Statistical self-similarity

Various means of constructing self-similar surfaces are known.33 Some of them do not allow one to produce different realizations of surface profiles, for example, by making use of the Weierstrass function. These methods should be avoided in the current context because it would be difficult to make statistically meaningful statements without averaging over a set of statistically independent simulations. An appropriate method through which to construct self-similar surfaces i s to use a representation of the height profile h(x) via its Fourier transform h(q). 

The set S is statistically self-similar when S is the union of N distinct subsets each of which is scaled down by r from the original and is congruent to r(S) in all statistical respects. For... 

Statistical self-similarity holds when the normalized grain-size distribution and the number-of-sides distribution remain constant during growth. A grain structure at a later time then looks statistically similar to itself at an earlier time, except for a uniform magnification, and the structures are therefore scaled with respect to each other. 

Monte Carlo methods, direct tracking methods, and vertex models, where the evolution of the two-dimensional grain structure is described in terms of the motion of the vertices. After initial transients, all of these simulations exhibit statistical self-similarity during growth and an average grain area that increases linearly with time according to Eq. 15.35. 

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

S. Lee, R. Rao, Scale-based formulations of statistical self-similarity in images, Proceedings of the International Conference on Image Processing, ICIP, vol. 4 (2004), pp. 2323-2326... 

The simplest fractals are mathematical constructs that replicate a given structure at all scales, thus forming a scale-invariant structure which is self-similar. Most natural phenomena, such as colloidal aggregates, however, form a statistical self-similarity over a reduced scale of applicability. For example, a colloidal aggregate would not be expected to contain (statistical) self-similarity at a scale smaller than the primary particle size or larger than the size of the aggregate. 

The statistical self-similarity of an object is expressed mathematically as a power law. For example, the number of particles expected to be found within some distance r from the center of an aggregate follows the power law ... 

Most simply, the object looks similar at different magnifications. Some degree of self-similarity (exact or statistical) should be displayed in the object or distributions of objects within a certain range of dilations. 

The frequency distribution of individual lake areas has a strong positive skew, fits a hyperbolic probability distribution, and has a statistical property known as self-similarity (Hamilton et al. 1991, Sippel et al. 1992). 

This theory was first developed for colloidal aggregate networks and was later adapted to fat crystal networks (52-54). In colloidal systems (with a disordered distribution of mass and statistical self-similar patterns), the mass of a fractal aggregate (or the distribution of mass within a network), M, is related to the size of the object or region of interest (R) in a power-law fashion ... 

Self-similarity is expected to be one of the important concepts to understand statistics and motion in Hamiltonian systems. However, the Poincare-Birkhoff theorem and the two models introduced by Aizawa et al. and Meiss et al. are based on the two-dimensionality of the phase spaces, and they cannot be directly applied to high-dimensional systems. As far as I know, existence of the self-similarity has not been clearly exhibited, since visualizing the selfsimilarity is not easy due to the high-dimensionality of Poincare sections, which has 2N — 2 dimension for systems with N degrees of freedom. 

In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. 

The main feature of the definition in Eq. (76) is that max ln —> 0. Whence in general the Hausdorff-Besicovitch dimension is a local characteristic. For deterministic self-similar sets, the local Hausdorff-Besicovitch dimension coincides with the dimension of the set itself. For statistically homogeneous sets, however, the local Hausdorff-Besicovitch dimension may not coincide with the dimension of the whole set. 

During self-similar growth of the model, the similarity dimension of the statistically homogeneous fraction that now arises can be estimated right at the percolation threshold from... 

Figure 3. Comparison of the trajectories of a Gaussian (left) and a Levy (right) process, the latter with index a = 1.5. While both trajectories are statistically self-similar, the Levy walk trajectory possesses a fractal dimension, characterizing the island structure of clusters of smaller steps, connected by a long step. Both walks are drawn for the same number of steps ( 7000).

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