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

¹  Departamento de Matemática Aplicada y Estadística, ETSI Aeronáuticos, Universidad Politécnica de Madrid Madrid, Spain, EU
²  Departament de Teoria del Senyal i Comunicacions, Universitat Politècnica de Catalunya - Barcelona, Spain, EU
³  Departamento de Matemática Aplicada a los Recursos Naturales, ETSI Montes, Universidad Politécnica de Madrid Madrid, Spain, EU

Received: 7 January 2009
Accepted: 14 April 2009

Abstract

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 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.