Evaluation of the dispersional analysis method for fractal time series

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.