Luciano Zunino

Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review

Entropy is a powerful tool for the analysis of time series, as it allows describing the probability distributions of the possible state of a system, and therefore the information encoded in it. Nevertheless, important information may be codified also in the temporal dynamics, an aspect which is not usually taken into account. The idea of calculating entropy based on permutation patterns (that is, permutations defined by the order relations among values of a time series) has received a lot of attention in the last years, especially for the understanding of complex and chaotic systems. Permutation entropy directly accounts for the temporal information contained in the time series; furthermore, it has the quality of simplicity, robustness and very low computational cost. To celebrate the tenth anniversary of the original work, here we analyze the theoretical foundations of the permutation entropy, as well as the main recent applications to the analysis of economical markets and to the understanding of biomedical systems.

Time Scales of a Chaotic Semiconductor Laser With Optical Feedback Under the Lens of a Permutation Information Analysis

We analyze the intrinsic time scales of the chaotic dynamics of a semiconductor laser subject to optical feedback by estimating quantifiers derived from a permutation information approach. Based on numerically and experimentally obtained times series, we find that permutation entropy and permutation statistical complexity allow the extraction of important characteristics of the dynamics of the system. We provide evidence that permutation statistical complexity is complementary to permutation entropy, giving valuable insights into the role of the different time scales involved in the chaotic regime of the semiconductor laser dynamics subject to delay optical feedback. The results obtained confirm that this novel approach is a conceptually simple and computationally efficient method to identify the characteristic time scales of this relevant physical system.

Wavelet entropy of stochastic processes

We compare two different definitions for the wavelet entropy associated to stochastic processes. The first one, the normalized total wavelet entropy (NTWS) family [S. Blanco, A. Figliola, R.Q. Quiroga, O.A. Rosso, E. Serrano, Time–frequency analysis of electroencephalogram series, III. Wavelet packets and information cost function, Phys. Rev. E 57 (1998) 932–940; O.A. Rosso, S. Blanco, J. Yordanova, V. Kolev, A. Figliola, M. Schurmann, E. Basar, Wavelet entropy: a new tool for analysis of short duration brain electrical signals, J. Neurosci. Method 105 (2001) 65–75] and a second introduced by Tavares and Lucena [Physica A 357(1) (2005) 71–78]. In order to understand their advantages and disadvantages, exact results obtained for fractional Gaussian noise (-1<α<1) and fractional Brownian motion (1<α<3) are assessed. We find out that the NTWS family performs better as a characterization method for these stochastic processes.

Characterizing the Hyperchaotic Dynamics of a Semiconductor Laser Subject to Optical Feedback Via Permutation Entropy

The time evolution of the output of a semiconductor laser subject to delayed optical feedback can exhibit high-dimensional chaotic fluctuations. In this contribution, our aim is to quantify the degree of unpredictability of this hyperchaotic time evolution. To that end, we estimate permutation entropy, a novel information-theory-derived quantifier particularly robust in a noisy environment. The permutation entropy is defined as a functional of a symbolic probability distribution, evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the chaotic system. This measure quantifies the diversity of orderings present in the associated time series. In order to evaluate the performance of this novel quantifier, we compare with the results obtained by using a more standard chaos quantifier, namely the Kolmogorov-Sinai entropy. Here, we present numerical results showing that the permutation entropy, evaluated at specific time-scales involved in the chaotic regime of the semiconductor laser subject to optical feedback, give valuable information about the degree of unpredictability of the chaotic laser dynamics. The influence of additive observational noise on the proposed tool is also investigated.

Wavelet entropy and fractional Brownian motion time series

We study the functional link between the Hurst parameter and the normalized total wavelet entropy when analyzing fractional Brownian motion (fBm) time series—these series are synthetically generated. Both quantifiers are mainly used to identify fractional Brownian motion processes [L. Zunino, D.G. Perez, M. Garavaglia, O.A. Rosso, Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform, Fractals 12(2) (2004) 223–233]. The aim of this work is to understand the differences in the information obtained from them, if any.

Permutation min-entropy: An improved quantifier for unveiling subtle temporal correlations

The aim of this letter is to introduce the permutation min-entropy as an improved symbolic tool for identifying the existence of hidden temporal correlations in time series. On the one hand, analytical results obtained for the fractional Brownian motion stochastic model theoretically support this hypothesis. On the other hand, the analysis of several computer-generated and experimentally observed time series illustrate that the proposed symbolic quantifier is a versatile and practical tool for identifying the presence of subtle temporal structures in complex dynamical systems. Comparisons against the results obtained with other tools confirm its usefulness as an alternative and/or complementary measure of temporal correlations.

Complexity–entropy causality plane: A useful approach for distinguishing songs

Nowadays we are often faced with huge databases resulting from the rapid growth of data storage technologies. This is particularly true when dealing with music databases. In this context, it is essential to have techniques and tools able to discriminate properties from these massive sets. In this work, we report on a statistical analysis of more than ten thousand songs aiming to obtain a complexity hierarchy. Our approach is based on the estimation of the permutation entropy combined with an intensive complexity measure, building up the complexity–entropy causality plane. The results obtained indicate that this representation space is very promising to discriminate songs as well as to allow a relative quantitative comparison among songs. Additionally, we believe that the here-reported method may be applied in practical situations since it is simple, robust and has a fast numerical implementation.

Fraunhofer diffraction by Cantor fractals with variable lacunarity

Abstract Optical diffraction by fractal openings is increasingly being studied because it allows the properties and parameters that characterize these objects to be determined. Allain and Cloitre published the first results showing that the resulting analysis of the distribution of intensity obtained from diffraction experiments through fractal deterministic pupils permits the self-similar dimension and other geometrical characteristic of these structures to be obtained directly. In this work the lacunarity effect ε, dimension d and order k of growth of the Cantor fractal about the distribution of intensities of the diffraction pattern are studied, solved analytically and characterized. In particular we note the influence of lacunarity because this is one of the first studies in which this geometric fractal parameter is taken into consideration. The selfsimilarity of the diffraction figure at different orders is also proved. The results of this study allow us to say that an intimate relation exists between the distribution of the diffracted waves and the parameters that describe this kind of fractal geometry.

Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator.

In the dynamics of optical systems, one commonly needs to cope with the problem of coexisting deterministic and stochastic components. The separation of these components is an important, although difficult, task. Often the time scales at which determinism and noise dominate the systems dynamics differ. In this Letter we propose to use information-theory-derived quantifiers, more precisely, permutation entropy and statistical complexity, to distinguish between the two behaviors. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrate that the time scales at which deterministic or noisy behavior dominate can be identified. Supporting numerical simulations prove the accuracy of this identification.

Modeling turbulent wave-front phase as a fractional Brownian motion: a new approach.

We introduce a new, general formalism to model the turbulent wave-front phase by using fractional Brownian motion processes. Moreover, it extends results to non-Kolmogorov turbulence. In particular, generalized expressions for the Strehl ratio and the angle-of-arrival variance are obtained. These are dependent on the dynamic state of the turbulence.