Scaling collapse and structure functions: Identifying self-affinity in finite length time series

Abstract

Empirical determination of the scaling properties and exponents of time series presents a formidable challenge in testing, and developing, a theoretical understanding of turbulence and other out-of-equilibrium phenomena. We discuss the special case of self affine time series in the context of a stochastic process. We highlight two complementary approaches to the differenced variable of the data: i) attempting a scaling collapse of the Probability Density Functions which should then be well described by the solution of the corresponding Fokker-Planck equation and ii) using structure functions to determine the scaling properties of the higher order moments. Using the example of a Lévy flight, we demonstrate a method for conditioning the time series that recovers the underlying self affine scaling in a finite length time series.

Submitted Nonlinear Processes in Geophysics, 18 pages including 8 figures

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• Kolmogorov and Irosnikov-Kraichnan scaling in the anisotropic turbulent solar wind
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• The signature of evolving turbulence in quiet solar wind as seen by ULYSSES
• Approximated maximum likelihood estimation in multifractal random walks
• Quantifying the Anisotropy and Solar Cycle Dependence of "$1/f$" Solar Wind Fluctuations Observed by Ace
• Assessing market uncertainty by means of a time-varying intermittency parameter for asset price fluctuations