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 F(l), which quantifies the fluctuations of the time series around the local trend, which is substracted (detrended). If the time series exhibits scaling properties, then F(l)& SIM;l alpha asymptotically, and the scaling exponent alpha is typically estimated as the slope of a linear fitting in the logF(l) vs. log(l) plot. In this way, alpha 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 logF(l) vs. log(l) plots with two different slopes, alpha 1 at short scales and alpha 2 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 F(l) is asymptotic, we question the use of alpha 1 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 alpha valid for all scales, DFA provides an alpha 1 value that systematically overestimates the true exponent alpha. In addition, second, when artificial time series with two different scaling exponents at short and large scales are considered, the alpha 1 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 alpha 1 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 alpha 1 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 alpha 1 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 alpha 1 value would not characterize properly the short scale behavior of the dynamical system.

Automated summary

Detrended Fluctuation Analysis (DFA) is a method for quantifying correlations and scaling properties in real-world time series. However, when the scaling behavior differs at short and long scales, using alpha1 to describe the short-scale properties is problematic.