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

TL;DR

This paper investigates how finite observation intervals can cause apparent non-stationarity in estimated scaling exponents of time series, especially for heavy-tailed processes, and proposes a semi-empirical method to determine minimal data length for reliable estimates.

Contribution

It introduces a semi-empirical approach to estimate the minimum data length needed for accurate scaling exponent estimation in finite interval time series, highlighting effects of heavy tails.

Findings

Heavy-tailed processes show slow convergence to theoretical scaling behavior.

Proposed semi-empirical estimate guides data length requirements.

Finite sample effects can mimic non-stationarity in scaling exponents.

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.