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In fractal geometry, the generalized Hurst exponent, named H in honor of both Harold Edwin Hurst (1880-1978) and Ludwig Otto Hölder (1859-1937) by Benoît Mandelbrot, is referred to as the "index of dependence," and is the relative tendency of a time series to either strongly regress to the mean or 'cluster' in a direction. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" H is the eighth letter of the Latin alphabet H may also refer to H (S-train, a train service in Metropolitan Copenhagen Denmark Harold Edwin Hurst (1880-1978 was a British hydrologist Hurst's (1951 study on measuring the long-term storage capacity of reservoirs documented the presence of Long-range dependence Otto Ludwig Hölder ( December 22, 1859 - August 29, 1937) was a German Mathematician born in Stuttgart. Benoît B Mandelbrot (born 20 November 1924 is a French mathematician, best known as the father of fractal geometry. In Statistics, Signal processing, and many other fields a time series is a sequence of Data points measured typically at successive times spaced at (often [1]

H was 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. Hydrology (from Greek Yδωρ hudōr, "water" and λόγος logos, "study" is the study of the movement distribution and quality of The Nile (النيل, Ancient Egyptian iteru or Ḥ'pī, Coptic piaro or phiaro) is a major north-flowing River The Hurst exponent is non-deterministic in that it expresses what is actually observed in nature; it is not calculated so much as it is estimated. The accuracy and fidelity of H are also limited in the extreme for small measurements of roughness by the Heisenberg Uncertainty Principle, a boundary for all measurements. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain

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:

Hq = H(q),

for a time series

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

may be defined by the scaling properties of its structure functions Sq(τ):

S_q = \langle |g(t + \tau) - g(t)|^q \rangle_T \sim \tau^{qH(q)}, \,

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

T \gg \tau,\,

usually the largest time scale of the system. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations,

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. In Fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a Fractal appears to fill space as 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 τ (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

Hq = ½,

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

Hq = 0. In Science, Brownian noise ( also known as Brown noise or red noise, is the kind of Signal noise produced by Brownian motion Pink noise or 1/f noise is a signal or process with a Frequency spectrum such that the power spectral density is Proportional White noise is a random signal (or process with a flat Power spectral density.

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

Hq = q/α for q < α and Hq = 1 for q ≥ α. In Probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a four parameter family of continuous

Note added Oct. 2007: in the above definition two separate requirements are mixed together as if they would be one. Here are the two independent requirements: (i) stationarity of the increments, x(t+T)-x(t)=x(T) in distribution. this is the condition that yields long time autocorrelations. (ii) Self-similarity of the stochastic then yields variance scaling, but is not needed for long time memory. E. g. , both Markov processes (i. e. , memory-free processes) and fractional Brownian motion scale at the level of 1-point densities (simple averages), but neither scales at the level of pair correlations or, correspondingly, the 2-point probability density.

An efficient market requires a martingale condition, and unless the variance is linear in the time this produces nonstationary increments, x(t+T)-x(t)≠x(T). Martingales are Markovian at the level of pair correlations, meaning that pair correlations cannot be used to beat a martingale market. Stationary increments with nonlinear variance, on the other hand, induce the long time pair memory of fBm that would make the market beatable at the level of pair correlations. Such a market would necessarily be far from "efficient".

References

References for note added Oct. , 2007: Joseph L. McCauley, Kevin E. Bassler, and Gemunu H. Gunaratne , Martingales, Detrending Data, and the Efficient Market Hypothesis, Physica A37, 202, 2008.

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