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

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. Introduction Detrended Fluctuation Analysis (DFA) is a technique commonly applied to time series as a means of approximating the Hurst exponent, which indicates the degree of long-range temporal correlations present [1–4]. Long-range temporal correlations (LRTCs) occur in time series with an autocorrelation function that decays as a power law function of the lag [5]. The presence of LRTCs suggests that the underlying signal is governed by non-local behaviour, with all scales contributing to system behaviour. LRTCs have been detected in various biological time series and natural phenomena [1–3,6–9], see a review in [10]. In neurophysiological signals, it has been argued that LRTCs facilitate essential functions such as memory formation, rapid information transfer, and the efficient neural network reorganisation that promotes learning [11–17]. DFA produces estimates of the magnitude of detrended fluctuations at different scales (window sizes) of a time series and assesses the scaling relationship between estimates and time scales. Estimation of the Hurst exponent through DFA assumes self-similarity in the time series. If the signal is self-similar, then the detrended fluctuations will increase as a power law function of window size, and the relationship between the two can be visualised as a straight line on a log-log fluctuation plot [1, 2]. DFA returns the slope of the plot as its exponent with no check as to whether the self-similarity of the time series is supported by there being a linear fluctuation plot. At present there is no method which establishes the linearity of a DFA plot and an important shortcoming of the typically used method (see below) is that