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Hurst exponent

Contents

Idea

As Wikipedia states:

In fractal geometry, the generalized Hurst exponent, named H in honor of both Harold Edwin Hurst (1880–1978) and Ludwig Otto Hölder (1859–1937) by Benoît Mandelbrot (1924-2010), is referred to as the “index of dependence,” and is the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction.

H was originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river’s volatile rain and drought conditions that had been observed over a long period of time. The Hurst exponent is non-deterministic in that it expresses what is actually observed in nature; it is not calculated so much as it is estimated. The Hurst exponent is used as a measure of the long term memory of time series, i.e. the autocorrelation of the time series. Where a value of 0 < H < 0.5 indicates a time series with negative autocorrelation (e.g. a decrease between values will probably be followed by an increase), and a value of 0.5 < H < 1 indicates a time series with positive autocorrelation (e.g. an increase between values will probably be followed by another increase). A value of H=0.5 indicates a true random walk, where it is equally likely that a decrease or an increase will follow from any particular value (e.g. the time series has no memory of previous values)

Details

Mathematics

Applications

Most commonly i time series analysis. It is used in two dimensions in digital image processing?, to do image enhancements.

Reference