Hurst exponent

In its turn, p and Hurst exponent H are connected between themselves like this [5] ... 

Successive increments of mathematical fractal random processes are independent of the time step. Here D = 1.5 corresponds to a completely uncorrelated random process r = 0, such as Brownian motion, and D = 1.0 corresponds to a completely correlated process r= 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 

Using DEA, we have established that there are statistical processes for which 8 = H and statistical processes for which 8 H, both of which scale. However, there is a third class of processes for which the scaling index is a function of the Hurst exponent, but the relation is not one of their being equal. This third class is the Levy random walk process (Levy diffusion) introduced by Shlesinger et al. [65] in their discussion of the application of Levy statistics to the understanding of turbulent fluid flow. 

If we make the association of the order of the fractional operator with the Hurst exponent... 

Consequently we have that p(0) = H so that x(0) = 2 — H, as it should because of the well-known relation between the fractal dimension and the global Hurst exponent Dq = 2 — H. 

The Hurst exponent H characterizes the roughness of the saturated interface. As for the local slope p, it is literally the approximate average slope of the local structure. Its value is related to the intersection of C2(r, t) with the vertical axis. For normal scaling, it does not depend on time. - If p does depend on time, the scaling becomes anomalous. As one can see, to be able to characterize a growing interface, one needs the height-height correlation function C2(r, t), and more specifically, the critical exponents FI and p. 

Hurst exponent Hamiltonian transition matrix n X n identity matrix... 

This exponent corresponds to a symmetric a-stable Levy process 8 (0, a Levy flight, which is self-similar with Hurst exponent H = Xja. It follows from (3.89) that the mesoscopic density of particles is the solution to the space-fractional diffusion equation [371] ... 

The theory of fractal dimension may be used in bioimpedance signal analysis, for example, for studying time series. Such analysis is often done by means of Hurst s rescaled range analysis (R/S analysis), which characterizes the time series by the so-called Hurst exponent H = 2 — D. Hurst found that the rescaled range often can be described by the empirical relation... 

One key advantage of the box-counting dimension Dbc over the similarity dimension Ds is that Due can be used to evaluate the dimension of self-affine sets. In these sets, however. Due is not uniquely defined instead, it assumes two different values a local or small-scale value and a global or large-scale value [e.g. [10 (p. 187), 31 (p. 55), 35 (p. 8)]. In the case of the fractional Brownian motion (Section 2.2.5), the local Due value is equal to the Hausdorff dimension and is given hy2- H, where H is the Hurst exponent, whereas the global value of Dbc = 1 [e.g. 10 (p. 189)]. 

Software. TruSoft International s BENOIT fractal analysis system enables the user to measure the fractal dimension and the Hurst exponent of data sets using eleven different methods. 

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