Multifractal Sets Research Articles

On using escort distributions in digital image analysis

Methods of digital image analysis find wide application both in scientific research and in many branches of industry. During the last decades, interest has grown in images with multifractal structure which are obtained in biology, medicine, chemistry and studying the soil. The mathematics of fractals and fractal geometry are well known and studied. However, despite this, a common approach to designing the methods of practical investigation of such images has not been developed until now. The main purpose of this work is to propose the using of multifractal formalism as the mathematical tool for the statistical description of multifractal sets. Such a description adequately depicts the chaotic behavior of the majority of real systems. The method for calculation of Rényi and singularity spectra based on using parametrized spectra, which are obtained from escort (zooming) distributions of an initial measure, is considered. The method for comparing images based on using vectors of divergences calculated for the sequence of escort distributions is proposed. The role of parametrized spectra as the tool for the approximation of any part of the singularity spectrum is substantiated. An estimation of the rate of growth of the divergence vector is obtained. Main theoretical results are confirmed by numerical experiments with images of biomedical preparations. These show the ability of the implemented methods to find subtle distinctions in image structure for a simple choice of an initial measure.

A spectral algorithm to simulate nonstationary random fields on spheres and multifractal star-shaped random sets

An extension of the turning arcs algorithm is proposed for simulating a random field on the two-dimensional sphere with a second-order dependency structure associated with a locally varying Schoenberg sequence. In particular, the correlation range as well as the fractal index of the simulated random field, obtained as a weighted sum of Legendre waves with random degrees, may vary from place to place on the spherical surface. The proposed algorithm is illustrated with numerical examples, a by-product of which is a closed-form expression for two new correlation functions (exponential-Bessel and hypergeometric models) on the sphere, together with their respective Schoenberg sequences. The applicability of our findings is also described via the emulation of three-dimensional multifractal star-shaped random sets.

Identification of healthy and pathological heartbeat dynamics based on ECG-waveform using multifractal spectrum

Human body surface electrocardiogram (ECG) is non-stationary and frequency-varying by nature that belongs to a typical nonlinear signal. Therefore, traditional linear and time–frequency​ analysis methods cannot fully disclose its nonlinear nature. Meanwhile, physiological complexity of heartbeat signal may vary with age, diseases, drug administrations, or even behavioral modifiers. To test the intrinsic relationships among them, we first put forward a theoretical model for nonlinear time series analysis and then took the generally accepted multifractal sets to verify it. Upon that, we then investigated the multifractal singularity spectrum areas of synchronous 12-lead ECG signals taken from crowds with different age stages, healthy conditions and drug medications. Our results suggested that aging and diseases can not only decrease multifractal complexity of the signals, but also increase inhomogeneity of it. With aging and deepening lesion, fractal-like structure of the heartbeat system is damaged or even structurally changed, which lead to the declination of physiological complexity and at the same time the increasement of irregularity and anisotropy of ECG signal’s propagation. In addition, the mean value of multifractal spectrum area of human 12-lead ECG signals also reflect the self-discipline regulation of human autonomic nervous system. The value descends with age growing or drug intervention to restrain sympathetic nerves. That suggest self-discipline control function weakens when people are getting old, or they are under repressed heartbeat activities with lower heart rate and lower blood pressure. Then, complexity of heartbeat signal declines and even tends to turn from multifractality to monofractality, which means drops off of human individual adaptability.

SOME TOPOLOGICAL RESULTS ON LEVEL SETS AND MULTIFRACTAL SETS

In this paper, we study topological properties of some level sets and some multifractal sets induced by Rademacher’s series and Takagi’s series, respectively. By using symbolic space, we obtain necessary and sufficient conditions for them to be residual.

ON THE ‐COMPLETENESS AND ‐COMPLETENESS OF MULTIFRACTAL DECOMPOSITION SETS

The purpose of this paper twofold. Firstly, we establish -completeness and -completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the -completeness and -completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if is a self-similar measure satisfying the strong separation condition and is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of defined by are -complete provided they are non-empty.

Clustering of particles in turbulence due to phoresis.

We demonstrate that diffusiophoretic, thermophoretic, and chemotactic phenomena in turbulence lead to clustering of particles on multifractal sets that can be described using one single framework, valid when the particle size is much smaller than the smallest length scale of turbulence l_{0}. To quantify the clustering, we derive positive pair correlations and fractal dimensions that hold for scales smaller than l_{0}. For scales larger than l_{0} the pair-correlation function is predicted to show a stretched exponential decay towards 1. In the case of inhomogeneous turbulence we find that the fractal dimension depends on the direction of inhomogeneity. By performing experiments with particles in a turbulent gravity current we demonstrate clustering induced by salinity gradients in conformity to the theory. The particle size in the experiment is comparable to l_{0}, outside the strict validity region of the theory, suggesting that the theoretical predictions transfer to this practically relevant regime. This clustering mechanism may provide the key to the understanding of a multitude of processes such as formation of marine snow in the ocean and population dynamics of chemotactic bacteria.

Fractal dimension computation from equal mass partitions.

Numerical methods which utilize partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets. However, it is known that mass-oriented methods generate relatively good results for computing generalized dimensions for important cases where the box-counting method is known to fail. Here, we revisit two mass-oriented methods and discuss their strengths and limitations.

Fractional Quantum Field Theory on Multifractals Sets

Problem statement: This study is a contribution to the general program of describing complex dynamical systems using the tool of fractional calculus of variations. Approach: Following our previous work, fractional quantum field theory based on the fractional actionlike variational approach supported by Saxena-Kumbhat fractional integrals functionals, fractional derivative of order (α, β) and dynamical fractional exponent on multi-fractal sets is considered. Results: In order to build the required theory, we introduce the Saxena-Kumbhat hypergeometric fractional functionals determined on the functions on a multifractal sets. We prove, developing the corresponding fractional calculus of variations, that a hierarchy of differential equations can be developed from the extended fractional Lagrangian formalism. Besides, a generalization of the resulting Hamiltonian and Lagrangian dynamics on the complex plane is addressed. Conclusion: The new complexified dynamics guides to a new dynamics which may differ totally from the classical mechanics cardinally and may bring new appealing consequences. Some additional interesting results are explored and discussed in some details.

From fractal to multifractal super fibres and wool fibres with exceptional mechanical properties

From the concept of fractal superfibre with fractal cross-section, multifractal superfibre with multifractal cross-section is introduced in the present paper. Various single fractal sets, e.g. three, seven- and eleven-circle fractal sets, have combined to form double fractal sets, e.g. three-/seven-circle fractal set and three-/eleven-circle fractal set. The fractal dimensions of the double fractal and multifractal sets are calculated. By taking the double fractal superfibre as an idealised model of fibre reinforced bar, the authors estimate its strength, stiffness and fracture energy. The theory can best describe the exceptional mechanical properties of the wool fibre.

Memory effects in the statistics of interoccurrence times between large returns in financial records

We study the statistics of the interoccurrence times between events above some threshold Q in two kinds of multifractal data sets (multiplicative random cascades and multifractal random walks) with vanishing linear correlations. We show that in both data sets the relevant quantities (probability density functions and the autocorrelation function of the interoccurrence times, as well as the conditional return period) are governed by power laws with exponents that depend explicitly on the considered threshold. By studying a large number of representative financial records (market indices, stock prices, exchange rates, and commodities), we show explicitly that the interoccurrence times between large daily returns follow the same behavior, in a nearly quantitative manner. We conclude that this kind of behavior is a general consequence of the nonlinear memory inherent in the multifractal data sets.

The effects of multifractality on the statistics of return intervals

We study the statistics of the return intervals in multifractal data sets with and without linear correlations. In the absence of linear correlations, we find that the nonlinear correlations inherent in multifractal data yield (i) a power-law decay of the autocorrelation function of the return intervals, (ii) a power-law increase of the conditional return period as function of the previous return interval, and (iii) a power-law decay of the probability density function of the return intervals. These features remain unchanged in the presence of linear long-term correlations. Deviations observed in the asymptotic behaviour are probably due to finite size effects. We compare our results with those obtained for uncorrelated and for monofractal long-term correlated data, and demonstrate significant differences. Applications can be found in studying the dynamics of several processes characterised by multifractality, such as turbulence, climate dynamics, heartbeat dynamics, stock market dynamics, and tele-traffic in large networks.

On the Occurence of Extreme Events in Long-term Correlated and Multifractal Data Sets

We review recent studies of the statistics of return intervals (i) in long-term correlated monofractal records and (ii) in multifractal records in the absence (or presence) of linear long-term correlations. We show that for the monofractal records which are long-term power-law correlated with exponent γ, the distribution density of the return intervals follows a stretched exponential with the same exponent γ and the return intervals are long-term correlated, again with the same exponent γ. For the multifractal record, significant differences in scaling behavior both in the distribuiton and correlation behavior of return intervals between large events of different magnitudes are demonstrated. In the absence of linear long-term correlations, the nonlinear correlations contribute strongly to the statistics of the return intervals such that the return intervals become long-term correlated even though the original data are linearly uncorrelated (i.e., the autocorrelation function vanishes). The distribution density of the return intervals is mainly described by a power law.

Multifractality and randomness in the unstable plastic flow near the lower strain-rate boundary of instability

The unstable plastic flow of an AlMg alloy, associated with the Portevin-Le Chatelier effect, was studied near the lower strain-rate boundary of instability using multifractal analysis. Self-similarity of deformation curves, indicating long-range time correlations of stress serrations, was detected within the strain-rate range where serrations are commonly ascribed to the occurrence of uncorrelated deformation bands. The deformation curves display a wide range of shapes that are characterized by different groupings of serrations. Multifractal analysis provides a method to quantify the observed complexity and compare it to known Portevin-Le Chatelier effect regimes. The measurement noise effect on the multifractal spectra determined from experimental data was mimicked by superposing multifractal Cantor sets with random noise. Such tests using standard multifractal data sets justify the separation of self-similar and random components of the serrated deformation curves. Furthermore, these results shed light on the general problem of the effect of experimental noise on the apparent multifractal properties of physical fractals.

Mass exponent spectrum analysis of human ECG signals and its application to complexity detection

The complexity of electrocardiogram (ECG) signal may reflect the physiological function and healthy status of the heart. In this paper, we introduced two novel intermediate parameters of multifractality, the mass exponent spectrum curvature and area, to characterize the nonlinear complexity of ECG signal. These indicators express the nonlinear superposition of the discrepancies of singularity strengths from all the adjacent points of the spectrum curve and thus overall subsets of original fractal structure. The evaluation of binomial multifractal sets validated these two variables were entirely effective in exploring the complexity of this time series. We then studied the ECG mass exponent spectra taken from the cohorts of healthy, ischemia and myocardial infarction (MI) sufferer based on a large sets of 12 leads’ recordings, and took the statistical averages among each crowd. Experimental results suggest the two values from healthy ECG are apparently larger than those from the heart diseased. While the values from ECG of MI sufferer are much smaller than those from the other two groups. As for the ischemia sufferer, they are almost of moderate magnitude. Afterward, we compared these new indicators with the nonlinear parameters of singularity spectrum. The classification indexes and results of total separating ratios (TSR, defined in the paper) both indicated that our method could achieve a better effect. These conclusions may be of some values in early diagnoses and clinical applications.

Effect of Nonlinear Correlations on the Statistics of Return Intervals in Multifractal Data Sets

We study the statistics of return intervals between events above a certain threshold in multifractal data sets without linear correlations. We find that nonlinear correlations in the record lead to a power-law (i) decay of the autocorrelation function of the return intervals, (ii) increase in the conditional return period, and (iii) decay in the probability density function of the return intervals. We show explicitly that all the observed quantities depend both on the threshold value and system size, and hence there is no simple scaling observed. We also demonstrate that this type of behavior can be observed in real economic records and can be used to improve considerably risk estimation.

Higher Multifractal Analysis for Simultaneous Actions of Fuchsian Groups

We consider groups Γ1, Γ2, …, Γ d acting on the hyperbolic plane H 2 and by the method of Series for coding hyperbolic geodesics we can do simultaneous multifractal decompositions from the action of these groups. We obtain a variational formula for a dimension of the multifractal sets which generalizes that presented in our previous article for d = 1.

Higher-dimensional multifractal value sets for conformal infinite graph directed Markov systems

We give a description of the level sets in the higher dimensional multifractal formalism for infinite conformal graph directed Markov systems. If these systems possess a certain degree of regularity this description is complete in the sense that we identify all values with non-empty level sets and determine their Hausdorff dimension. This result is also partially new for the finite alphabet case.

On the Hausdorff Measure of Irregular Multifractal Sets in Hyperbolic Flows

Article On the Hausdorff Measure of Irregular Multifractal Sets in Hyperbolic Flows was published on March 1, 2007 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 8, issue 1).

Observed Structure of Addresses in IP Traffic

We investigate the structure of addresses contained in IPv4 traffic--specifically, the structural characteristics of destination IP addresses seen on Internet links, considered as a subset of the address space. These characteristics have implications for algorithms that deal with IP address aggregates, such as routing lookups and aggregate-based congestion control. Several example address structures are well modeled by multifractal Cantor-like sets with two parameters. This model may be useful for simulations where realistic IP addresses are preferred. We also develop concise characterizations of address structures, including active aggregate counts and discriminating prefixes. Our structural characterizations are stable over short time scales at a given site, and different sites have visibly different characterizations, so that the characterizations make useful fingerprints of the traffic seen at a site. Also, changing traffic conditions, such as worm propagation, significantly alter these fingerprints.

Hausdorff measures, dimensions and mutual singularity

Let ( X , d ) (X,d) be a metric space. For a probability measure μ \mu on a subset E E of X X and a Vitali cover V V of E E , we introduce the notion of a b μ b_{\mu } -Vitali subcover V μ V_{\mu } , and compare the Hausdorff measures of E E with respect to these two collections. As an application, we consider graph directed self-similar measures μ \mu and ν \nu in R d \mathbb {R}^{d} satisfying the open set condition. Using the notion of pointwise local dimension of μ \mu with respect to ν \nu , we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen’s multifractal Hausdorff measures are mutually singular.