The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations 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 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 name "Hurst exponent", or "Hurst coefficient", derives from Harold Edwin Hurst (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, the generalized Hurst exponent has been denoted by H or Hq in honor of both Harold Edwin Hurst and Ludwig Otto Hölder (1859–1937) by Benoît Mandelbrot (1924–2010). H is directly related to fractal dimension, 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 that the decay in autocorrelation is slower than exponential, following a power law; for the series it means that a high value tends to be followed by another high value and that future excursions to more high values do occur. 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, also following a power law. A value of H=0.5 indicates short-memory, with (absolute) autocorrelations decaying exponentially quickly to zero.
Definition The Hurst exponent, H, is defined in terms of the asymptotic behaviour of the rescaled range as a function of the time span of a time series as follows;
E [ R ( n ) S ( n ) ] = C n H as n → ∞ , {\displaystyle \mathbb {E} \left[{\frac {R(n)}{S(n)}}\right]=Cn^{H}{\text{ as }}n\to \infty \,,}
where
R ( n ) {\displaystyle R(n)} is the range of the first n {\displaystyle n} cumulative deviations from the mean
S ( n ) {\displaystyle S(n)} is the series (sum) of the first n standard deviations
E [ x ] {\displaystyle \mathbb {E} \left[x\right]\,} is the expected value
n {\displaystyle n} is the time span of the observation (number of data points in a time series)
C {\displaystyle C} is a constant.
Relation to Fractal Dimension For self-similar time series, H is directly related to fractal dimension, 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 (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 and frequency domain.
Rescaled range (R/S) analysis 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 nonoverlapping shorter time series of length n, where n takes values N, N/2, N/4, ... (in the convenient case that N is a power of 2). The average rescaled range is then calculated for each value of n. For each such time series of length n {\displaystyle n} , X = X 1 , X 2 , … , X n {\displaystyle X=X_{1},X_{2},\dots ,X_{n}\,} , the rescaled range is calculated as follows:
Calculate the mean; m = 1 n ∑ i = 1 n X i . {\displaystyle m={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\,.}
Create a mean-adjusted series; Y t = X t − m for t = 1 , 2 , … , n . {\displaystyle Y_{t}=X_{t}-m\quad {\text{ for }}t=1,2,\dots ,n\,.}
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