Hurst Exponent - Estimating The Exponent

Estimating The Exponent

To estimate the Hurst exponent, one must first estimate the dependence of the rescaled range on the time span n of observation. A time series of full length N is divided into a number of shorter time series of length n = N, N/2, N/4, ... The average rescaled range is then calculated for each value of n.

For a (partial) time series of length, the rescaled range is calculated as follows:

1. Calculate the mean;

2. Create a mean-adjusted series;

3. Calculate the cumulative deviate series ;

4. Compute the range ;

5. Compute the standard deviation ;

6. Calculate the rescaled range and average over all the partial time series of length

The Hurst exponent is estimated by fitting the power law to the data. This can be done by plotting the logarithm of as a function of, and fitting a straight line; the slope of the line gives . Such a graph is called a pox plot. A more principled approach fits the power law in a maximum-likelihood fashion.