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

Abstract

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.

Appendix

The equations of the chaotic systems used in this study are listed as follows. The sets of parameters were chosen so that the systems describe chaotic attractors.

Duffing system:

$$ \left\{\begin{array}{l} \dot{x} = y \\ \dot{y} = - x^3 - cy + F \cos(z) \\ \dot{z} = \Upomega \\ \end{array}\right., $$

(6)

where c = 0.05, F = 7.5 and $\Upomega = 1.$

Lorenz system:

$$ \left\{\begin{array}{l} \dot{x} = a (y - x)\\ \dot{y} = x(b - z) - y\\ \dot{z} = xy + c z\\ \end{array}\right., $$

(7)

where a = 10, b = 28 and c = −8/3.

Rossler system:

$$ \left\{\begin{array}{l} \dot{x} = -(y + z)\\ \dot{y} = x + a y\\ \dot{z} = b + z (x - c)\\ \end{array}\right., $$

(8)

where a = 0.2, b = 0.2 and c = 5.7.

Henon map:

$$ \left\{\begin{array}{l} x_t = a - x_{t-1}^2 + b y_{t-1} \\ y_t = x_{t-1} \end{array}\right., $$

(9)

where a = 1.4 and b = 0.3.