Hurst Exponent - Generalized Exponent

Generalized Exponent

The basic Hurst exponent can be related to the expected size of changes, as a function of the lag between observations, as measured by E(|X_t+τ-X_t|2). For the generalized form of the coefficient, the exponent here is replaced by a more general term, denoted by q.

There are a variety of techniques that exist for estimating H, however assessing the accuracy of the estimation can be a complicated issue. Mathematically, in one technique, the Hurst exponent can be estimated such that:

H_q = H(q),

for a time series

g(t) (t = 1, 2,...)

may be defined by the scaling properties of its structure functions S_q:

where q > 0, is the time lag and averaging is over the time window

usually the largest time scale of the system.

Practically, in nature, there is no limit to time, and thus H is non-deterministic as it may only be estimated based on the observed data; e.g., the most dramatic daily move upwards ever seen in a stock market index can always be exceeded during some subsequent day.

H is directly related to fractal dimension, D, 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.

In the above mathematical estimation technique, the function H(q) contains information about averaged generalized volatilities at scale (only q = 1, 2 are used to define the volatility). In particular, the H₁ exponent indicates persistent (H₁ > ½) or antipersistent (H₁ < ½) behavior of the trend.

For the BRW (brown noise, 1/f²) one gets

H_q = ½,

while for the pink noise (1/f) and white noise we have

H_q = 0.

For the popular Lévy stable processes and truncated Lévy processes with parameter α it has been found that

H_q = q/α for q < α and H_q = 1 for q ≥ α.

A method to estimate from non-stationary time series is called detrended fluctuation analysis. When is a non-linear function of q the time series is a multifractal system.