Hurst Exponent

The Hurst exponent is a scaling exponent used to quantify long-range dependence and roughness in a time series. In the rescaled-range definition, if is the statistical range of the cumulative sums of deviations from the sample mean in a block of length and is its sample standard deviation, then

as grows. Brownian motion has . In models for which this scaling describes correlations between increments, a process with is a persistent process, while one with is an antipersistent process. For a self-affine graph with stationary increments, its fractal dimension is commonly related by . Here denotes the expectation value.

In rescaled-range analysis, is estimated from the slope obtained by least squares fitting of against over a selection of block sizes.

For example, consider the eight observations 2, 4, 3, 6, 8, 7, 9, and 5. Their sample mean is , and the cumulative sums of their deviations from the sample mean range from to . Consequently, . The sample standard deviation is , so this block contributes

An estimate of repeats this computation for many blocks at several block lengths and fits the slope of the resulting logarithms rather than using a single block.