On the Validity of Detrended Fluctuation Analysis at Short Scales

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 [Formula: see text] , which quantifies the fluctuations of the time series around the local trend, which is substracted (detrended). If the time series exhibits scaling properties, then [Formula: see text] asymptotically, and the scaling exponent [Formula: see text] is typically estimated as the slope of a linear fitting in the [Formula: see text] vs. [Formula: see text] plot. In this way, [Formula: see text] 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 [Formula: see text] vs. [Formula: see text] plots with two different slopes, [Formula: see text] at short scales and [Formula: see text] 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 [Formula: see text] is asymptotic, we question the use of [Formula: see text] 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 [Formula: see text] valid for all scales, DFA provides an [Formula: see text] value that systematically overestimates the true exponent [Formula: see text]. In addition, second, when artificial time series with two different scaling exponents at short and large scales are considered, the [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] value would not characterize properly the short scale behavior of the dynamical system.