability distribution function (PDF) the node degree distribution obtained via the horizontal visibility graph
The Shannon entropy values are normalized with its maximum value for N~100,000, that corresponds to the entropy of the gaussian white noise

Summary

Introduction

Temporal sequences of measurements or observations, are one of the basic tools for investigating natural phenomena. One should judiciously extract information about the dynamics of the underlying process. Time series arising from chaotic systems share with those generated by stochastic processes several properties that make them very similar. Examples of these properties are: a wide-band power spectrum (PS), a delta-like autocorrelation function, and an irregular behavior of the measured signals. As irregular and apparently unpredictable behavior is often observed in natural time series, the question that immediately emerges is whether the system is chaotic (low-dimensional deterministic) or stochastic. The complex behavior could be modeled by a system dominated by a very large number of excited modes which are in general better described by stochastic or statistical approaches

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Conclusion