Elsevier

Physics Letters A

Volume 372, Issues 27–28, 30 June 2008, Pages 4768-4774
Physics Letters A

Permutation entropy of fractional Brownian motion and fractional Gaussian noise

https://doi.org/10.1016/j.physleta.2008.05.026Get rights and content

Abstract

We have worked out theoretical curves for the permutation entropy of the fractional Brownian motion and fractional Gaussian noise by using the Bandt and Shiha [C. Bandt, F. Shiha, J. Time Ser. Anal. 28 (2007) 646] theoretical predictions for their corresponding relative frequencies. Comparisons with numerical simulations show an excellent agreement. Furthermore, the entropy-gap in the transition between these processes, observed previously via numerical results, has been here theoretically validated. Also, we have analyzed the behaviour of the permutation entropy of the fractional Gaussian noise for different time delays.

Introduction

Entropic studies almost always assume that the underlying probability distribution is given. This is not at all the case if one is dealing with an input signal regarded as a time series. Part of the concomitant analysis involves extracting the probability distribution from the data and, rarely, a univocal procedure imposes itself. One recent and successful method is that introduced by Bandt and Pompe [1]. The Bandt and Pompe method (BPM) for evaluating the probability distribution is based on the details of the attractor reconstruction procedure. It is the only one among those in popular use that takes into account the temporal structure of the time series generated by the physical process under study. A notable result from the Bandt and Pompe approach is a notorious improvement in the performance of the information quantifiers obtained using the probability distribution generated by their algorithm [2], [3], [4], [5], [6], [7], [8]. Of course, one must assume with the BPM that the system fulfills a very weak stationary condition and that enough data are available for a correct attractor reconstruction. The permutation entropy is just the celebrated Shannon entropic measure evaluated using the BPM to extract the associated probability distribution.

We are interested in the characterization of stochastic processes through this quantifier. In particular, we have chosen the fractional Brownian motion and its noise, the fractional Gaussian noise, for the analysis. The former is a ubiquitous non-stationary model for many physical phenomena which have empirical spectra of power-law type, 1/fα, with 1<α<3. Thus, the characterization of these processes has become of interest in different and heterogeneous scientific fields, like physics, biology, finance, telecommunications and music [9], [10], [11], [12]. It should be stressed that both processes, fBm and fGn, were jointly introduced in the seminal work of Mandelbrot and Van Ness published in 1968 [13]. Moreover, many authors have made use of the physical connection between fBm and fGn for modelling and synthesis purposes [14], [15], [16], [17].

In a previous effort [18], the normalized permutation entropy of the fractional Gaussian noise and fractional Brownian motion was numerically computed. A clear entropy-gap was observed in the transition between these two stochastic processes, that does not depend upon neither the length of the associated time series nor the embedding dimension. Curiously enough, this is a new result. Previous approaches that employ probability distributions based on a wavelet description fail to detect such gap [19].

In this Letter we have worked out theoretical curves for the above-mentioned normalized permutation entropy of the fractional Gaussian noise and fractional Brownian motion. To such an end we have used theoretical results published recently by Bandt and Shiha [20]. This allows the previously observed entropy-gap to be now conclusively classified as a real phenomenon, not a numerical artifact. Also, we have analyzed the behaviour of the normalized permutation entropy of the fractional Gaussian noise for different time delays. Finally, the curves we worked out using the Bandt and Shiha results were compared with those obtained from numerical simulations of the two stochastic processes under analysis.

The reminder of the Letter is organized as follows. In Section 2 we describe the Bandt and Pompe probability distribution and its associated permutation entropy. In Section 3 we give a brief review of the two stochastic processes under analysis: the fractional Gaussian noise and fractional Brownian motion. The theoretical curves and the comparison with their numerical simulations counterparts are presented in Section 4. Discussions and conclusions are the subject of the last section. Finally, in Appendix A we give some details concerning the Bandt and Shiha theoretical results that we use throughout the Letter.

Section snippets

The Bandt and Pompe approach

Given a time series {xt:t=1,,M}, an embedding dimension D>1, and a time delay τ, consider the ordinal patterns of order D [1], [2], [21] generated bys(xs(D1)τ,xs(D2)τ,,xsτ,xs). To each time s we are assigning a D-dimensional vector that results from the evaluation of the time series at times s,sτ,,s(D1)τ. Clearly, the greater the D value, the more information about the past is incorporated into the ensuing vectors. By the ordinal pattern of order D related to the time s we mean the

Fractional Brownian motion and fractional Gaussian noise

Fractional Brownian motion (fBm) is the only family of processes which is Gaussian, self-similar,1 and endowed with stationary increments—see Ref. [19] and references therein. The normalized family of these Gaussian processes, {BH(t),t>0}, is the one with BH(0)

Theoretical results

The Bandt and Pompe probability distribution associated to different stochastic processes has been recently analyzed by Bandt and Shiha [20]. They provide us with theoretical expressions for the relative frequencies p(πi) for stationary Gaussian process and fractional Brownian motion with arbitrary time delay τ, and D=3 and D=4.2 For further details about these

Conclusions

We have worked out theoretical curves for the normalized permutation entropy of two well-known and widely used stochastic processes, the fractional Gaussian noise and fractional Brownian motion. The entropy-gap in the transition between these processes, previously observed on the basis of numerical simulations, has been theoretically validated. This gap was not observed in a previous approach that employed a wavelet probability distributions [19]. Thus, we conclude that the Bandt and Pompe

Acknowledgements

This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) (PIP 0029/98; 5687/05 and 6036/05), Argentina, Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) (FONDECYT project No. 11060512), Chile, and Pontificia Universidad Católica de Valparaíso (PUCV) (Project No. 123.788/2007), Chile. O.A.R. gratefully acknowledges support from Australian Research Council (ARC) Centre of Excellence in Bioinformatics, Australia.

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