Evaluating rescaled range analysis for time series

The trend-corrected method gives better estimates ofH on Brownian fractal signals of knownH whenH≥0.5, that is, for signals with positive correlations between neighboring elements, and Hurst's 1951 conclusion that many natural phenomena exhibit not random but correlated time series is strongly affirmed.

Abstract

Rescaled range analysis is a means of characterizing a time series or a one-dimensional (1-D) spatial signal that provides simultaneously a measure of variance and of the long-term correlation or "memory," The trend-corrected method is based on the statistical self-similarity in the signal: in the standard approach one measures the ratio R/S on the range R of the sum of the deviations from the local mean divided by the standard deviation S from the mean. For fractal signals R/S is a power law function of the length tau of each segment of the set of segments into which the data set has been divided. Over a wide range of tau's the relationship is: R/S = a tau H, where kappa is a scalar and the H is the Hurst exponent. (For a 1-D signal f(t), the exponent H = 2 - D, with D being the fractal dimension.) The method has been tested extensively on fractional Brownian signals of known H to determine its accuracy, bias, and limitations. R/S tends to give biased estimates of H, too low for H > 0.72, and too high for H or = 0.5, that is, for signals with positive correlations between neighboring elements. Rescaled range analysis has poor convergence properties, requiring about 2,000 points for 5% accuracy and 200 for 10% accuracy. Empirical corrections to the estimates of H can be made by graphical interpolation to remove bias in the estimates. Hurst's 1951 conclusion that many natural phenomena exhibit not random but correlated time series is strongly affirmed.