Definition of the fluctuation function in the detrended fluctuation analysis and its variants

Bastien Berthelot (), Eric Grivel (), Pierrick Legrand () and Audrey Giremus ()
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Bastien Berthelot: THALES AVS France
Eric Grivel: Bordeaux University-INP Bordeaux-IMS-UMR CNRS 5218
Pierrick Legrand: Bordeaux University-IMB UMR CNRS 5251-ASTRAL team, INRIA
Audrey Giremus: Bordeaux University-IMS-UMR CNRS 5218

The European Physical Journal B: Condensed Matter and Complex Systems, 2021, vol. 94, issue 11, 1-20

Abstract: Abstract The detrended fluctuation analysis (DFA) and its variants are popular methods to analyze the self-similarity of a signal. Two steps characterize them: firstly, the trend of the centered integrated signal is estimated and removed. Secondly, the properties of the so-called fluctuation function which is an approximation of the standard deviation of the resulting process is analyzed. However, it appears that the statistical mean was assumed to be equal to zero to obtain it. As there is no guarantee that this assumption is true a priori, this hypothesis is debatable. The purpose of this paper is to propose two alternative definitions of the fluctuation function. Then, we compare all of them based on a matrix formulation and the filter-based interpretation we recently proposed. This analysis will be useful to show that the approach proposed in the original paper remains a good compromise in terms of accuracy and computational cost. Graphic abstract

Date: 2021
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DOI: 10.1140/epjb/s10051-021-00231-7

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