Multifractal Formalism Research Articles

Multifractal analysis of Hewitt-Stromberg measures with respect to gauge control functions

This study provides a general multifractal formalism that overcomes the limitations of the traditional one. The generic Hewitt-Stromberg measures are used to introduce and study a multifractal formalism. The generic Hewitt-Stromberg dimensions' upper and lower bounds are estimated, producing results even at places $q$ where the upper and lower multifractal Hewitt-Stromberg dimension functions diverge.

Multifractal perturbations to multiplicative cascades promote multifractal nonlinearity with asymmetric spectra.

Biological and psychological processes have been conceptualized as emerging from intricate multiplicative interactions among component processes across various spatial and temporal scales. Among the statistical models employed to approximate these intricate nonlinear interactions across scales, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, several fundamental questions persist concerning the proper interpretations of multifractal evidence of nonlinear cross-scale interactivity. Does multifractal spectrum width depend on multiplicative interactions, constituent noise processes participating in those interactions, or both? We conducted numerical simulations of cascade time series featuring component noise processes characterizing a range of nonlinear temporal correlations: nonlinearly multifractal, linearly multifractal (obtained via the iterative amplitude adjusted wavelet transform of nonlinearly multifractal), phase-randomized linearity (obtained via the iterative amplitude adjustment Fourier transform of nonlinearly multifractal), and phase and amplitude randomized (obtained via shuffling of nonlinearly multifractal). Our findings show that the multiplicative interactions coordinate with the nonlinear temporal correlations of noise components to dictate emergent multifractal properties. Multiplicative cascades with stronger nonlinear temporal correlations make multifractal spectra more asymmetric with wider left sides. However, when considering multifractal spectral differences between the original and surrogate time series, even multiplicative cascades produce multifractality greater than in surrogate time series, even with linearized multifractal noise components. In contrast, additivity among component processes leads to a linear outcome. These findings provide a robust framework for generating multifractal expectations for biological and psychological models in which cascade dynamics flow from one part of an organism to another.

The relative multifractal analysis of a vector function in a metric space

ABSTRACT Scale invariance is a widely used concept for analysing real data from many different applications, and multifractal analysis has become the standard and powerful tool in signal processing. It characterizes the data by globally and geometrically describing the fluctuations of the local regularity. In this paper, we study the relative vectorial multifractal formalism based on vectorial multifractal Hewitt-Stromberg measures. We use a function defined on balls in a metric space and Banach valued, which is more general than measures. This leads to results on the simultaneous behaviour of possibly many local Hölder exponents. As an application, we study the relative multifractal formalism of branching random walk on the Galton-Watson tree.

Multifractal spectral features enhance classification of anomalous diffusion.

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Multifractal Analysis of Measures Arising from Random Substitutions

We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation wordLq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum of a random substitution and show that this coincides with the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum of the corresponding frequency measure for all q≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\ge 0$$\\end{document}. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.

Multifractal organization of EEG signals in multiple sclerosis

Quantifying the complex/multifractal organization of the brain signals is crucial to fully understanding the brain processes and structure. In this contribution, we performed the multifractal analysis of the electroencephalographic (EEG) data obtained from a controlled multiple sclerosis (MS) study, focusing on the correlation between the degree of multifractality, disease duration, and disability level. Our results reveal a significant correspondence between the complexity of the time series and multiple sclerosis development, quantified respectively by scaling exponents and the Expanded Disability Status Scale (EDSS). Namely, for some brain regions, a well-developed multifractality and little persistence of the time series were identified in patients with a high level of disability, whereas the control group and patients with low EDSS were characterized by persistence and monofractality of the signals. The analysis of the cross-correlations between EEG signals supported these results, with the most significant differences identified for patients with EDSS >1 and the combined group of patients with EDSS ≤1 and controls. No association between the multifractality and disease duration was observed, indicating that the multifractal organization of the data is a hallmark of developing the disease. The observed complexity/multifractality of EEG signals is hypothetically a result of neuronal compensation – i.e., of optimizing neural processes in the presence of structural brain degeneration. The presented study is highly relevant due to the multifractal formalism used to quantify complexity and due to scarce resting-state EEG evidence for cortical reorganization associated with compensation.

New Appliance Signatures for NILM Based on Mono-Fractal Features and Multi-Fractal Formalism

New Appliance Signatures for NILM Based on Mono-Fractal Features and Multi-Fractal Formalism

Note on the Multifractal Formalism of Covering Number on the Galton-Watson Tree

We consider, for t in the boundary of Galton-Watson tree (∂T), the covering number Nn(t) by cylinder of generation n. For a suitable set I and a sequence (sn,γ), we establish almost surely, and uniformly on γ, the Hausdorff and packing dimensions of the set {t ∈ ∂T : Nn(t) − nb ∼ sn,γ} for b ∈ I.

MIXED MULTIFRACTAL SPECTRA OF HOMOGENEOUS MORAN MEASURES

There are only two kinds of measures in which the mixed multifractal formalism applies, which are self-similar and self-conformal measures. This paper studies the validity and non-validity of the mixed multifractal formalism of other kinds of measures, called irregular/homogeneous Moran measures.

Chaotic dynamics of diatomic systems in an electromagnetic field: Dynamical and topological invariants

An advanced combined quantum-dynamic and chaos-geometric method for analysis, modelling, and forecasting of the chaotic dynamics of diatomic molecules in an intense electromagnetic field is presented. The method is based on the use of the non-stationary theory of the Schrödinger equation in the approximation of the density functional and the methods of the theory of chaos and dynamic systems for the analysis of time series of polarization and other characteristics of diatomic molecules in an intense electromagnetic field. In particular, the latter includes the Gottwald-Melbourne test, the correlation integral method , fractal and multifractal formalism, average mutual information, false nearest neighbours, surrogate data algorithms, analysis on the basis of the Lyapunov's exponents, Kolmogorov entropy, nonlinear forecast models based on algorithms of optimized predicted trajectories, B-spline approximations. As an illustration, the advanced data for the dynamical and topological invariants (correlation dimension, embedding dimension, Kaplan-York dimension, Lyapunov's exponents, Kolmogorov entropy, etc.) for the diatomic ZrO molecule in a linearly polarized electromagnetic field are listed.

Scaling and Complexity of Stress Fluctuations Associated with Smooth and Jerky Flow in FeCoNiTiAl High-Entropy Alloy

Recent observations of jerky flow in high-entropy alloys (HEA) revealed a high role of self-organization of dislocations in their plasticity. The present work reports the first results of the investigation of stress fluctuations during plastic deformation of an FeCoNiTiAl alloy, examined in a wide temperature range covering both smooth and jerky flow. These fluctuations, which accompany the overall deformation behavior representing an essentially slower stress evolution controlled by the work hardening, were processed using complementary approaches comprising Fourier spectral analysis, refined composite multiscale entropy, and multifractal formalisms. The joint analysis at distinct scales testified that even a macroscopically smooth plastic flow is accompanied by nonrandom fluctuations, disclosing the self-organized dynamics of dislocations. Qualitative changes in such a fine-scale “noise” were found with varying temperature. The observed diversity is significant for understanding the relationships between different scales of plasticity of HEAs and crystal materials in general.

Multifractal Characterization of Texts for Pattern Recognition: On the Complexity of Morphological Structures in Modern and Ancient Languages.

The study of languages' structure and their organization in a set of well-defined relation schemes is a delicate matter. In the last decades, the convergence of traditional conflicting views by linguists is supported by an interdisciplinary approach that involves not only genetics or bio-archelogy but nowadays even the science of complexity. In light of this new and useful approach, this study proposes an in-depth analysis of the complexity underlying the morphological organization, in terms of multifractality and long-range correlations, of several modern and ancient texts pertaining to various linguistic strains (including ancient Greek, Arabic, Coptic, Neo-Latin and Germanic languages). The methodology is grounded on the mapping procedure between lexical categories belonging to text excerpts and time series, which is based on the rank of the frequency occurrence. Through the well-known MFDFA technique and a specific multifractal formalism, several multifractal indexes are then extracted for characterizing texts and the multifractal signature has been adopted for characterizing several language families, such as Indo-European, Semitic and Hamito-Semitic. The regularities and differences in the linguistic strains are assessed within a multivariate statistical framework and corroborated with a Machine Learning approach that is dedicated, in turn, to investigate the predictive power of the multifractal signature pertinent to text excerpts. The obtained results show a strong presence of persistence, i.e., memory, in the morphological structure of analyzed texts and we claim that this property has a role in characterizing the studied linguistic families. In fact, for example, the proposed analysis framework - grounded on complexity indexes - is able to easily distinguish ancient Greek texts from Arabic ones, as they belong to different language strains, i.e., indo-European and Semitic, respectively. The proposed approach has been proven effective and can be adopted for further comparative studies and for designing new informetrics for further advances in the fields of information retrieval and Artificial Intelligence.

Practical Aspects of Using Multifractal Formalism to Assess the Morphology of Biological Tissues.

The objects of the study were histological images of lung tissues of Wistar rats without pathology and with detected pathological changes, obtained at 50×, 100×, 200× magnifications. Image processing was carried out using the ImageJ/Fiji universal software. The multifractal spectrum of the images, processed to obtain a linear contour, was calculated with the use of FracLac - a module for ImageJ. This module was used to determine the scaling exponent (the function of the Rényi exponent, τ(q)) and the singularity spectrum itself. The singularity spectra for tissues with no pathology have signs of multifractality. The image spectrum of tissue with pathology is shifted to the left relative to the spectrum characteristic of tissue without pathology. A decrease in the spectral height in the presence of pathology indicates a "simplification" of the alveolar pattern, which is presumably associated with the presence of widespread vasculitis, since it causes areas of hemorrhage to appear on the image; this leads to leveling the contour of the alveolar pattern, reducing the surface area of the alveoli and emerging areas inflamed by erythrocytes. At lower magnification, images with pathology lose signs of multifractality. Correct results of assessing multifractal spectra of histological images can be achieved at 200× magnification and preprocessing to obtain linear contours. Significant differences between the morphological structure of lung tissues with and without pathology are observed when comparing the height, width, and position of the spectrum relative to the origin.

Generalized Sν$S^\nu$ spaces

AbstractIn this work, we introduce a generalization of the spaces underlying a multifractal formalism for non‐concave spectra. We prove among other things that the essential topological properties of the spaces can be transposed in this context; in particular, these new spaces are metric.

VHCF ultrasonic tests on EN AW‐6082 aluminum alloy samples over a wide dimensional range

AbstractIn the present paper, a very high cycle fatigue test campaign carried out with an ultrasonic testing machine up to 1010 cycles is presented. The aluminum alloy hourglass and dog‐bone samples have dimensions spanning over a very wide range, with the diameter in the middle cross‐section comprised between 3 and 30 mm. From the experimental results, the influence of structural size can be clearly detected, and it is possible to observe a transition between small scales, where the size effect is more pronounced, and larger scales, where the size effect is vanishing. This phenomenon can be explained in an effective way by adopting a multifractal formalism, which is equipped with the probabilistic treatment of the statistical dispersion of experimental data.

The Frisch-Parisi conjecture II: Besov spaces in multifractal environment, and a full solution

Frisch-Parisi conjecture claims the existence of Baire function spaces in which Baire typical functions share the same multifractal behavior, prescribed in advance, and obey a multifractal formalism. In this paper, we introduce a family B of heterogeneous Besov spaces, which generalize the standard Besov spaces - they are obtained by replacing the Lebesgue measure (which plays a key role in the definition of the standard Besov spaces) by multifractal Radon measures belonging to some class constructed in the companion paper [1]. We find a characterization of the elements of B in terms of wavelet coefficients, and then describe the multifractal properties (singularity spectrum, validity of the multifractal formalism) of their Baire typical functions. This allows us to fully solve the Frisch-Parisi conjecture inside B.

The Frisch-Parisi conjecture I: Prescribed multifractal behavior, and a partial solution

In this work and its companion [1], we construct Baire function spaces in which typical elements share the same prescribed multifractal behavior and obey a multifractal formalism, providing a solution to the so-called Frisch-Parisi conjecture for functions, an inverse problem raised by S. Jaffard. In this first part, a family Ed of almost-doubling fully supported capacities on Rd with prescribed singularity spectra is constructed. With each μ∈Ed we associate a Baire function space Bμ(Rd) (a generalisation of Hölder-Zygmund spaces) in which typical functions share the same singularity spectrum as μ. This yields a partial solution to the conjecture. In [1], we introduce and study a family B={Bqμ,p(Rd)}μ∈Ed,(p,q)∈[1,+∞]2 of heterogeneous Besov spaces that contains {Bμ(Rd)}μ∈Ed and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside B.

The role of teleconnections and solar activity on the discharge of tropical river systems within the Niger basin

The behavior of tropical river systems is driven by some internal and external factors. Understanding the role of these external forces, such as large-scale oscillations, on river discharge will provide insight into their dynamic complexities and modelling. In this study, the role of teleconnection patterns and solar activity on river discharges within the Niger basin was considered using both linear (correlation) and nonlinear (multifractal and joint recurrence analysis) statistical approaches. Correlation analysis suggests the existence of a linear relationship between tropical teleconnection patterns in the Atlantic and Pacific oceans with river discharge in the Niger basin. Nonlinear relationships were investigated using multifractal formalism and joint recurrence quantification analysis. A strong nonlinear relationship was observed between the teleconnection patterns and river discharge in Diola while other stations (Koulikoro, Ansongo, Niamey, Mopti, Kirango) showed no such relationship. The observation at Diola is attributed to its location (coastal) among other things. The multifractal strengths were found in the range of 0.58-2.86, suggesting fractal correlations between the parameters. There was no conclusive evidence of a linear and nonlinear relationship between solar activity and tropical river discharge within the Niger basin.

Sparse sampling and dilation operations on Gibbs capacities, and multifractal formalism

Sparse sampling and dilation operations on Gibbs capacities, and multifractal formalism

Multifractal analysis of Gaussian multiplicative chaos and applications

Let $M_{\gamma}$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D \subset \mathbb{R}^d$, $d \geq 1$. We find an explicit formula for its singularity spectrum by showing that $M_{\gamma}$ satisfies almost surely the multifractal formalism, i.e., we prove that its singularity spectrum is almost surely equal to the Legendre-Fenchel transform of its $L^q$-spectrum. Then, applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.