Deriving dispersional and scaled windowed variance analyses using the correlation function of discrete fractional Gaussian noise

Abstract

Methods for estimating the fractal dimension, D, or the related Hurst coefficient, H, for a one-dimensional fractal series include Hurst's method of rescaled range analysis, spectral analysis, dispersional analysis, and scaled windowed variance analysis (which is related to detrended fluctuation analysis). Dispersional analysis estimates H by using the variance of the grouped means of discrete fractional Gaussian noise series (DfGn). Scaled windowed variance analysis estimates H using the mean of grouped variances of discrete fractional Brownian motion (DfBm) series. Both dispersional analysis and scaled windowed variance analysis have small bias and variance in their estimates of the Hurst coefficient. This study demonstrates that both methods derive their accuracy from their strict mathematical relationship to the expected value of the correlation function of DfGn. The expected values of the variance of the grouped means for dispersional analysis on DfGn and the mean of the grouped variance for scaled windowed variance analysis on DfBm are calculated. An improved formulation for scaled windowed variance analysis is given. The expected values using these analyses on the wrong kind of series (dispersional analysis on DfBm and scaled windowed variance analysis on DfGn) are also calculated.