SciELO - Scientific Electronic Library Online

 
vol.7 número1What season suits you best? Seasonal light changes and cyanobacterial competitionDensity distribution of partióles upon jamming after an avalanche in a 2D silo índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

  • No hay articulos citadosCitado por SciELO

Links relacionados

  • No hay articulos similaresSimilares en SciELO

Compartir


Papers in physics

versión On-line ISSN 1852-4249

Pap. Phys. vol.7 no.1 La Plata jun. 2015

 

ARTICULOS

Noise versus chaos in a causal Fisher-Shannon plane

 

Osvaldo A. Rosso,1'2* Felipe Olivares,3 Angelo Plastino4

*Email: oarosso@gmail.com

1 Insitituto Tecnológico de Buenos Aires, Av. Eduardo Madero 399, C1106ACD Ciudad Autónoma de Buenos Aires, Argentina.
2 Instituto de Física, Universidade Federal de Alagoas, Maceió, Alagoas, Brazil.
3 Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, La Plata, Argentina.
4 Instituto de Física, IFLP-CCT, Universidad Nacional de La Plata, La Plata, Argentina.


Received: 20 November 2014, Accepted: 1 April 2015
Edited by: C. A. Condat, G. J. Sibona
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.070006


 

We revisit the Fisher-Shannon representation plane , evaluated using the Bandt and Pompe recipe to assign a probability distribution to a time series. Several stochastic dynamical (noises with power spectrum) and chaotic processes (27 chaotic maps) are analyzed so as to illustrate the approach. Our main achievement is uncovering the informational properties of the planar location.

 

I. Introduction

Temporal sequences of measurements (or observations), that is, timeseries (TS), are the basic elements for investigating natural phenomena. From TS, one should judiciously extract information on dynamical systems. Those TS arising from chaotic systems share with those generated by stochastic processes several properties that make them very similar: (1) a wideband power spectrum (PS), (2) a deltalike autocorrelation function, (3) irregular behavior of the measured signals, etc. Now, irregular and apparently unpredictable behavior is often observed in natural TS, which makes interesting the establishment of whether the underlying dynamical process is of either deterministic or stochastic character in order to i) model the associated phenomenon and ii) determine which are the relevant quantifiers.

Chaotic systems display \sensitivity to initial conditions" and lead to non-periodic motion (chaotic time series). Long-term unpredictability arises despite the deterministic character of the trajectories (two neighboring points in the phase space move away exponentially rapidly). Let x1(t) and x2(t) be two such points, located within a ball of radius R at time t. Further, assume that these two points cannot be resolved within the ball due to poor instrumental resolution. At some later time t0, the distance between the points will typically grow to for a chaotic dynamics, the largest Lyapunov exponent. When this distance at time t0 exceeds R, the points become experimentally distinguishable.

This implies that instability reveals some information about the phase space population that was not available at earlier times [1]. One can then think of chaos as an information source. The associated rate of generated information can be cast in precise fashion via the Kolmogorov-Sinai's entropy [2, 3].

One question often emerges: is the system chaotic (low-dimensional deterministic) or stochastic? one is able to show that the system is dominated by low-dimensional deterministic chaos, then only few (nonlinear and collective) modes are required to describe the pertinent dynamics [4]. not, then 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.

Several methodologies for evaluation of Lyapunov exponents and Kolmogorov-Sinai entropies for timeseries' analysis liave been proposed (see Ref. [5]), but their applicability involves taking into account constraints (stationarity, time series length, parameters valúes election for the meíhodology, etc.) which in general make the ensuing results non-conclusive. Thus, one wishes for new tools able to distinguish chaos (determinism) from noise (stochastic) and this leads to our present interest in the computation of quantifiers based on Information Theory, for instance, "entropy", "statistical complexity", "Fisher information", etc.

These quantifiers can be used to detect determinism in time series [6-11]. Different Information Theory based measures (normalized Shannon entropy, statistical complexity, Fisher information) allow for a better distinction between deterministic chaotic and stochastic dynamics whenever "causal" information is incorporated via the Bandt and Pompe's (BP) methodology [12]. For a review of BP's methodology and its applications to physics, biomedical and econophysic signáis, see [13].

Here we revisit, for the purposes previously detailed, the socalled causality Fisher-Shannon entropy plane, [14], which allows to quantify the global versus local characteristic of the time series generated by the dynamical process under study. The two functionals H and F are evaluated using the Bandt and Pompe permutation approach. Several stochastic dynamics (noises with , power spectrum) and chaotic processes (27 chaotic maps) are analyzed so as to illustrate the methodology. We will encounter that significant information is provided by the planar location.

II. Shannon entropy and Fisher information measure

Given a continuous probability distribution function (PDF) 1, its associated Shannon Entropy S [15] is

a measure of "global character" that is not too sensitive to strong changes in the distribution taking place on a smallsized región. Such is not the case with Fisher's Information Measure (FIM) F [16,17], which constitutes a measure of the gradient contení of the distribution /(x), thus being quite sensitive even to tiny localized perturbations. It reads FIM can be variously interpreted as a measure of the ability to estimate a parameter, as the amount of information that can be extracted from a set of measurements, and also as a measure of the state of disorder of a system or phenomenon [17]. In the previous defnition of FIM (Eq. (2)), the division by f(x) is not convenient f(x) ! 0 at certain xvalues. We avoid this we work with real probability amplitudes [16, 17], which is a simpler form (no divisors) and shows that F simply measures the gradient content in (x). The gradient operator signifcantly in uences the contribution of minute local variations to FIM's value. Accordingly, this quantifer is called a \local" one [17].

Let now be a discrete probability distribution, with N the number of possible states of the system under study. The concomitant problem of information-loss due to discretization has been thoroughly studied and, in particular, it entails the loss of FIM's shift-invariance, which is of no importance for our present purposes [10, 11]. In the discrete case, we defne a \normalized" Shannon entropy as

It has been extensively discussed that this discretization is the best behaved in a discrete environment [18]. Here, the normalization constant Fo reads

our system lies in a very ordered state, which occurs when almost all the pn - valúes are zeros, we have a normalized Shannon entropy H ~ 0 and a normalized Fisher's Information Measure T ~ 1. On the other hand, when the system under study is represented by a very disordered state, that is when all the pi - valúes oscillate around the same valué, we obtain H ~ 1 while T ~ 0. One can state that the general FIM-behavior of the present discrete versión (Eq. (4)), is opposite to that of the Shannon entropy, except for periodic motions [10, 11]. The local sensitivity of FIM for discrete-PDFs is reflected in the fact that the specific "i-ordering" of the discrete valúes pn must be seriously taken into account in evaluating the sum in Eq. (4). This point was extensively discussed by us in previous works [10,11]. The summands can be regarded as a kind of "distance" between two contiguous probabilities. Thus, a different ordering of the pertinent summands would lead to a different FIM-value, hereby its local nature. In the present work, we follow the lexicographic order described by Lehmer [22] in the generation of Bandt-Pompe PDF.

III. Description of our chaotic and stochastic systems

Here we study both chaotic and stochastic systems, selected as illustrative examples of different classes of signáis, namely, (a) 27 chaotic dynamic maps [9,19] and (b) truly stochastic processes, noises withpower spectrum [9].

i. Chaotic maps

In the present work, we consider 27 chaotic maps described by J. C. Sprott in the appendix of his book [19]. These chaotic maps are grouped as

a)    Noninvertihle maps: (1) Logistic map; (2) Sine map; (3) Tent map; (4) Linear congruential generator; (5) Cubic map; (6) Ricker's population model; (7) Gauss map; (8) Cusp map; (9) Pinchers map; (10) Spence map; (11) Sinecircle map;

b)    Dissipative maps: (12) Hénon map; (13) Lozi map; (14) Delayed logistic map; (15) Tinkerbell map; (16) Burgers' map; (17) Holmes cubic map; (18) Dissipative standard map; (19) Ikeda map; (20) Sinai map; (21) Discrete predatorprey map,

c)    Conservative maps: (22) Chirikov standard map; (23) Hénon area-preserving quadratic map; (24) Arnold's cat map; (25) Gingerbreadman map; (26) Chaotic web map; (27) Lorenz three-dimensional chaotic map;

Even when the present list of chaotic maps is not exhaustive, it could be taken as representative of common chaotic systems [19].

ii. Noises withpower spectrum

IV. Results and discussion

 

Summing up, we have presented an extensive series of numerical simulations/computations and have contrasted the characterizations of deterministic chaotic and noisystochastic dynamics, as represented by time series of finite length. Surprisingly enough, one just has to look at the different planar locations of our two dynamical regimes. The planar location is able to tell us whether we deal with chaotic or stochastic time series.

 

Acknowledgements - O. A. Rosso and A. Plastino were supported by Consejo Nacional de Investigaciones Científicas y T´ecnicas (CONICET), Argentina. O. A. Rosso acknowledges support as a FAPEAL fellow, Brazil. F. Olivares is supported by Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina.

1 H D I Abarbanel, Analysis of observed chaotic data, Springer-Verlag, New York (1996).         [ Links ]

2 A N Kolmogorov, A new metric invariant for transitive dynamical systems and automorphisms in lebesgue sapces, Dokl. Akad. Nauk. (USSR) 119, 861 (1959).

3 Y G Sinai, On the concept of entropy for a dynamical system, Dokl. Akad. Nauk. (USSR) 124, 768 (1959).         [ Links ]

4 A R Osborne, A Provenzale, Finite correlation dimension for stochastic systems with power-law spectra, Physica D 35, 357 (1989).

5 H Kantz, T Scheiber, Nonlinear time series analysis, Cambridge University Press, Cambridge, UK (2002).

6 O A Rosso, H A Larrondo, M T Martín, A Plastino, M A Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99, 154102 (2007).         [ Links ]

7 O A Rosso, L C Carpi, P M Saco, M Gomez Ravetti, A Plastino, H A Larrondo, Causality and the entropycomplexity plane: Robustness and missing ordinal patters, Physica A 391, 42 (2012).

8 O A Rosso, L C Carpi, P M Saco, M G´omez Ravetti, H A Larrondo, A Plastino, The Amig´o paradigm of forbidden/missing patterns: A detailed analysis, Eur. Phys. J. B 85, 419 (2012).

9 O A Rosso, F Olivares, L Zunino, L De Micco, A L L Aquino, A Plastino, H A Larrondo, Characterization of chaotic maps using the permutation Bandt-Pompe probability distribution, Eur. Phys. J. B 86, 116 (2013).         [ Links ]

10 F Olivares, A Plastino, O A Rosso, Ambiguities in Bandt-Pompe's methodology for local entropic quantifiers, Physica A, 391, 2518 (2012).

11 F Olivares, A Plastino, O A Rosso, Contrasting chaos with noise via local versus global information quantifiers, Phys. Lett A 376, 1577 (2012).         [ Links ]

12 C Bandt, B Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett. 88, 174102 (2002).

13 M Zanin, L Zunino, O A Rosso, D Papo, Permutation entropy and its main biomedical and econophysics applications: A review, Entropy 14, 1553 (2012).

14 C Vignat, J F Bercher, Analysis of signals in the Fisher-Shannon information plane, Phys. Lett. A 312, 27 (2003).         [ Links ]

15 C Shannon, W Weaver, The mathematical theory of communication, University of Illinois Press, Champaign, USA (1949).         [ Links ]

16 R A Fisher, On the mathematical foundations of theoretical statistics, Philos. Trans. R. Soc. Lond. Ser. A 222, 309 (1922).

17 B R Frieden, Science from Fisher information: A Unification, Cambridge University Press, Cambridge, UK (2004).

18 P Sanchez-Moreno, R J Yañez, J S Dehesa, Discrete densities and Fisher information, In: Proceedings of the 14th International Conference on Difference Equations and Applications, Eds. M. Bohner, et al., Pag. 291, UgurBahce sehir University Publishing Company, Istanbul, Turkey (2009).

19 J C Sprott, Chaos and time series analysis, Oxford University Press, New York, USA (2003).

20 H A Larrondo, Matab program: noisefk:m (http://www.mathworks.com/matlabcentral/fileexchange/35381) (2012).

21 M Matsumoto, T Nishimura, Mersenne twister: A 623-dimensionally uniform pseudorandom number gererator, ACM T. Model. Comput. S. 8, 3 (1998).         [ Links ]

22 http://www.keithschwarz.com/interesting/code/factoradic-permutation/FactoradicPermutation.hh.html

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons