Consistency of detrended fluctuation analysis

O. Løvsletten
Phys. Rev. E 96, 012141 – Published 21 July 2017
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Abstract

The scaling function F(s) in detrended fluctuation analysis (DFA) scales as F(s)sH for stochastic processes with Hurst exponent H. This scaling law is proven for stationary stochastic processes with 0<H<1 and nonstationary stochastic processes with 1<H<2. For H<0.5, it is observed that the asymptotic (power-law) autocorrelation function (ACF) scales as s1/2. It is also demonstrated that the fluctuation function in DFA is equal in expectation to (i) a weighted sum of the ACF and (ii) a weighted sum of the second-order structure function. These results enable us to compute the exact finite-size bias for signals that are scaling and to employ DFA in a meaningful sense for signals that do not exhibit power-law statistics. The usefulness is illustrated by examples where it is demonstrated that a previous suggested modified DFA will increase the bias for signals with Hurst exponents 1<H1.5. As a final application of these developments, an estimator F̂(s) is proposed. This estimator can handle missing data in regularly sampled time series without the need of interpolation schemes. Under mild regularity conditions, F̂(s) is equal in expectation to the fluctuation function F(s) in the gap-free case.

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  • Received 14 July 2016

DOI:https://doi.org/10.1103/PhysRevE.96.012141

©2017 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

O. Løvsletten

  • Department of Mathematics and Statistics, UiT Artic University of Norway, 9037 Tromsø, Norway

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Issue

Vol. 96, Iss. 1 — July 2017

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