Detrended fluctuation analysis of scaling crossover effects

Detrended fluctuation analysis (DFA) is one of the most frequently used fractal time series algorithms. DFA has also become the tool of choice for analysis of the short-time fluctuations despite the fact that its validity in this domain has never been demonstrated. We adopt an Ornstein-Uhlenbeck Langevin equation to generate a time series which exhibits short-time powerlaw scaling and incorporates the fundamental property of physiological control systems —negative feedback. To determine the scaling exponent, we derive the analytical expressions for the standard deviation of the solution X(t) of this equation using both the ensemble of statistically independent trajectories and the ensemble obtained by partitioning a single trajectory. The latter approach is used in DFA and many other physiological applications. Surprisingly, the formulas for the standard deviations are different for these two ensembles. We demonstrate that the partitioning amounts to building up deterministic trends that satisfy the “trend + signal” decomposition assumption which is characteristic of DFA. Consequently, the dependence of the rms of DFA residuals F (τ) on the length τ of data window is the same for both ensembles. The growth of F (τ) is significantly different from that of the standard deviation of X(t). While the DFA estimate of the shorttime scaling exponent is correct, the polynomial detrending delays the approach of F (τ) to the asymptotic value by as much as an order of magnitude. This delay may underlie the gradual change of the DFA scaling index typically observed in time series that exhibit crossover between the shortand long-time scaling. editor’s choice Copyright c © EPLA, 2010 Detrended fluctuation analysis (DFA) [1] is one of the most frequently used algorithms for fractal analysis of experimental time series. Bashan et al. [2] mention that there are about 500 papers on DFA. This method was developed to improve on the original scaled window variance analysis of Mandelbrot [3,4]. In particular, DFA allows one to detect long-time, power-law scaling of the second moment of the time series fluctuations in the presence of additive, polynomial non-stationarities. Physiological time series, such as electroencephalogram (EEG), electrocardiogram (ECG), waveforms of arterial blood pressure (ABP) and cerebral blood flow velocity are notoriously non-stationary, which was, in part, the reason for the rapid adoption of this algorithm. The initial application of DFA to heart rate variability, namely to inter-beat interval time series, revealed the existence of two scaling (a)E-mail: [email protected] regimes with the crossover taking place at approximately 10 heart beats [5]. Interestingly enough, the short-time scaling exponent turned out to be clinically significant. For example, this measure was the most accurate predictor of all-cause mortality in a cohort of 446 survivors of acute myocardial infarction [6]. The “two exponent” approach was used to quantify heart rate variability in various physiological conditions [7–11], dynamics of arterial blood pressure [12,13] and cerebral blood flow [14]. This prompts the question: Is DFA applicable in the short-time limit? Before we pose a second question, let us briefly describe the DFA algorithm. Given a time series {ξj}j=1, the zero-centered time series is aggregated: