  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.
