Statistical Self-similarity Research Articles

Quantum spectrum as a time series: Fluctuation measures

The fluctuations in the quantum spectrum could be treated like a time series. In this framework, we explore the statistical self-similarity in the quantum spectrum using the detrended fluctuation analysis (DFA) and random matrix theory (RMT). We calculate the Hausdorff measure for the spectra of atoms and Gaussian ensembles and study their self-affine properties. We show that DFA is equivalent to the Delta3 statistics of RMT, unifying two different approaches. We exploit this connection to obtain theoretical estimates for the Hausdorff measure.

Self-similar random field models in discrete space

Self-similar random fields are of interest in various areas of image processing since they fit certain types of natural patterns and textures. Current treatments of self-similarity in continuous two-dimensional (2-D) space use a definition that is a direct extension of the one-dimensional definition, which requires invariance of the statistics of a random process to time scaling. Current discrete-space 2-D approaches do not consider scaling, but, instead, are based on ad hoc formulations, such as digitizing continuous random fields. In this paper, we show that the current statistical self-similarity definition in continuous space is restrictive and provide an alternative, more general definition. We also provide a formalism for discrete-space statistical self-similarity that relies on a new scaling operator for discrete images. Within the new framework, it is possible to synthesize a wider class of discrete-space self-similar random fields and texture images.

A GIS Numerical Framework to Study the Process Basis of Scaling Statistics in River Networks

A new geographic information system (GIS) numerical framework (NF), called CUENCAS, for flows in river networks is presented. The networks are extracted from digital elevation models (DEMs). The program automatically partitions a basin into hillslopes and channel links that are required to correspond to these features in an actual terrain. To investigate the appropriate DEM resolution for this correspondence, we take a high-resolution DEM at 10-m pixel size, and create DEMs at eight different resolutions in increments of 10 m by averaging. The extracted networks from 10-30 m remain about the same, even though there is a tenfold reduction in the number of pixels. By contrast, the extracted networks show increasing distortions of the original network from 40-90 m DEMs. We show the presence of statistical self-similarity (scaling) in the probability distributions of drainage areas in a Horton-Strahler framework using CUENCAS. The NF for flows takes advantage of the hillslope-link decomposition of an actual terrain and specifies mass and momentum balance equations and physical parameterizations at this scale. These equations are numerically solved. An application of NF is given to test different physical assumptions that produce statistical self-similarity in spatial peak flow statistics in a Horton-Strahler framework.

Image denoising based on wavelets and multifractals for singularity detection

This paper presents a very efficient algorithm for image denoising based on wavelets and multifractals for singularity detection. A challenge of image denoising is how to preserve the edges of an image when reducing noise. By modeling the intensity surface of a noisy image as statistically self-similar multifractal processes and taking advantage of the multiresolution analysis with wavelet transform to exploit the local statistical self-similarity at different scales, the pointwise singularity strength value characterizing the local singularity at each scale was calculated. By thresholding the singularity strength, wavelet coefficients at each scale were classified into two categories: the edge-related and regular wavelet coefficients and the irregular coefficients. The irregular coefficients were denoised using an approximate minimum mean-squared error (MMSE) estimation method, while the edge-related and regular wavelet coefficients were smoothed using the fuzzy weighted mean (FWM) filter aiming at preserving the edges and details when reducing noise. Furthermore, to make the FWM-based filtering more efficient for noise reduction at the lowest decomposition level, the MMSE-based filtering was performed as the first pass of denoising followed by performing the FWM-based filtering. Experimental results demonstrated that this algorithm could achieve both good visual quality and high PSNR for the denoised images.

The scaling properties of dissipation in incompressible isotropic three-dimensional magnetohydrodynamic turbulence

The statistical properties of the dissipation process constrain the analysis of large scale numerical simulations of three-dimensional incompressible magnetohydrodynamic (MHD) turbulence, such as those of Biskamp and Müller [Phys. Plasmas 7, 4889 (2000)]. The structure functions of the turbulent flow are expected to display statistical self-similarity, but the relatively low Reynolds numbers attainable by direct numerical simulation, combined with the finite size of the system, make this difficult to measure directly. However, it is known that extended self-similarity, which constrains the ratio of scaling exponents of structure functions of different orders, is well satisfied. This implies the extension of physical scaling arguments beyond the inertial range into the dissipation range. The present work focuses on the scaling properties of the dissipation process itself. This provides an important consistency check in that we find that the ratio of dissipation structure function exponents is that predicted by the She and Leveque [Phys. Rev. Lett 72, 336 (1994)] theory proposed by Biskamp and Müller. This supplies further evidence that the cascade mechanism in three-dimensional MHD turbulence is nonlinear random eddy scrambling, with the level of intermittency determined by dissipation through the formation of current sheets.

A generalized model for the effective thermal conductivity of porous media based on self-similarity

A generalized model for the effective thermal conductivity of porous media is derived based on the fact that statistical self-similarity exists in porous media. The proposed model assumes that porous media consist of two portions: randomly distributed non-touching particles and self-similarly distributed particles contacting each other with resistance. The latter are simulated by Sierpinski carpets with side length L = 13 and cutout size C = 3, 5, 7 and 9, respectively, depending upon the porosity concerned. Recursive formulae are presented and expressed as a function of porosity, ratio of areas, ratio of component thermal conductivities and contact resistance, and there is no empirical constant and every parameter has a clear physical meaning. The model predictions are compared with the existing experimental data, and good agreement is found in a wide range of porosity of 0.14–0.80, and this verifies the validity of the proposed model.

Weathering: Toward a Fractal Quantifying

Weathering occurs over a wide range of scales. To link features through these scales is a major challenge for interdisciplinary weathering studies. Fractal approach seems to be specially useful for this purpose. We introduce a multistep fractal weathering assessment scheme devoted to extract fractal weathering classifiers from texture analysis of the mineral's image. Our scheme enables to quantitatively estimate the global and local information about the geometry of the weathering pattern. This information is basic to develop geometrical indices of weathering, which can significantly enrich the common qualitative and semiquantitative weathering assessment schemes. To justify the fractal approach, a strong statistical self-similarity has been documented for both the weathering and fresh features of two common silica minerals: quartz and biogenic A-opal (phytolith) over four orders of length scales. The procedure is fast, drastically reduces thresholding bias, promises to be universal, it is valid for genetically different minerals and rock types, scale independent, and specially useful for monitoring the changes in the mineral's roughness during the alteration. Two of the proposed classifiers seem to be potentially useful for direct application in the field and be used by nonspecialist.

Fractal geometry model for effective thermal conductivity of three-phase porous media

An approximate model, a fractal geometry model, for the effective thermal conductivity of three-phase/unsaturated porous media is proposed based on the thermal-electrical analogy technique and on statistical self-similarity of porous media. The proposed thermal conductivity model is expressed as a function of porosity (related to stage n of Sierpı́nski carpet), ratio of areas, ratio of component thermal conductivities, and saturation. The recursive algorithm for the thermal conductivity by the proposed model is presented and found to be quite simple. The model predictions are compared with the existing measurements. Good agreement is found between the present model predictions and the existing experimental data. This verifies the validity of the proposed model.

Fractal behavior in magnetoconductance in coupled quantum dot systems

Fractal behavior in magnetoconductance fluctuations in coupled quantum dots has been studied by means of exact and statistical self-similarity. The fractal dimensions from the different features are not coincident exactly but show the similar gate voltage dependences, where the values increase with increasing negative gate voltage. Moreover, results of statistical fractal dimensions obtained from two types of dot-array samples show the same dependence on the gate voltage variation. It seems a common feature of coupled-dot systems that the fractal dimension increases by decreasing the inter-dot coupling.

Attached Eddies and Production Spectra in the Atmospheric Logarithmic Layer

We investigate the production components of turbulent spectra within logarithmic layers over flat ground. This assumes that the turbulence giving rise to these spectra consists of active coherent structures (eddies) that are attached to the ground, and whose properties display perfect statistical self-similarity under inner scaling. That is, we take the extreme view that active coherent structures not only contribute to turbulence production spectra but explain the whole of them, so that neither detached eddies nor unstructured motions make any significant contribution. Perfect self-similarityis held to apply only to eddies that are themselves formed totally within the log layer, so the theory applies in the limit of spectra obtained at the hearts of very deep log layers. The model predicts that spectral variance and covariance should become independent of wavenumber at small wavenumbers. This asymptotic behaviour is observed in all neutral spectra from the Kansas experiment. The model also interprets the various positions of the spectral peaks observed at Kansas and in aircraft flights over the sea as consequences of the eddies being aggregated into files aligned with the wind. The observed spectra are therefore consistent with large-scale wedge-like structures being the principal component of active turbulence in the neutral atmospheric surface layer.

Scaling, asymmetry and a Fokker–Planck model of the fast and slow solar wind as seen by WIND

The solar wind plasma is a natural laboratory for studies of plasma turbulence. Long, evenly sampled satellite data sets are natural candidates for statistical studies and these are often performed in the context of magnetohydrodynamic (MHD) turbulence. In this paper, scaling properties of solar wind bulk plasma parameters are discussed. Low order probability density function (PDF) asymmetry analysis is applied to data that combines slow and fast solar wind. Results are compared to those obtained using a PDF rescaling technique. A break in scaling, identified by the rescaling method, is confirmed to occur at a temporal scale of ∼26 hours. Low asymmetry levels of the fluctuations PDF are identified for the quantities that also exhibit self-similar statistics. A generalized structure function analysis is then applied to the kinetic energy density obtained from slow and fast solar wind streams. The applicability of a Fokker–Planck model for slow and fast wind ρv2 fluctuations is investigated.

Densities of some Poisson [formula omitted]-martingales and random covering numbers

The asymptotic behavior of the logarithm of the density of some T -martingales (in the sense of Kahane theory (Chinese Ann. Math. Ser. B 8 (1) (1987) 1–12)) is described in detail even in absence of statistical self-similarity. Poisson intensities of the form Lebesgue⊗ μ on R× R + ∗ are involved in the construction of these martingales. We prove that there are three possible behaviors according to the fact that α ̄ = lim sup ε→0(− logε) −1∫ [ε,1)ℓ dμ(ℓ) is zero, positive and finite, or infinite. This problem is closely related to the asymptotic behaviors of covering numbers in Poisson covering of the line and Dvoretzky covering of the circle. To cite this article: J. Barral, A. Fan, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Heterogenous multiscale method for locally self-similar problems

We present a multiscale method for a class of problems that are locally self-similar in scales and hence do not have scale separation. Our method is based on the framework of the heterogeneous multiscale method (HMM). At each point where macroscale data is needed, we perform several small scale simulations using the microscale model, then using the results and local self- similarity to predict the needed data at the scale of interest. We illustrate this idea by computing the effective macroscale transport of a percolation network at the percolation threshold. 1. Introduction and HMM Upscaling In the last several years, there has been a tremendous growth of activity on developing multiscale methods in a number of application areas. For some reviews and perspectives, we refer to (4, 5, 8). The primary goal is to develop computational techniques that can extract accurately the macroscale behavior of the system under consideration, at a cost that is substantially less than the cost of solving the microscale problem. Obviously this can only be done if the problem under consideration has some special features that can be taken advantage of. Up to now, with the exception of renormalization group methods (1), all other existing multiscale techniques assume that there is scale separation in the problem, and this property is used in an essential way in order to design efficient multiscale methods (5, 8). While many problems, particularly problems with multi-physics, do have scale separation, there are other important problems with multiscales that do not have this property. The most well-known example is the fully developed turbulent flow whose active scales typically span continuously from the large energy pumping scale to the small energy dissipation scale, the ratio of these two scales was estimated by Kolomogorov to be of O(Re 3/4 )( 2, 6). HereRe is the Reynolds number of the flow which can easily reach 10 9 for atmospheric turbulence. In the absence of scale separation, we must seek other special features of these problems in order to develop efficient computational methods. The special feature that we will focus on in this paper is (statistical) self-similarity in scales, namely that the averaged properties of the system at different scales are related to each other by simple scaling factors. This feature is approximately satisfied by many important problems such as turbulent flows and sub-surface transport in the inertial range of scales. We will call these type D problems (for the definition of type A, type B and type C problems, see (3)). To develop an efficient computational technique, it is most convenient to use the framework of the heterogeneous multiscale method (HMM) (3). HMM has two main components: • Selecting the macroscale solver;

Complexity of forest communities:a case study of three different forest types in Heishiding Nature Reserve, Guangdong

The complexity of a forest community is defined as having the average amount of its information by eliminating the uncertainties of species and layers of a tree individual randomly selected from all trees in the forest community. The joint entropy H(X, Y) is proposed to measure the complexity of a forest community H(X, Y) = H(X) + H(Y|X), in which H(X) = - and H(Y|X) = - , where S stands for the number of tree species (X), N for the total number of individuals in the forest community, n_i for the number of the ith tree species, and n_(ij) for the number of the ith tree species in the jth layer. H(X) is defined as the compositional complexity of tree species and H(Y|X) as the structural complexity of tree species. The higher the H(X, Y) value, the greater the complexity in the forest community. A case study is presented based on the survey data from three types of forest communi- ties in Heishiding Nature Reserve, Guangdong Province. Three sampling plots were established, each with a size of 60 m×60 m, representing coniferous forest, mixed coniferous broad-leaved forest and ever- green broad-leaved forest. Each plot was divided into 36 quadrats with a size of 10 m×10 m. The data for all trees with DBH≥1 cm were gathered, including their coordinates in the sampling plots. Tree sizes were divided into four categories based on their DBH: DBH≥1, 5, 10, and 30 cm. Using computer sim- ulation, 13 types of quadrat sizes (12 m×12 m, 16 m×16 m,…, 60 m×60 m) within a plot were ob- jectively selected based on the method of nested quadrat sampling. The results show that the order of H(X, Y) of three typical forest types is as follows: evergreen broad-leaved forestmixed coniferous broad-leaved forestconiferous forest. At the same time, the fractal relationships between H(X, Y) and sampling size among the three forest types reveal that H(X, Y) has a statistical self-similarity feature.

Numerical Simulation on Damage and Failure of Brittle Rocks under Different Confining Pressures

A numerical analysis has been conducted to evaluate the effect of confining pressure on the damage evolution and the fracture process of brittle rock materials by using the rock failure process analysis code (RFPA2D). A fractal dimension D associated with material damage and a damage variable ω are defined to describe the damage process of the specimen subjected to tensile or compressive load. We find that the spatial distribution of microfractures is fractal and has very good statistical self-similarity in the specimen under different confined conditions. The fractal dimension D increases progressively with loading. Meanwhile, the damage variable ω develops from 0 to 1, and the specimen failure is characterized by the critical damage variable ωc with 0.4⩽ωc⩽0.8. The initial damage is delayed by the confining pressure. The confining pressure plays an important role in the mode of rock failure in numerical simulations. The tensile microfractures dominate near the brittle splitting of the specimens under no...

A self-similarity model for effective thermal conductivity of porous media

A new model, the self-similarity model, for effective thermal conductivity of porous media is proposed based on the thermal–electrical analogy technique and on statistical self-similarity of porous media. The proposed thermal conductivity model is expressed as a function of porosity (related to stage n of Sierpinski carpet), ratio of areas, ratio of component thermal conductivities, and contact resistance. A recursive algorithm for the thermal conductivity is obtained using the proposed model and is found to be quite simple. The predictions of the model are compared with those of other models and with existing measurements and good agreement was found. This proves that the proposed model is a valid one.

Kolmogorov's third hypothesis and turbulent sign statistics.

The breakdown of turbulent eddies can be characterized by sets of "multipliers," defined as ratios of velocity increments at successively smaller scales. These quantities were introduced by Kolmogorov, who hypothesized their self-similar statistics and independence at distant scales. Here we report experimental and numerical results on the statistics of these multipliers, for both their magnitude and sign. We show that the multipliers at adjacent scales are not independent but that their correlations decay rapidly in scale separation. New scaling laws are thereby predicted and verified for both roughness and sign of turbulent velocity increments. The sign oscillations per cascade step are found to decrease at points of increasing roughness or singularity of the velocity.

Singularity spectra of fractional Brownian motions as a multi-fractal

Fractional Brownian motion acts as a random process with statistical self-similarity in time and self-affinity in shape. From these properties, the complicated patterns can be suitably represented by it with a minimal parameter and less memory. By considering its statistical property through the power spectrum density we can see that this process is not stationary, even though its differential motion is stationary. So in this paper, by taking the wavelet transfom instead of Fourier transformation we investigate its multi-fractal spectrum as a multi-fractal model.

Discrete-time models for statistically self-similar signals

Wide-sense statistical self-similarity in continuous-time random processes is defined through invariance of its first-order and second-order statistics to scaling in time. Since scaling has an unambiguous definition in continuous-time but not in discrete-time, researchers have provided various definitions of discrete-time self-similarity without reference to scaling. This paper proposes a discrete-time continuous-dilation scaling operator and develops a framework based on it for formulating statistical self-similarity from first principles in a manner analogous to the continuous-time development. Relationship between the resulting model and fractional order transfer function systems is presented. The potential for using this model in applications involving long-range dependent phenomena is explored.

Statistical self-similarity in Rhines’ concept of unique multiphase diffusion paths on the ternary gibbs’ isotherm

For ternary systems with infinite boundary values encompassing supersaturation, interface instability is inevitable with the emergence of internal microstructural order parameters that exhibit autonomy from the initial and/or boundary conditions in accord with tie-line degrees of freedom. A well-known description of statistical self-similarity in such a nonisotropic system following Rhine’s experiments is Wagner’s one-dimensional theory of dilute multidisperse internal oxidation and its generalization provided by the author and Morral and co-workers. These treatments yield unique self-similar or parabolic solutions transferable to the Gibbs isotherm for solute distribution and relative precipitate fraction. This article is concerned with the more general free-boundary dispersion problem involving both microstructure mode and wave number selection. It offers a vacancy-based argument within a scalar time-dependent, Ginzburg-Landau reaction formalism as to why observations of average parabolic relaxation extend to a much wider range of constitutions, morphology, and volume fractions than considered by Wagner. The idea of mode selection is introduced in some detail; a cusp catastrophe being explicitly identified with the triplet of virtual diffusion paths corresponding to special composition terminals. The possible role of dissipation principles in removing such complex degeneracies is briefly discussed. We present a semiquantitative analysis of Coates’ experiments in Cu-Zn-Ni together with an explanation of the observed plastic turbulence in terms of Curie’s theorem and the Kirkendall effect.