Pseudo-nonstationarity in the scaling exponents of finite interval time series

Abstract

The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of non-stationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ~1/N as N goes to infinity for certain statistical estimators; however, the convergence to this behaviour will depend on the details of the process, and may be slow. We study the variation in the scaling of second order moments of the time series increments with N, for a variety of synthetic and `real world' time series; and find that in particular for heavy tailed processes, for realizable N, one is far from this 1/N limiting behaviour. We propose a semi-empirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some `real world' time series.

arXiv comment12 pages, 5 figures, accepted Physical Review E

References in corpus (10)

• Crackling Noise
• Observation of inertial energy cascade in interplanetary space plasma
• Power law burst and inter-burst interval distributions in the solar wind: turbulence or dissipative SOC ?
• A method for extracting the scaling exponents of a self-affine, non-Gaussian process from a finite length timeseries
• Finite size scaling in the solar wind magnetic field energy density as seen by WIND
• Scaling and a Fokker-Planck model for fluctuations in geomagnetic indices and comparison with solar wind epsilon as seen by WIND and ACE
• Self-similar signature of the active solar corona within the inertial range of solar wind turbulence
• Extremum statistics: A framework for data analysis
• The signature of evolving turbulence in quiet solar wind as seen by ULYSSES
• Accounting for outliers and calendar effects in surrogate simulations of stock return sequences

Cited by in corpus (5)

• Global scale-invariant dissipation in collisionless plasma turbulence
• Nature of the MHD and kinetic scale turbulence in the magnetosheath of Saturn: Cassini observations
• Methods for characterising microphysical processes in plasmas
• Helios 2 observations of solar wind turbulence decay in the inner heliosphere
• The scale-free texture of the fast solar wind