ANALYZING EXACT FRACTAL TIME-SERIES - EVALUATING DISPERSIONAL ANALYSIS AND RESCALED RANGE METHODS

Precise reference signals are required to evaluate methods for charact erizing a fractal time series. Here we use fGp (fractional Gaussian pr ocess) to generate exact fractional Gaussian noise (fGn) reference sig nals for one-dimensional time series. The average autocorrelation of m ultiple realizations of fGn converges to the theoretically expected au tocorrelation. Two methods commonly used to generate fractal time seri es, an approximate spectral synthesis (SSM) method and the successive random addition (SRA) method, do not give the correct correlation stru ctures and should be abandoned. Time series from fGp were used to test how well several versions of rescaled range analysis (RIS) and disper sional analysis (Disp) estimate the Hurst coefficient(0 < H < 1.0). Di sp is unbiased for H < 0.9 and series length N greater than or equal t o 1024, but underestimates H when H > 0.9. R/S-detrended overestimates H for time series with H < 0.7 and underestimates H for H > 0.7. Esti mates of H((H) over cap)) from all versions of Disp usually have lower bias and variance than those from R/S. All versions of dispersional a nalysis, Disp, now tested on fGp, are better than we previously though t and are recommended for evaluating time series as long-memory proces ses.