Complexity–entropy analysis of daily stream flow time series in the continental United States

  • Francesco SerinaldiEmail author
  • Luciano Zunino
  • Osvaldo A. Rosso
Original Paper


Complexity–entropy causality plane (CECP) is a diagnostic diagram plotting normalized Shannon entropy \({\cal H}_S\) versus Jensen–Shannon complexity \({\cal C}_{JS}\) that has been introduced in nonlinear dynamics analysis to classify signals according to their degrees of randomness and complexity. In this study, we explore the applicability of CECP in hydrological studies by analyzing 80 daily stream flow time series recorded in the continental United States during a period of 75 years, surrogate sequences simulated by autoregressive models (with independent or long-range memory innovations), Theiler amplitude adjusted Fourier transform and Theiler phase randomization, and a set of signals drawn from nonlinear dynamic systems. The effect of seasonality, and the relationships between the CECP quantifiers and several physical and statistical properties of the observed time series are also studied. The results point out that: (1) the CECP can discriminate chaotic and stochastic signals in presence of moderate observational noise; (2) the signal classification depends on the sampling frequency and aggregation time scales; (3) both chaotic and stochastic systems can be compatible with the daily stream flow dynamics, when the focus is on the information content, thus setting these results in the context of the debate on observational equivalence; (4) the empirical relationships between \({\mathcal H}_S\) and \({\mathcal C}_{JS}\) and Hurst parameter H, base flow index, basin drainage area and stream flow quantiles highlight that the CECP quantifiers can be considered as proxies of the long-term low-frequency groundwater processes rather than proxies of the short-term high-frequency surface processes; (6) the joint application of linear and nonlinear diagnostics allows for a more comprehensive characterization of the stream flow time series.


Stream flow Complexity–entropy causality plane Permutation entropy Permutation statistical complexity Bandt and Pompe method Hurst parameter 



Francesco Serinaldi acknowledges financial support from the Willis Research Network. Luciano Zunino and Osvaldo A. Rosso were supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina. Osvaldo A. Rosso gratefully acknowledges the support from FAPEAU fellowship, Brazil. The authors thank Dr. Bellie Sivakumar (University of New South Wales, Australia) for his useful remarks on an earlier version of this paper, and two anonymous reviewers for their comments and suggestions. The analyses were performed in R (R Development Core Team 2009) with the help of the contributed packages fractal (Constantine and Percival 2007), fArma (Wuertz et al. 2008), tseriesChaos (Di Narzo 2007), tsDyn (Di Narzo and Aznarte 2007) and msProcess (Gong et al. 2009). The authors and maintainers of this software are gratefully acknowledged.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Francesco Serinaldi
    • 1
    • 2
    Email author
  • Luciano Zunino
    • 3
    • 4
  • Osvaldo A. Rosso
    • 5
    • 6
  1. 1.School of Civil Engineering and GeosciencesNewcastle UniversityNewcastle Upon TyneUK
  2. 2.Willis Research NetworkLondonUK
  3. 3.Centro de Investigaciones Ópticas (CONICET La Plata-CIC)GonnetArgentina
  4. 4.Departamento de Ciencias Básicas, Facultad de IngenieríaUniversidad Nacional de La Plata (UNLP)La PlataArgentina
  5. 5.Laboratorio de Sistemas Complejos, Facultad de IngenieríaUniversidad de Buenos AiresCiudad Autónoma de Buenos AiresArgentina
  6. 6.Instituto de FísicaUniversidade Federal de AlagoasMaceióBrazil

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