Multifractal Measures Research Articles

Intermittency and interrelationships between turbulence scaling exponents: Phase-randomization tests

A unified framework for velocity structure functions, sign-singular cancellation measures, multifractal measures of the energy dissipation and absolute values of the velocity derivative (“vorticity”), and the two-scale velocity structure functions is suggested. The scaling behavior of these measures within the inertial subrange is considered and tested using one-dimensional (1-D) velocity measurements from a high Re jet flow. In these tests we used the phase-randomization procedure that appears to be an effective tool in identifying the effects of turbulence intermittency and spectral energy flux. A number of relationships between turbulence scaling exponents are derived and tested.

Left-sided multifractality in a binary random multiplicative cascade.

In this paper we study a binary random multiplicative cascade. Specifically, the cascade is used to produce and study left-sided multifractal random measures. Extensive numerical simulations of the random cascade process were undertaken and f(alpha) spectra obtained and compared with the analytical results. We believe that this model and approach can serve as a simple and fundamental tool in the analysis and understanding of physical systems possessing an underlying multiplicative structure.

Joint Multifractal Analysis of Crop Yield and Terrain Slope

Quantifying the spatial variability of crop yields and yield‐affecting factors are important issues in precision agriculture. Topography is frequently one of the most important factors affecting yields, and topographical data are much easier to obtain than time and labor‐consuming measurements of soil properties. In this study, yield variability and the relationships between yields and terrain slopes were analyzed using theories of multifractal and joint multifractal measures. Corn (Zea mays L.) and soybean [Glycine max (L.) Merr.] yield data from 1994 to 1998 were collected via yield monitors from a central 6.6 ha section of an agricultural field in eastern Indiana. Slopes were derived from a field terrain map using a GIS. Multifractal analysis of yield and slope maps revealed that both yield and slope distributions were multifractal measures. Hence, joint multifractal analysis was applied to evaluate the effect of slope on crop‐yield spatial variability. Joint multifractal analysis facilitated (i) the ability to differentiate between yield distributions corresponding to field locations with high and low slopes, and (ii) the ability to make inferences about slope distributions that affect grain yield the most. Multifractal analysis revealed that during four growing seasons with moderate and dry weather conditions, larger yields were observed at low slope locations while a wide range of yield values was observed at sites with moderate and high slopes. During the wet growing season, lower yields prevailed at locations with low slopes. Joint multifractal theory was useful for the study of yield/topography relationships and was an applicable tool for the analysis of spatially distributed data.

Multifractal phase transitions in the non-debye relaxation processes

The multifractal measures of the relaxation-time distributions are analytically obtained for some typical non-Debye dielectric relaxation processes. The characteristics of the corresponding multifractal thermodynamics are discussed. It is shown that the probability of the relaxation times near the poles of their distribution function is fractal scaling. The corresponding Lipschitz-Hold singular exponent is, or can be, determined by the so-called shape parameters in the empirical dielectric relaxation formulas. The relationship to some analytical proofs of the empirical dielectric formulas based upon the fractal models is also analyzed. Some generalized multifractal phase transitions with interesting features are reported in this paper. The recent experiment results on the molten-crystal transition in organic glass systems are also discussed to support our conclusions.

Multifractal phase transition in the Davidson–Cole relaxation process

A multifractal measure for the relaxation time distribution in Davidson–Cole dielectric relaxation is analytically obtained. A general first-order multifractal phase transition is reported. It is characterized by the existence of three different phases and the corresponding triple point.

Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures

Let be a Borel probability measure on R . We study the Hausdorff dimension and the packing dimension of the multifractal Hausdorff measurehq;t and the multifractal packing measure p q;t introduced in [L. Olsen, A multifractal formalism, Advances in Mathematics 116 (1995), 82^196]. Let b denote the multifractal Hausdorff dimension function and let B denote the multifractal packing dimension function introduced in [Olsen, op_ cit_]. For a fixed q 2 R, we obtain bounds for the Hausdorff dimension and the packing dimension of hq;b q† and p q;B q† in terms of the subdifferential of b and B at q. For q ˆ 1, our result reduces to yD‡B 1† lim inf r&0 log B x; r† log r lim sup r&0 log B x; r† log r yDyB 1† for -a.a. x † where DyB 1† and D‡B 1† denote the left and right derivative of B at 1. Inequality ( ) improves a similar result obtained independently by Y: Heurteaux and S:-Z: Ngai. It follows from ( ) that if the mulifractal box dimension spectrum (or Lq spectrum) of is differentiable at 1 then y 0 1† equals the entropy dimension (or information dimension) of . This result has been conjectured in the physics literature and proved rigorously in certain special cases.

Self-organization and competition in the immune response to cancer invasion: a phase-orientated computational model of oncogenesis.

Recent studies indicate that fractal dimensions can uncover aspects of cellular dynamics prior to pathological manifestation. In this respect we are interested in building a computational model of oncogenesis able to generate patterns with the same fractal dimension spectrum as the in vivo tumor. A new theoretical model incorporating a systemic view of oncogenesis in a computational model was proposed. The tumor growth is viewed as competition for resources between the two self-organizing subsystems: the neoplastic and the immune. Numerical simulations revealed that tumor escape can be uncovered in some earlier stage of the immune-system-tumor interaction using multifractal measures. The described computational model is able to simulate also the case of immune, surgical, chemical and radiotherapeutical treatment, as well as their effects. T The software used is available on request from the authors. Sorinel Oprisan, University of New Orleans, Department of Psychology, Ne w Orleans, LA 70148, USA. soprisan@uno.edu

The gliding box method for multifractal modeling

The so-called ‘gliding box’ algorithm proposed originally for lacunarity analysis has been extended to multifractal modeling and provides an alternative to the ‘box-counting’ method for implementing multifractal modeling. This method can be used to implement several multifractal techniques including but not limited to the moment and multiplier methods. The results obtained by the ‘gliding box’ and ‘box-counting’ methods for multifractal modeling show the gliding box method may provide better results with less uncertainty when the number of samples and data resolution are limited. Both methods have been applied with the aid of geographic information systems (GIS) to remote sensing data. It is relatively straightforward to implement the gliding box method with the aid of GIS for comparison with the box-counting method. The pixel values of the Landsat TM imagery band 5 from the Mitchell-Sulphurets Area, northwestern BC, Canada, were analyzed as multifractal measure using both box-counting and gliding box algorithm yielding the characteristic values τ(0)≈−2, τ(1)≈0, and τ(2)≈1.92.

Multifractality and spatial statistics

The concepts of fractals and multifractals have been increasingly applied in various fields of science for describing complexity and self-similarity in nature. Fractals and multifractals are a natural consequence of self-similarity resulting from scale-independent processes. In the present paper, a theoretical investigation is developed to illustrate: (1) the characteristics of multifractality as measured by the parameter τ″( q); (2) relationships between multifractality and spatial statistics including semivariogram and autocorrelation in geostatistics, indexes used in lacunarity analysis and correlation coefficients. It can be shown that these statistics primarily are related to multifractality as determined by τ″(1). This is an important result because not only does it provide the link between multifractals and spatial statistics but it also shows that statistics based on second-order moments are restrictive in that they only characterize a multifractal measure around the mean value. In applications where extreme values need to be taken into account, the entire multifractal spectrum should be used rather than local properties of the spectrum around the mean only; alternatively, statistics defined on the basis of higher-order moments can be employed for analysis of extreme values. These theoretical results are illustrated by means of application to Landsat TM imagery (bands 1 to 7) from the Mitchell–Sulphurets mineral district, northwestern British Columbia, Canada. It is shown that variations of the TM data bands 1 to 3 in this area can be approximated by fractals but for those of bands 4, 5 and 7, multifractal models with different fractal spectra must be used. The multifractality of these images is evaluated and several methods of spatial analysis are applied to the dataset including a new version of principal component analysis in conjunction with the newly defined statistical parameters based on multifractality.

A MODIFIED DOUBLE TRACE MOMENT METHOD OF MULTIFRACTAL ANALYSIS

The Double Trace Moment method for the estimation of parameters of universal multifractal measures is revisited. It is found that the method applies strictly to bare measures but is approximate for dressed measures, which are the only observable ones. The approximation stems from the fact that the double dressed moments used by the method do not scale as required to produce unbiased estimates of the parameters. Modified moments are proposed, which correctly scale for both bare and dressed measures. The original and modified methods are compared theoretically and numerically. Conditions under which the original method gives accurate and heavily biased estimates are identified.

Multifractal Measures and a Weak Separation Condition

We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the well-known class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of the two-scale dilation equations. Our main purpose in this paper is to prove the multifractal formalism under such condition.

Intermittent Lower Bound on Quantum Diffusion

An intermittent lower bound on quantum diffusion is proven in presence of a multifractal spectral measure.

Convolutions and the Geometry of Multifractal Measures

AbstractThis paper relates multifractal features of a measure μ on IRn to those of the projection of the measure onto m‐ dimensional subspaces. We achieve this through the study of appropriately defined convolution kernels. This provides a unified approach to projections of measures and leads to new results on multifractal properties as well as alternative derivations of some existing formulae. These include formulae and estimates for the local dimensions and generalised q‐ dimensions of projected measures as well as more precise information about the limiting behaviour of multifractal expressions. We consider briefly how similar ideas may be applied to sections of a measure by (n ‐ m)‐dimensional planes.

Convolutions and the Geometry of Multifractal Measures

Convolutions and the Geometry of Multifractal Measures

Measurability of multifractal measure functions and multifractal dimension functions

Measurability of multifractal measure functions and multifractal dimension functions

The Average (n) Sphere Spans a Four-Dimensional Manifold

Let ν be a probability measure on Ω. We define the upper and lower multifractal box dimension (the measure ν with respect to μ) on a probability space and investigate the relation between the multifractal box dimension and the multifractal Hausdorff dimension, the multifractal pre-packing dimension . Then, we generalize the dimension inequalities of multifractal Hausdorff measures and multifractal packing measures in a probability space.

Multifractal formalism for self-similar bridges

We derive the thermodynamics of self-similar paths (or bridges) joining the two points and of the plane. These paths may be constituted with both macroscopic and microscopic fragments, each deserving its specific statistics, while remaining continuous. Such discontinuous paths are also studied with some information related to the statistics of their jumps. If the bridges under study are bound to be non-decreasing -paths, this study coincides with the one of multifractal measures on the unit interval. Relaxing this condition leads to an extension of the multifractal formalism whose main lines are derived here.

An evaluation of Landsat TM spectral data and SAR-derived textural information for lithological discrimination in the Red Sea Hills, Sudan

The effectiveness of spectral and textural information in the identification of surface rock types in an arid region, the Red Sea Hills of Sudan, is evaluated using spectral information from the six Landsat TM optical bands and textural features derived from Shuttle Imaging Radar-C (SIR-C) C-band HH polarization data. An initial classification is derived from Landsat TM data alone using three classification algorithms, Gaussian maximum likelihood, a multi-layer feed-forward neural network and a Kohonen self-organizing feature map (SOM), to generate lithological maps, with classification accuracy being measured using a confusion matrix approach. The feed-forward neural net produced the highest overall classification accuracy of 57 per cent and was, therefore, selected for the second experiment, in which texture measures from SIR-C C-band HH-polarized synthetic aperture radar (SAR) data are added to selected TM spectral features. Four methods of measuring texture are employed, based on the Fourier power spectrum, grey level co-occurrence matrix (GLCM), multi-fractal measures, and the multiplicative autoregressive random field (MAR) model. The use of textural information together with a subset of the TM spectral features leads to an increase in classification accuracy to almost 70 per cent. Both the MAR model and the GLCM matrix approach perform better than Fourier and multi-fractal based methods of texture characterization.

The thermodynamic formalism of random walks: Relevance for chaotic diffusion and multifractal measures

We apply the thermodynamic formalism to discrete random walks on infinite lattices. This can be considered as a new application of the thermodynamic formalism, but it also provides a very general framework for the thermodynamics of chaotic systems and multifractals. In contrast to finite Markov models which are often discussed in this context, nonanalytic behavior of characteristic quantities is encountered for models with broken translation invariance. In particular, we present new mechanisms for second-order phase transitions in the Lyapunov fluctuation spectra for chaotic diffusion and the singularity spectra of multifractal measures.

Multifractal properties of power-law time sequences: Application to rice piles

We study the properties of time sequences extracted from a self-organized critical system, within the framework of the mathematical multifractal analysis. To this end, we propose a fixed-mass algorithm, well suited to deal with highly inhomogeneous one dimensional multifractal measures. We find that the fixed mass (dual) spectrum of generalized dimensions depends on both the system size L and the length N of the sequence considered, being however stable when these two parameters are kept fixed. A finite-size scaling relation is proposed, allowing us to define a renormalized spectrum, independent of size effects.We interpret our results as an evidence of extremely long-range correlations induced in the sequence by the criticality of the system