Rescaled range analysis and the fractal dimension of pi

P. Vanouplines

What is the rescaled range analysis?

Hurst (1965) developed the rescaled range analysis, a statistical method to analyse long records of natural phenomena. There are two factors used in this analysis: firstly the range R, this is the difference between the minimum and maximum 'accumulated' values or cumulative sum of X(t,tau) of the natural phenomenon at discrete integer-valued time t over a time span tau, and secondly the standard deviation S, estimated from the observed values Xi(t). Hurst found that the ratio R/S is very well described for a large number of natural phenomena by the following empirical relation:

where tau is the time span, and H the Hurst exponent. The coefficient c was taken equal to 0.5 by Hurst. R and S are defined as:

and

where:

and

This method handles observations in time. The graphical representation uses time in the abscissa, and the observed value in the ordinate. This is comparable to what we have with the pi-digits: the position in the decimal expansion is represented on the x-axis, the (derived) value of the digit on the y-axis.

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Last updated on Sunday 19 November 1995.
©Patrick Vanouplines.