On the Validity of Detrended Fluctuation Analysis at Short Scales.

Pedro Carpena,Manuel Gómez-Extremera,Pedro A Bernaola-Galván

doi:10.3390/e24010061

Abstract

Detrended Fluctuation Analysis (DFA) has become a standard method to quantify the correlations and scaling properties of real-world complex time series. For a given scale ℓ of observation, DFA provides the function , which quantifies the fluctuations of the time series around the local trend, which is substracted (detrended). If the time series exhibits scaling properties, then asymptotically, and the scaling exponent is typically estimated as the slope of a linear fitting in the vs. plot. In this way, measures the strength of the correlations and characterizes the underlying dynamical system. However, in many cases, and especially in a physiological time series, the scaling behavior is different at short and long scales, resulting in vs. plots with two different slopes, at short scales and at large scales of observation. These two exponents are usually associated with the existence of different mechanisms that work at distinct time scales acting on the underlying dynamical system. Here, however, and since the power-law behavior of is asymptotic, we question the use of to characterize the correlations at short scales. To this end, we show first that, even for artificial time series with perfect scaling, i.e., with a single exponent valid for all scales, DFA provides an value that systematically overestimates the true exponent . In addition, second, when artificial time series with two different scaling exponents at short and large scales are considered, the value provided by DFA not only can severely underestimate or overestimate the true short-scale exponent, but also depends on the value of the large scale exponent. This behavior should prevent the use of to describe the scaling properties at short scales: if DFA is used in two time series with the same scaling behavior at short scales but very different scaling properties at large scales, very different values of will be obtained, although the short scale properties are identical. These artifacts may lead to wrong interpretations when analyzing real-world time series: on the one hand, for time series with truly perfect scaling, the spurious value of could lead to wrongly thinking that there exists some specific mechanism acting only at short time scales in the dynamical system. On the other hand, for time series with true different scaling at short and large scales, the incorrect value would not characterize properly the short scale behavior of the dynamical system.

Highlights

Since a great diversity of real-world dynamical systems exhibit observable time series outputs characterized by scaling properties and complex correlations structure, many techniques have been developed in the last two decades to analyze this kind of time series and quantify adequately their properties, with Detrended Fluctuation Analysis (DFA) [1]
The curvature produces a systematic overestimation of α1, which is in all cases larger than the correct exponent α, α1 > α. We show that this overestimation is not due to effects produced by the finite time series length, but an intrinsic limitation of DFA, which only recovers the true scaling exponent α at larger scales of observation
Many authors, especially in the field of physiology in the analysis of cardiac signals, study the scaling properties of the experimental time series by applying separately DFA at short and large scales of observation, characterizing the time series by two exponents, α1 and α2, corresponding to short and large scales, respectively. If both exponents are different, and this happens very often, the difference is attributed to the existence of different mechanisms controlling the underlying dynamical system which act at different time scales, short and long range

Summary

Introduction

In this case, we show the overestimation effect described above, and systematically quantify it as a function of the true scaling exponent, and of the fitting range considered to estimate it. In many real-world time series, the autocorrelation function is not convenient to determine the exponent γ (or H), since C (r ) is noisy and very sensitive to the time series size N [16,27], and it is only properly estimated for large N, very often not available in real experiments This is the reason motivating the use of indirect methods to quantify correlations and scaling, such as Detrended Fluctuation Analysis (DFA), which is one of the most widely used. In the case of non-stationary time series, we have used the standard DFA algorithm (4)

Detrended Fluctuation Analysis at Short Scales

Behavior of α1 in Time Series with Perfect Scaling

Behavior of α1 for Time Series with Scaling Crossovers

Discussion and Conclusions