Multifractal Measures Research Articles

MULTIFRACTAL STRUCTURE OF NON-COMPACTLY SUPPORTED INFINITE MEASURES

Many important definitions in the theory of multifractal measures on ℝd, such as the Lq-spectrum, L∞-dimensions, and the Hausdorff dimension of a measure, cannot be applied to non-compactly supported or infinite measures. We propose definitions that extend the original definitions to positive Borel measures on ℝd which are finite on bounded sets, and recover many important results that hold for compactly supported finite measures. In particular, we prove that if the Lq-spectrum is differentiable at q = 1, then the derivative is equal to the Hausdorff dimension of the measure.

The equivalence of multifractal measures on cookie-cutter-like sets

In this paper, we concentrate on the properties of the multifractal centered Hausdorff measure and the multifractal packing measure on a class of cookie-cutter-like (CCL) sets. We conclude that these two multifractal measures are equivalent on the CCL sets satisfying the strong separation condition. Then we obtain some relevant conclusions about the image measure of a σ -invariant ergodic Borel probability measure.

Empirical multifractal moment measures of self-similar measure for q < 0*

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L q - spectrum for q < 0 is generally considered significantly more difficult since the L q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0.

A Fractal Interaction Model for Winding Paths through Complex Distributions: Application to Soil Drainage Networks

Water interacts with soil through pore channels putting mineral constituents and pollutants into solution. The irregularity of pore boundaries and the heterogeneity of distribution of soil minerals and contaminants are, among others, two factors influencing that interaction and, consequently, the leaching of chemicals and the dispersion of solute throughout the soil.This paper deals with the interaction of irregular winding dragging paths through soil complex distributions. A mathematical modelling of the interplay between multifractal distributions of mineral/pollutants in soil and fractal pore networks is presented.A Hölder path is used as a model of soil pore network and a multifractal measure as a model of soil complex distribution, obtaining a mathematical result which shows that the Hölder exponent of the path and the entropy dimension of the distribution may be used to quantify such interplay. Practical interpretation and potential applications of the above result in the context of soil are discussed. Since estimates of the value of both parameters can be obtained from field and laboratory data, hopefully this mathematical modelling might prove useful in the study of solute dispersion processes in soil.

Lp-variations for multifractal fractional random walks

A multifractal random walk (MRW) is defined by a Brownian motion subordinated by a class of continuous multifractal random measures M[0, t], 0≤t≤1. In this paper we obtain an extension of this process, referred to as multifractal fractional random walk (MFRW), by considering the limit in distribution of a sequence of conditionally Gaussian processes. These conditional processes are defined as integrals with respect to fractional Brownian motion and convergence is seen to hold under certain conditions relating the self-similarity (Hurst) exponent of the fBm to the parameters defining the multifractal random measure M. As a result, a larger class of models is obtained, whose fine scale (scaling) structure is then analyzed in terms of the empirical structure functions. Implications for the analysis and inference of multifractal exponents from data, namely, confidence intervals, are also provided.

On the equivalence of the multifractal centred Hausdorff measure and the multifractal packing measure

In this work, we focus on the multifractal centred Hausdorff measure and the multifractal packing measure in . We find that for 0 < s < t < 2, q < 1, then there exist a set and a probability measure μ on such that and, in addition,

Regularities of multifractal measures

First, we prove the decomposition theorem for the regularities of multifractal Hausdorff measure and packing measure in ”d. This decomposition theorem enables us to split a set into regular and irregular parts, so that we can analyze each separately, and recombine them without affecting density properties. Next, we give some properties related to multifractal Hausdorff and packing densities. Finally, we extend the density theorem in [6] to any measurable set.

Multifractal analysis of a measure of multifractal exact dimension

We focus on the multifractal generalization of the centered Hausdorff measure and dimension. We analyze the correlation among different approaches to the definition of the multifractal exact dimension of locally finite and Borel regular measures on the basis of fractal analysis of essential supports of these measures. Using characteristic multifractal measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures.

Fractal Analysis of Yield Maps

Agricultural crop yield data usually are highly nonlinear and complex. Basic mathematical and statistical techniques are sometimes insufficient to describe the nature, trend, or cause of variations in yield. This article investigates fractal analysis of agricultural yield maps and describes a method of applying fractal analysis to multiple years of yield data. It also shows how patterns of yield variation can be described by fractal geometry. Crop yield was measured and mapped for an agricultural field for five consecutive years. In order to obtain a sufficiently dense set of points necessary for valid fractal analysis, a method was proposed to transfer the data points from to . Analysis of the resulting data set revealed multifractality of the yield variations. It was shown that multifractal measures such as the Rnyi spectrum can be used to quantify and compare global and local yield variations.

Dynamics of the multifractal measures of EEG

Event Abstract Back to Event Dynamics of the multifractal measures of EEG Peter D. Dojnow1* and N. Vitanov2 1 Institute of Inorganic and General Chemistry, Bulgaria 2 Bulgarian Academy of Sciences, Bulgaria A typical approach of numeric estimation of the properties of an EEG epoch as a whole gives estimates averaged toward the time. The EEG as a nonstationary signals possess variable multifractal characteristics in the time domain that's why a modified MDFA algorithm for analysis of short time series (about hundreds samples) is applied to the nonoverlapping pieces of EEG. This reveals the dynamics of the multifractal measures - generalized Hurst exponent and the width of the singular spectrum as a quantity of the multifractality. On a multichannel EEG records this approach gives a new correlation and synchronization detection tool for investigation of the interaction between different parts of the brain. The dynamics of the brain activity during performance of a cognitive task can be visualized by a mapping of the obtained quantities. Conference: 10th International Conference on Cognitive Neuroscience, Bodrum, Turkey, 1 Sep - 5 Sep, 2008. Presentation Type: Poster Presentation Topic: Neuroinformatics of Cognition Citation: Dojnow PD and Vitanov N (2008). Dynamics of the multifractal measures of EEG. Conference Abstract: 10th International Conference on Cognitive Neuroscience. doi: 10.3389/conf.neuro.09.2009.01.334 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 15 Dec 2008; Published Online: 15 Dec 2008. * Correspondence: Peter D Dojnow, Institute of Inorganic and General Chemistry, Sofia, Bulgaria, Dojnow@svr.igic.bas.bg Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Peter D Dojnow N. Vitanov Google Peter D Dojnow N. Vitanov Google Scholar Peter D Dojnow N. Vitanov PubMed Peter D Dojnow N. Vitanov Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Multifractality and intermittency in the solar wind

Abstract. Within the complex dynamics of the solar wind's fluctuating plasma parameters, there is a detectable, hidden order described by a chaotic strange attractor which has a multifractal structure. The multifractal spectrum has been investigated using Voyager (magnetic field) data in the outer heliosphere and using Helios (plasma) data in the inner heliosphere. We have also analyzed the spectrum for the solar wind attractor. The spectrum is found to be consistent with that for the multifractal measure of the self-similar one-scale weighted Cantor set with two parameters describing uniform compression and natural invariant probability measure of the attractor of the system. In order to further quantify the multifractality, we also consider a generalized weighted Cantor set with two different scales describing nonuniform compression. We investigate the resulting multifractal spectrum depending on two scaling parameters and one probability measure parameter, especially for asymmetric scaling. We hope that this generalized model will also be a useful tool for analysis of intermittent turbulence in space plasmas.

An improved multifractal method for pavement cracks extraction

PurposeThis paper aims to provide an improved multifractal method to extract the pavement cracks in the complicated background. Furthermore, the pavement surface images with or without crack can also be distinguished by this method.Design/methodology/approachThe framework of analyzing the image singularity is based on the sub‐pixel multifractal measure (SPMM). Performing the SPMM can give the sub‐pixel local distribution of the image gradient and a more precise singularity exponent distribution in the image. Meantime, using the singularity exponents and the most singular manifold (MSM), the image can be decomposed into a series of sets with different statistical and physical properties automatically and easily. One can extract the cracks according to the MSM.FindingsThe example shows that the physical and geometrical properties of the pavement images can be obtained by analyzing the distribution of singularity exponents and the greatest singularity exponent. The simulation results show that the SPMM has higher quality factor in the image edge detection. And the MSM detected this way reflects the most important information of the image.Originality/valuePerforming the SPMM can give a more precise singularity exponent distribution in the image.

Heterogeneous ubiquitous systems in ℝd and Hausdorff dimension

Let {x n }n∈ℕ be a sequence in [0, 1] d , {λ n }n∈ℕ a sequence of positive real numbers converging to 0, and δ > 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the form $$ S{\left( \delta \right)} = {\bigcap\limits_{N \in \mathbb{N}} \; {{\bigcup\limits_{n \geqslant N} {B{\left( {x_{n} ,\lambda ^{\delta }_{n} } \right)}} }} }. $$ Let μ be a positive Borel measure on [0, 1]d , ρ 2 (0, 1] and α > 0. Consider the finer limsup-set $$ S_{\mu } {\left( {p,\delta ,\alpha } \right)} = {\bigcap\limits_{N \in \mathbb{N}} \; {{\bigcup\limits_{n \geqslant N:\mu {\left( {B{\left( {x_{n} ,\lambda ^{p}_{n} } \right)}} \right)} \sim \lambda ^{{p\alpha }}_{n} } {B{\left( {x_{n} ,\lambda ^{\delta }_{n} } \right)}} }} }. $$ We show that, under suitable assumptions on the measure μ, the Hausdorff dimension of the sets S μ (ρ, δ, α) can be computed. Moreover, when ρ < 1, a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension of S μ (ρ, δ, α). Our results apply to several classes of multifractal measures, and S(δ) corresponds to the special case where μ is a monofractal measure like the Lebesgue measure. The computation of the dimensions of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Diophantine approximation conditioned by (or combined with) b-adic expansion properties, by averages of some Birkhoff sums and branching randomwalks, as well as by asymptotic behavior of random covering numbers.

Return times for stochastic processes with power-law scaling

An analytical study of the return time distribution of extreme events for stochastic processes with power-law correlation has been carried out. The calculation is based on an epsilon expansion in the correlation exponent: C(t)=/t/-1+epsilon. The fixed point of the theory is associated with stretched exponential scaling of the distribution; analytical expressions have been provided in the preasymptotic regime. Also, the permanence time distribution appears to be characterized by stretched exponential scaling. The conditions for application of the theory to non-Gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed.

The exact rate of convergence of the [formula omitted]-spectra of self-similar measures for [formula omitted

The L q -spectrum of a Borel measure is one of the key objects in multifractal analysis, and it is widely believed that L q -spectrum associated with a fractal measure encode important information about the underlying dynamics and geometry. The study of the L q -spectrum therefore plays a fundamental role in the understanding of dynamical systems or fractal measures. For q ⩾ 0 Olsen [L. Olsen, Empirical multifractal moment measures and moment scaling functions of self-similar multifractals, Math. Proc. Cambridge Philos. Soc. 133 (2002) 459–485] recently determined the exact rate of convergence of the L q -spectra of a self-similar measure satisfying the Open Set Condition (OSC). Unfortunately, nothing is known about the rate of convergence for q < 0 . Indeed, the problem of analysing L q -spectra for q < 0 is generally considered significantly more difficult since the L q -spectra are extremely sensitive to small variations in the distribution of μ for q < 0 . The purpose of this paper is to overcome these obstacles and to investigate the more difficult problem of determining the exact rate of convergence of the multifractal L q -spectra of a self-similar measure satisfying the OSC for q < 0 .

Multifractal characteristics of soil particle size distribution under different land-use types on the Loess Plateau, China

Soil particle-size distribution (PSD) is one of the most important physical attributes due to its great influence on soil properties related to water movement, productivity, and soil erosion. The multifractal measures were useful tools in characterization of PSD in soils with different taxonomies. Land-use type largely influences PSD in a soil, but information on how this occurs for different land-use types is very limited. In this paper, multifractal Rényi dimension was applied to characterize PSD in soils with the same taxonomy and different land-use types. The effects of land use on the multifractal parameters were then analyzed. The study was conducted on the hilly-gullied regions of the Loess Plateau, China. A Calcic Cambisols soil was sampled from five land-use types: woodland, shrub land, grassland, terrace farmland and abandoned slope farmland with planted trees (ASFP). The result showed that: (1) entropy dimension ( D 1) and entropy dimension/capacity dimension ratio ( D 1/ D 0) were significantly positively correlated with finer particle content and soil organic matter. (2) D 0, D 1 and D 1/ D 0 were significantly influenced by land use. Land use could explain 24.6–58.5% of variability of D 0, D 1/ D 0 and D 1, which may be potential parameters to reflect soil physical properties and soil quality influenced by land use.

A multifractal formalism for vector-valued random fields based on wavelet analysis: application to turbulent velocity and vorticity 3D numerical data

Extreme atmospheric events are intimately related to the statistics of atmospheric turbulent velocities. These, in turn, exhibit multifractal scaling, which is determining the nature of the asymptotic behavior of velocities, and whose parameter evaluation is therefore of great interest currently. We combine singular value decomposition techniques and wavelet transform analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method is calibrated on synthetic self-similar 2D vector-valued multifractal measures and monofractal 3D vector-valued fractional Brownian fields. We report the results of some application of the TWTMM method to turbulent velocity and vorticity fields generated by direct numerical simulations of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship \({D_{{\bf v}}(h+1)=D_{\varvec{\omega}}(h)},\) between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.

A Chaos Theoretic Analysis of Motion and Illumination in Video Sequences

Accurate and robust image motion detection has been of substantial interest in the image processing and computer vision communities. Unfortunately, no single motion detection algorithm has been universally superior, yet biological vision systems are adept at motion detection. Recent research in neural signals have shown biological neural systems are highly responsive to signals which appear to be chaotic in nature. In this paper, we exploit these biological results and hypothesize that motion in images may produce changes in pixel amplitudes that are reminiscent of chaotic dynamical systems. In particular, we demonstrate that the trajectories of pixel amplitudes in phase space due to motion result in chaos-like behavior. We likewise demonstrate that the effects of spatio-temporally varying illumination produces phase space trajectories of the pixel amplitudes which are clearly non-chaotic. We review the research tying chaotic behavior to the fractal characteristics of phase space trajectories, and we investigate multi-fractal measures which can be used to classify the pixels in an image stream based on their fractal behavior in phase space. We finally apply these measures to the task of motion detection and segmentation and show they are effective in identifying moving objects while ignoring spatio-temporally varying illumination changes.

Modeling geophysical complexity: a case for geometric determinism

Abstract. It has been customary in the last few decades to employ stochastic models to represent complex data sets encountered in geophysics, particularly in hydrology. This article reviews a deterministic geometric procedure to data modeling, one that represents whole data sets as derived distributions of simple multifractal measures via fractal functions. It is shown how such a procedure may lead to faithful holistic representations of existing geophysical data sets that, while complementing existing representations via stochastic methods, may also provide a compact language for geophysical complexity. The implications of these ideas, both scientific and philosophical, are stressed.

Location of Q-wave myocardial infarction in the era of cardiac magnetic resonance imaging techniques: An update

Autonomic nervous system (ANS) is governed by complex interactions arising from feedback loops of nonlinear systems that operate over a wide range of temporal and spatial scales, enabling the organism to adapt to stress, metabolic changes and diseases. This study is aimed to assess multifractal and nonlinear characteristics of the ANS during ischemic events provoked by a prolonged percutaneous coronary intervention (PCI) procedure. Eighty-seven patients from the STAFF III database were used. Patients were classified into 2 groups: (1) with prior myocardial infarction (MI) and (2) without MI (noMI). R–R signals during three 3-min stages of the procedures were analyzed using multifractal and surrogate data techniques. Multifractal indices increased significantly from the pre-inflation stage to the post-deflation stage. These variations were more marked for the noMI group. Multifractal changes significantly correlated with both the decreased parasympathetic and the increased sympathetic modulations accounted by classical linear indices. Multifractal measures resulted to be a more powerful indicator than linear HRV indices in quantifying the ischemia-induced changes. Right coronary artery (RCA) occlusions provoke greater multifractal reactions throughout the PCI procedure. Our findings suggest reduced complex multifractal and nonlinear reactions of ANS activity in patients with prior MI in comparison to the noMI group, possibly due to degradation in the complexity of control mechanism of heart rate generation.