The Hurst exponent is used as a measure of
long-term memory
Long-term memory (LTM) is the stage of the Atkinson–Shiffrin memory model in which informative knowledge is held indefinitely. It is defined in contrast to short-term and working memory, which persist for only about 18 to 30 seconds. Long-t ...
of
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
. It relates to the
autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
s of the time series, and the rate at which these decrease as the lag between pairs of values increases.
Studies involving the Hurst exponent were originally developed in
hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
for the practical matter of determining optimum dam sizing for the
Nile river
The Nile, , Bohairic , lg, Kiira , Nobiin: Áman Dawū is a major north-flowing river in northeastern Africa. It flows into the Mediterranean Sea. The Nile is the longest river in Africa and has historically been considered the longest rive ...
's volatile rain and drought conditions that had been observed over a long period of time.
The name "Hurst exponent", or "Hurst coefficient", derives from
Harold Edwin Hurst
Harold Edwin Hurst (1 January 1880 – 7 December 1978) was a British hydrology, hydrologist from Leicester. Hurst's (1951) study on measuring the long-term storage capacity of reservoirs documented the presence of long-range dependence in hydrolo ...
(1880–1978), who was the lead researcher in these studies; the use of the standard notation ''H'' for the coefficient also relates to his name.
In
fractal geometry
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
, the generalized Hurst exponent has been denoted by
''H'' or ''H
q'' in honor of both Harold Edwin Hurst and
Ludwig Otto Hölder (1859–1937) by
Benoît Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
(1924–2010).
''H'' is directly related to
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
, ''D'', and is a measure of a data series' "mild" or "wild" randomness.
The Hurst exponent is referred to as the "index of dependence" or "index of long-range dependence". It quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction. A value ''H'' in the range 0.5–1 indicates a time series with long-term positive autocorrelation, meaning both that a high value in the series will probably be followed by another high value and that the values a long time into the future will also tend to be high. A value in the range 0 – 0.5 indicates a time series with long-term switching between high and low values in adjacent pairs, meaning that a single high value will probably be followed by a low value and that the value after that will tend to be high, with this tendency to switch between high and low values lasting a long time into the future. A value of ''H''=0.5 can indicate a completely uncorrelated series, but in fact it is the value applicable to series for which the autocorrelations at small time lags can be positive or negative but where the absolute values of the autocorrelations decay exponentially quickly to zero. This in contrast to the typically
power law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
decay for the 0.5 < ''H'' < 1 and 0 < ''H'' < 0.5 cases.
Definition
The Hurst exponent, ''H'', is defined in terms of the asymptotic behaviour of the
rescaled range The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978). Its purpose is to provide an assessment of how the apparent variability of a series changes with ...
as a function of the time span of a time series as follows;
:
where;
*
is the
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
of the first
cumulative deviations from the mean
*
is the series (sum) of the first n
standard deviations
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
*
is the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
*
is the time span of the observation (number of data points in a time series)
*
is a constant.
Relation to Fractal Dimension
For self-similar time series,
''H'' is directly related to
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
, ''D'', where 1 < ''D'' < 2, such that ''D'' = 2 - ''H''. The values of the Hurst exponent vary between 0 and 1, with higher values indicating a smoother trend, less volatility, and less roughness.
For more general time series or multi-dimensional process, the Hurst exponent and fractal dimension
can be chosen independently, as the Hurst exponent represents structure over asymptotically longer
periods, while fractal dimension represents structure over asymptotically shorter periods.
Estimating the exponent
A number of estimators of long-range dependence have been proposed in the literature. The oldest and best-known is the so-called
rescaled range The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978). Its purpose is to provide an assessment of how the apparent variability of a series changes with ...
(R/S) analysis popularized by Mandelbrot and Wallis
and based on previous hydrological findings of Hurst.
Alternatives include
DFA, Periodogram regression, aggregated variances, local Whittle's estimator, wavelet analysis, both in the
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the cas ...
and
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
.
Rescaled range (R/S) analysis
To estimate the Hurst exponent, one must first estimate the dependence of the
rescaled range The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978). Its purpose is to provide an assessment of how the apparent variability of a series changes with ...
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
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
;
:
2. Create a mean-adjusted series;
:
3. Calculate the cumulative deviate series
;
:
4. Compute the range
;
:
5. Compute the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
;
:
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
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
to the data. This can be done by plotting