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Boaretto BRR, Budzinski RC, Rossi KL, Prado TL, Lopes SR, Masoller C. Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks. Sci Rep 2021; 11:15789. [PMID: 34349134 PMCID: PMC8338970 DOI: 10.1038/s41598-021-95231-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2021] [Accepted: 07/15/2021] [Indexed: 02/07/2023] Open
Abstract
Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, [Formula: see text], that determines the strength of the correlation of the noise. To predict [Formula: see text] the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the [Formula: see text] value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same [Formula: see text] parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.
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Affiliation(s)
- B R R Boaretto
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - R C Budzinski
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - K L Rossi
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - T L Prado
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - S R Lopes
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - C Masoller
- Department of Physics, Universitat Politecnica de Catalunya, 08222, Barcelona, Spain.
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Classification of Fractal Signals Using Two-Parameter Non-Extensive Wavelet Entropy. ENTROPY 2017. [DOI: 10.3390/e19050224] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Wavelet-Tsallis Entropy Detection and Location of Mean Level-Shifts in Long-Memory fGn Signals. ENTROPY 2015. [DOI: 10.3390/e17127856] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Li Q, Fu Z, Yuan N. Beyond Benford's Law: Distinguishing Noise from Chaos. PLoS One 2015; 10:e0129161. [PMID: 26030809 PMCID: PMC4452586 DOI: 10.1371/journal.pone.0129161] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2015] [Accepted: 05/05/2015] [Indexed: 11/24/2022] Open
Abstract
Determinism and randomness are two inherent aspects of all physical processes. Time series from chaotic systems share several features identical with those generated from stochastic processes, which makes them almost undistinguishable. In this paper, a new method based on Benford's law is designed in order to distinguish noise from chaos by only information from the first digit of considered series. By applying this method to discrete data, we confirm that chaotic data indeed can be distinguished from noise data, quantitatively and clearly.
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Affiliation(s)
- Qinglei Li
- Laboratory for Climate and Ocean-Atmosphere Studies, Dept. of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China
| | - Zuntao Fu
- Laboratory for Climate and Ocean-Atmosphere Studies, Dept. of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China
- * E-mail: (ZTF); (NMY)
| | - Naiming Yuan
- Laboratory for Climate and Ocean-Atmosphere Studies, Dept. of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China
- Department of Geography, Climatology, Climate Dynamics, and Climate Change, Justus-Liebig University Giessen, Giessen, Germany
- * E-mail: (ZTF); (NMY)
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Ravetti MG, Carpi LC, Gonçalves BA, Frery AC, Rosso OA. Distinguishing noise from chaos: objective versus subjective criteria using horizontal visibility graph. PLoS One 2014; 9:e108004. [PMID: 25247303 PMCID: PMC4172653 DOI: 10.1371/journal.pone.0108004] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2013] [Accepted: 08/11/2014] [Indexed: 12/03/2022] Open
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 [Formula: see text], in which [Formula: see text] is the node degree and [Formula: see text] 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 [Formula: see text] chaotic maps, 2 chaotic flows and [Formula: see text] 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.
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Affiliation(s)
- Martín Gómez Ravetti
- Departamento de Engenharia de Produção, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil
- Departament de Física Fonamental, Universitat de Barcelona, Barcelona, Spain
| | - Laura C. Carpi
- Laboratório de Computação Científica e Análise Numérica (LaCCAN), Universidade Federal de Alagoas, Maceió, Alagoas, Brazil
| | - Bruna Amin Gonçalves
- Departamento de Engenharia de Produção, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil
- Instituto Politécnico. Centro Universitário UNA, Belo Horizonte, Minas Gerais, Brazil
| | - Alejandro C. Frery
- Laboratório de Computação Científica e Análise Numérica (LaCCAN), Universidade Federal de Alagoas, Maceió, Alagoas, Brazil
| | - Osvaldo A. Rosso
- Instituto de Física, Universidade Federal de Alagoas, Maceió, Alagoas – Brazil
- Instituto Tecnológico de Buenos Aires (ITBA), Ciudad Autónoma de Buenos Aires, Argentina
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Impairment of erythrocytes incubated in glucose medium: a wavelet-information theory analysis. Cell Biochem Biophys 2011; 60:329-34. [PMID: 21301991 DOI: 10.1007/s12013-011-9155-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Abstract
This study investigates the effects produced by an increased concentration of glucose in a suspending medium on the erythrocytes Information Theory quantifiers. Erythrocytes, which were obtained from eight healthy volunteers, were washed and incubated in vitro with glucose solutions at different concentrations. The measured Wavelet-based Information Theory quantifiers include the Relative Wavelet Energy (RWE), the Normalized Total Wavelet Shannon Entropy (NTWS), MPR-Statistical Complexity Measure (SCM) and entropy-complexity plane. The results show that the increase in glucose concentration does not produce significant changes on the RWE, while significant ones on the NTSE, which combined with SCM values allow to identify different behaviour for all the different populations in the entropy-complexity plane. Modification in the hemorheological properties of cells could be clearly detected with these Wavelet-based Information Theory quantifiers.
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Korol AM, D'Arrigo M, Foresto P, Pérez S, Martín MT, Rosso OA. Preliminary characterization of erythrocytes deformability on the entropy-complexity plane. Open Med Inform J 2010; 4:164-70. [PMID: 21611139 PMCID: PMC3099541 DOI: 10.2174/1874431101004010164] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2009] [Revised: 02/02/2010] [Accepted: 06/12/2010] [Indexed: 11/23/2022] Open
Abstract
We present an application of wavelet-based Information Theory quantifiers (Normalized Total Shannon Entropy, MPR-Statistical Complexity and Entropy-Complexity plane) on red blood cells membrane viscoelasticity characterization. These quantifiers exhibit important localization advantages provided by the Wavelet Theory. The present approach produces a clear characterization of this dynamical system, finding out an evident manifestation of a random process on the red cell samples of healthy individuals, and its sharp reduction of randomness on analyzing a human haematological disease, such as β-thalassaemia minor.
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Affiliation(s)
- Ana M Korol
- Facultad de Ciencias Bioquímicas y Farmacéuticas, Universidad Nacional de Rosario, Suipacha 531 (2000), Rosario, Argentina
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De Micco L, Larrondo HA, Plastino A, Rosso OA. Quantifiers for randomness of chaotic pseudo-random number generators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2009; 367:3281-3296. [PMID: 19620124 DOI: 10.1098/rsta.2009.0075] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
We deal with randomness quantifiers and concentrate on their ability to discern the hallmark of chaos in time series used in connection with pseudo-random number generators (PRNGs). Workers in the field are motivated to use chaotic maps for generating PRNGs because of the simplicity of their implementation. Although there exist very efficient general-purpose benchmarks for testing PRNGs, we feel that the analysis provided here sheds additional didactic light on the importance of the main statistical characteristics of a chaotic map, namely (i) its invariant measure and (ii) the mixing constant. This is of help in answering two questions that arise in applications: (i) which is the best PRNG among the available ones? and (ii) if a given PRNG turns out not to be good enough and a randomization procedure must still be applied to it, which is the best applicable randomization procedure? Our answer provides a comparative analysis of several quantifiers advanced in the extant literature.
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Affiliation(s)
- L De Micco
- Departamentos de Física y de Ingeniería Electrónica, Facultad de Ingeniería, Universidad Nacional de Mar del Plata, Juan B. Justo 4302, 7600 Mar del Plata, Argentina
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Rosso OA, Zunino L, Pérez DG, Figliola A, Larrondo HA, Garavaglia M, Martín MT, Plastino A. Extracting features of Gaussian self-similar stochastic processes via the Bandt-Pompe approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:061114. [PMID: 18233821 DOI: 10.1103/physreve.76.061114] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2007] [Indexed: 05/25/2023]
Abstract
By recourse to appropriate information theory quantifiers (normalized Shannon entropy and Martín-Plastino-Rosso intensive statistical complexity measure), we revisit the characterization of Gaussian self-similar stochastic processes from a Bandt-Pompe viewpoint. We show that the ensuing approach exhibits considerable advantages with respect to other treatments. In particular, clear quantifiers gaps are found in the transition between the continuous processes and their associated noises.
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Affiliation(s)
- O A Rosso
- Chaos & Biology Group, Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (UBA), Pabellón II, Ciudad Universitaria, 1428 Ciudad de Buenos Aires, Argentina.
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Rosso OA, Larrondo HA, Martin MT, Plastino A, Fuentes MA. Distinguishing noise from chaos. PHYSICAL REVIEW LETTERS 2007; 99:154102. [PMID: 17995170 DOI: 10.1103/physrevlett.99.154102] [Citation(s) in RCA: 178] [Impact Index Per Article: 10.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2007] [Indexed: 05/25/2023]
Abstract
Chaotic systems share with stochastic processes several properties that make them almost undistinguishable. In this communication we introduce a representation space, to be called the complexity-entropy causality plane. Its horizontal and vertical axis are suitable functionals of the pertinent probability distribution, namely, the entropy of the system and an appropriate statistical complexity measure, respectively. These two functionals are evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the system. Several well-known model-generated time series, usually regarded as being of either stochastic or chaotic nature, are analyzed so as to illustrate the approach. The main achievement of this communication is the possibility of clearly distinguishing between them in our representation space, something that is rather difficult otherwise.
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Affiliation(s)
- O A Rosso
- Centre for Bioinformatics, Biomarker Discovery and Information-Based Medicine, School of Electrical Engineering and Computer Science, The University of Newcastle, University Drive, Callaghan NSW 2308, Australia
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Chaudhury S, Cherayil BJ. Dynamic disorder in single-molecule Michaelis-Menten kinetics: The reaction-diffusion formalism in the Wilemski-Fixman approximation. J Chem Phys 2007; 127:105103. [PMID: 17867782 DOI: 10.1063/1.2768059] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Single-molecule equations for the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] mechanism of enzyme action are analyzed within the Wilemski-Fixman [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)] approximation after the effects of dynamic disorder--modeled by the anomalous diffusion of a particle in a harmonic well--are incorporated into the catalytic step of the reaction. The solution of the Michaelis-Menten equations is used to calculate the distribution of waiting times between successive catalytic turnovers in the enzyme beta-galactosidase. The calculated distribution is found to agree qualitatively with experimental results on this enzyme obtained at four different substrate concentrations. The calculations are also consistent with measurements of correlations in the fluctuations of the fluorescent light emitted during the course of catalysis, and with measurements of the concentration dependence of the randomness parameter.
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Affiliation(s)
- Srabanti Chaudhury
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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