The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion

By L. Lacasa, B. Luque, Jordi Luque, J. C. Nuño. 2009

371 citations recorded by the source corpus.

Fractional Brownian motion (fBm) has been used as a theoretical framework to study real-time series appearing in diverse scientific fields. Because of its intrinsic nonstationarity and long-range dependence, its characterization via the Hurst parameter, H, requires sophisticated techniques that often yield ambiguous results. In this work we show that fBm series map into a scale-free visibility graph whose degree distribution is a function of H. Concretely, it is shown that the exponent of the power law degree distribution depends linearly on H. This also applies to fractional Gaussian noises (fGn) and generic f−β noises. Taking advantage of these facts, we propose a brand new methodology to quantify long-range dependence in these series. Its reliability is confirmed with extensive numerical simulations and analytical developments. Finally, we illustrate this method quantifying the persistent behavior of human gait dynamics.

Subfields: Economics and Econometrics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7a42e4473530c8d877f9dadf10050d31a2b85d02

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Citations in the corpus

Statistical properties of visibility graph of energy dissipation rates in three-dimensional fully developed turbulence (2009, positive): “[1]. For nonstationary time series, such as stock price indexes, fractional Brownian motions, Conway series, and strike intervals, there is a linear dependence of the Hurst index on the tail exponent [18, 19]. However, no quantitative relation between the Hurst index of stationary time series and th”
Recurrence networks—a novel paradigm for nonlinear time series analysis (2009, methodology): “Visibility graphs have been used to study the behaviour of certain fractal as well as multifractal stochastic processes [68, 69], energy dissipation in three-dimensional turbulence [70] and the nonlinear properties of financial [71, 72] and environmental time series [73].”
Complex network approaches to nonlinear time series analysis (2019, background): “this empirical observation, VG analysis has been suggested to characterize fractional Brownian motions and f-noise, finding some heuristic relationship between γ and the process Hurst exponent H as γ = 3 − 2H, and γ = 5−2H for fractional Gaussian noise [61, 62].”
New diagnostic EEG markers of the Alzheimer’s disease using visibility graph (2010, background): “Recently, Lacasa et al. (2008, 2009) presented VG and showed that such a graph inherits some of the properties of the time series such as periodic series resulting in regular graphs, random series resulting in random graphs, and fractal series converting into scale-free networks.”