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Consistency of detrended fluctuation analysis.

O Løvsletten1

  • 1Department of Mathematics and Statistics, UiT Artic University of Norway, 9037 Tromsø, Norway.

Physical Review. E
|January 20, 2018
PubMed
Summary

Detrended fluctuation analysis (DFA) reveals how scaling functions F(s) relate to Hurst exponents (H) for various stochastic processes. New methods allow accurate bias computation and handling of missing data in time series analysis.

Area of Science:

  • * Statistical physics
  • * Time series analysis
  • * Signal processing

Background:

  • * Detrended fluctuation analysis (DFA) is a method to characterize the scaling properties of time series.
  • * The scaling function F(s) in DFA is expected to follow a power law F(s)∼s^{H}, where H is the Hurst exponent.
  • * Existing DFA methods have limitations with non-stationary processes and missing data.

Purpose of the Study:

  • * To rigorously analyze the scaling behavior of the DFA fluctuation function F(s).
  • * To develop methods for computing finite-size bias in DFA.
  • * To introduce a novel DFA estimator capable of handling missing data.

Main Methods:

  • * Theoretical analysis of the fluctuation function F(s) in relation to autocorrelation functions (ACF) and structure functions.

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  • * Derivation of exact finite-size bias for scaling signals.
  • * Development and theoretical validation of a new DFA estimator, F[over ̂](s), for time series with gaps.

    • Main Results:

    • * The scaling law F(s)∼s^{H} is proven for stationary (0
    • * For H<0.5, the autocorrelation function (ACF) scales as ∼s^{1/2}.
    • * The fluctuation function F(s) is shown to be a weighted sum of the ACF and the second-order structure function.
    • * The proposed estimator F[over ̂](s) is demonstrated to be equal in expectation to F(s) for gap-free data under mild conditions.
    • * Previous modified DFA methods were shown to increase bias for certain Hurst exponent ranges (1

    Conclusions:

    • * The theoretical framework provides a deeper understanding of DFA and its limitations.
    • * The developed methods enable more accurate and robust DFA, even with imperfect data.
    • * The new estimator offers a practical solution for analyzing time series with missing values without interpolation.