Hurst exponent (Rev #2, changes)

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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)~~

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)

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

category: mathematical methods

- Hurst exponent on Wikipedia
- John Russ CRC Press 2002 The imaging Process Handbook

hurst exponent?