Spurious Results of Fluctuation Analysis Techniques in Magnitude and Sign Correlations

Pedro Carpena,Concepción Carretero-Campos,Pedro Bernaola-Galván,Ana Coronado,Manuel Gómez-Extremera

doi:10.3390/e19060261

Pedro Carpena, Concepción Carretero-Campos + Show 3 more

Open Access

https://doi.org/10.3390/e19060261

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Journal: Entropy	Publication Date: Jun 7, 2017
Citations: 15	License type: CC BY 4.0

Affiliation: Universidad de Málaga

Abstract

Fluctuation Analysis (FA) and specially Detrended Fluctuation Analysis (DFA) are techniques commonly used to quantify correlations and scaling properties of complex time series such as the observable outputs of great variety of dynamical systems, from Economics to Physiology. Often, such correlated time series are analyzed using the magnitude and sign decomposition, i.e., by using FA or DFA to study separately the sign and the magnitude series obtained from the original signal. This approach allows for distinguishing between systems with the same linear correlations but different dynamical properties. However, here we present analytical and numerical evidence showing that FA and DFA can lead to spurious results when applied to sign and magnitude series obtained from power-law correlated time series of fractional Gaussian noise (fGn) type. Specifically, we show that: (i) the autocorrelation functions of the sign and magnitude series obtained from fGns are always power-laws; However, (ii) when the sign series presents power-law anticorrelations, FA and DFA wrongly interpret the sign series as purely uncorrelated; Similarly, (iii) when analyzing power-law correlated magnitude (or volatility) series, FA and DFA fail to retrieve the real scaling properties, and identify the magnitude series as purely uncorrelated noise; Finally, (iv) using the relationship between FA and DFA and the autocorrelation function of the time series, we explain analytically the reason for the FA and DFA spurious results, which turns out to be an intrinsic property of both techniques when applied to sign and magnitude series.

Highlights

Since the observation of the Hurst effect [1] in the Nile river, a huge number of dynamical systems whose observable outputs are time series with complex long-range power-law correlations and scaling properties have been identified
Detrended Fluctuation Analysis (DFA) has probably become the standard method of choice when analyzing complex time series and it has been used in hundreds of scientific articles
When Fluctuation Analysis (FA) and DFA are applied to these series, the results shown in Figure 2 and summarized in Equations (20) and (21) spuriously indicate a different behavior

Summary

Introduction

Since the observation of the Hurst effect [1] in the Nile river, a huge number of dynamical systems whose observable outputs are time series with complex long-range power-law correlations and scaling properties have been identified. Increment time series with identical linear correlations may well correspond to systems with different nonlinear and multifractal behavior [4,5] To overcome this problem and break the possible degeneration, the magnitude and sign decomposition method was proposed [4], consisting of studying separately the correlation properties of the magnitude and sign of the increment time series, typically using DFA or FA. We use fractional Gaussian noises with different correlation strengths as a model for typical increment time series, apply to them the magnitude and sign decomposition method and study their correlations by using both FA and DFA.

Fluctuation Analysis and Detrended Fluctuation Analysis

The Relationship of the Autocorrelation Function with FA and DFA

Magnitude and Sign Study Using FA and DFA

F A p u t

Exact Autocorrelation Function of Magnitude and Sign Series

Analytical FA and DFA Scaling on Sign and Magnitude Series

Conclusions