math

Hurst exponent

A dimensionless measure of long-term memory and self-similarity in a time series, indicating persistence or anti-persistence.
8 related articles

Description

The Hurst exponent is a single number that tells you how much a time series—like river levels, stock prices, or temperature records—'remembers' its own past. If the exponent is above 0.5, the series is persistent: a high value tends to be followed by another high value, and a low by a low, creating long-term trends. If it is below 0.5, the series is anti-persistent: it tends to flip direction, so a rise is likely followed by a fall. An exponent exactly 0.5 means the series is completely random, with no memory at all.

History

It was introduced by the British hydrologist Harold Edwin Hurst in 1951 while he was studying the Nile River's long-term flooding patterns, which led him to develop the rescaled range analysis.

How it works

The exponent is derived from how the range of cumulative deviations from the mean scales with the length of the time window. For a self-similar series, the rescaled range (the range divided by the standard deviation) grows as a power of the window length, and that power is the Hurst exponent. It is estimated by plotting the rescaled range against window size on a log-log scale and measuring the slope.

In practice

It is used in finance to assess market efficiency, in hydrology for dam design, and in climate science to detect long-term persistence in temperature or precipitation records.

Fun fact

Hurst's original data on Nile floods showed an exponent near 0.7, meaning the river's behavior was far more persistent than random, a finding that surprised statisticians of his time.

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