# Evaluating rescaled range analysis for time series

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DOI: 10.1007/BF02368250

- Cite this article as:
- Bassingthwaighte, J.B. & Raymond, G.M. Ann Biomed Eng (1994) 22: 432. doi:10.1007/BF02368250

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## 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 τ of each segment of the set of segments into which the data set has been divided. Over a wide range of τ's the relationship is:*R/S=aτ*^{M}, where*k* 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*<0.72. Hurst analysis without trend correction differs by finding the range*R* of accumulation of differences from the global mean over the total period of data accumulation, rather than from the mean over each τ. The trend-corrected method gives better estimates of*H* on Brownian fractal signals of known*H* when*H*≥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.