Evaluation of the dispersional analysis method for fractal time series

Abstract

Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is <0.5 for rough anticorrelated series, >0.5 for positively correlated series, and =0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 − D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (>0.8) and low H (<0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H.Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.

Beta- and gamma-frequency coupling between olfactory and motor brain regions prior to skilled, olfactory-driven reaching

“…Confirmation of structure in the nonstationary LFP time series-To verify that the LFPs contained significant structure despite their "noisy" appearance, we employed a method -calculation of the Hurst coefficient (Bassingthwaighte and Raymond 1995) for each trial's LFPs-designed to detect multi-scale structure in non-linear, nonstationary time series such as brain local field potentials. Virtually all our analyzed LFPs had Hurst coefficients ≥0.8, indicating a degree of long-term correlation well above the random value of 0.50.…”

Editing RR Series and computation of long-term scaling parameters

“…Three methods and their correspondent long term scaling parameters were considered: Detrended Fluctuation Analysis (DFA) [11], the slope of the spectrum at low frequencies [12], and Dispersional Analysis (DA) [13]. The indexes were selected as the most representatives among those usually employed in clinical studies.…”

Use of the Hurst Exponent for Analysis of Electrocortical Epileptiform Activity Induced in Rats by Administration of Camphor Essential Oil or 1,8-Cineole

“…In the case where H = = 0.5, the process has short-range correlations typical of ordinary uncorrelated Brownian motion. Among various methods for determining the H, like rescaled range statistics [3], wavelet methods [16], etc., we have chosen in this study dispersion analysis (DA) [17] and an aggregation variance method (AGV) [18] taking into account their simplicity and fast computability.…”