A maximum likelihood based technique for validating detrended fluctuation analysis (ML-DFA)

Abstract

Detrended Fluctuation Analysis (DFA) is widely used to assess the presence of long-range temporal correlations in time series. Signals with long-range temporal correlations are typically defined as having a power law decay in their autocorrelation function. The output of DFA is an exponent, which is the slope obtained by linear regression of a log-log fluctuation plot against window size. However, if this fluctuation plot is not linear, then the underlying signal is not self-similar, and the exponent has no meaning. There is currently no method for assessing the linearity of a DFA fluctuation plot. Here we present such a technique, called ML-DFA. We scale the DFA fluctuation plot to construct a likelihood function for a set of alternative models including polynomial, root, exponential, logarithmic and spline functions. We use this likelihood function to determine the maximum likelihood and thus to calculate values of the Akaike and Bayesian information criteria, which identify the best fit model when the number of parameters involved is taken into account and over-fitting is penalised. This ensures that, of the models that fit well, the least complicated is selected as the best fit. We apply ML-DFA to synthetic data from FARIMA processes and sine curves with DFA fluctuation plots whose form has been analytically determined, and to experimentally collected neurophysiological data. ML-DFA assesses whether the hypothesis of a linear fluctuation plot should be rejected, and thus whether the exponent can be considered meaningful. We argue that ML-DFA is essential to obtaining trustworthy results from DFA.