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.

Model selection for identifying power-law scaling

“…The proportions of realizations assigned as power-law as well as the α values closely resemble the results in [21].…”

“…When a signal is contaminated by a nontrivial, non-linear trend (e.g., a sinusoid), the fluctuation plots may have a scaledependent slope [15]. Then, the slope of the linear regression over a (too) large range does not necessarily represent the scaling behavior of interest [19,20,21].…”

“…We proposed an approach to assess (the presence of) linearity in the log-log fluctuation plots in DFA. Inspired by [21], we used a maximum likelihood estimate with the likelihood function L defined by means of the density functions of the mean-squared fluctuations. This definition allows for using the function L in assessing power-law behavior and estimating the scaling exponent α.…”

“…Due to its roots in information theory, the comparison between model and data is performed in terms of probability densities. In line with the traditional Kullback-Leibner approach, this boils down to evaluating terms like p(x) ln(q(x|θ))dx where p represents the 'truth' and q(x|θ) the probability density originating from an approximating model with parameters θ. Interestingly, in the work of Botcharova and co-workers [21] the log-likelihood function was defined as ln (L B ) = n log (F s ) ln (f θ ) with F s denoting a rescaled version of the fluctuation magnitudes Fi . The fitted model f θ was normalized such that it integrated to 1.…”

Identification of Criticality in Neuronal Avalanches: II. A Theoretical and Empirical Investigation of the Driven Case

“…DFA is one method by which to estimate the Hurst exponent and was chosen here as it has been shown to be an accurate estimate [45]. Moreover, it is a graphical approach and so can be used to check for crossover points [47] -see also the recently proposed ML-DFA [54] whereby the existence of crossover behaviour can be rigorously tested. As the Hurst exponent can only be estimated it is is considered best practice to check the consistency of the exponents using two methods [55].…”