Multifractal Properties Research Articles

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.

Quantifying the Pore Heterogeneity of Alkaline Lake Shale during Hydrous Pyrolysis by Using the Multifractal Method

Distinguishing itself from marine shale formations, alkaline lake shale, as a significant hydrocarbon source rock and petroleum reservoir, exhibits distinct multifractal characteristics and evolutionary patterns. This study employs a combination of hydrous pyrolysis experimentation, nitrogen adsorption analysis, and multifractal theory to investigate the factors influencing pore heterogeneity and multifractal dimension during the maturation process of shale with abundant rich alkaline minerals. Utilizing partial least squares (PLS) analysis, a comparative examination is conducted, elucidating the disparate influence of mineralogical composition on their respective multifractal dimensions. The findings reveal a dynamic evolution of pore characteristics throughout the maturation process of alkaline lake shale, delineated into three distinct stages. Initially, in Stage 1 (200 °C to 300 °C), both ΔD and H demonstrate an incremental trend, rising from 1.2699 to 1.3 and from 0.8615 to 0.8636, respectively. Subsequently, in Stages 2 and 3, fluctuations are observed in the values of ΔD and D, while the H value undergoes a pronounced decline to 0.85. Additionally, the parameter D1 exhibits a diminishing trajectory across all stages, decreasing from 0.859 to 0.829, indicative of evolving pore structure characteristics throughout the maturation process. The distinct alkaline environment and mineral composition of alkaline lake shale engender disparate diagenetic effects during its maturation process compared with other shale varieties. Consequently, this disparity results in contrasting evolutionary trajectories in pore heterogeneity and multifractal characteristics. Specifically, multifractal characteristics of alkaline lake shale are primarily influenced by quartz, potassium feldspar, clay minerals, and alkaline minerals.

Multifractal characterization of fried foods pore microstructure

This study aimed to unravel the multifractal properties of internal microstructure in relation to textural evolution of meat-analog (MA) based coated fried foods. Wheat and rice flour-based batters were used to coat MA model that was deep fried at 180 °C for 2, 4 and 6 min in canola oil. Internal pore microstructures were investigated by X-ray microtomography based multifractal analysis (MFA) and texture were assessed by mechanical texture analyzer. Results showed that batter coatings greatly influenced the formation of internal microstructure during frying as well as post-frying mass redistribution and textural evolution. The frying time impacted the evolution of micropores, and porosity showed a positive correlation with the fat content of MA-based fried products. Porous, low-moisture and high-fat containing coated fried MA-based product demonstrated higher textural stability. Internal microstructure (pore distribution) of fried MAs were heterogenous and possessed multifractal scaling behavior, as depicted by MFA outcomes (Singularity & Rényi spectra). Strong correlation observed between moisture-fat, texture, porosity and selected multifractal parameters (Δα, Δfα, R-L, D1, D2, ΔD).

Multifractal analysis of the perimeters of glaciers in the Svalbard Archipelago

The investigation of the evolution of glaciers largely relies on the characterisation of extensive quantities like their mass, area, and perimeter. In this work we use fractal and multifractal analysis to investigate the non-extensive structural properties of the perimeters of glaciers in the Svalbard Archipelago. We show that the perimeters of the glaciers exhibit a fractal structure with a fractal dimension Df≃1.25 , independently from the area of the glaciers. The investigation of the multifractal properties of the perimeters shows that small glaciers exhibit a more pronounced multifractal structure, as witnessed by the larger range of generalized dimensions Dq needed to characterise them. The range ΔD q of generalised dimensions required to characterise the multifractal perimeter of a glacier exhibits a power-law dependence with exponent −1.2 from the area, and represents a non-extensive parameter able to grab effectively the dependence of the multifractal structure of the perimeters on the size of glaciers. The comparison with similar results obtained in a previous study performed on glaciers in the Lombardy region of the Italian Alps confirms the robustness of the analysis performed, which does not appear to be affected by the morphology of the substrate or by climate conditions.

Multifractal detrended fluctuation analysis on the fracture surface of polycarbonate and acrylonitrile-butadiene-styrene alloy

Multifractal detrended fluctuation analysis (MF-DFA) has been carried out to evaluate the multifractal properties of the fracture morphology for Polycarbonate (PC) and Acrylonitrile-butadiene-styrene (ABS) alloy, and the relationship between the long-range correlation parameter of the samples fracture - Hurst exponent and fractal dimension was explored and analyzed. The results suggest that the existence of multifractal properties within the micrometer scale domain we investigated for the fracture surface of PC/ABS alloy samples. The Hurst exponent and fractal dimension can both be used to measure and characterize the fracture surface morphology of the PC/ABS alloys. As the content of compatibilizer PP-g-MAH increased, the tensile strength of PC/ABS alloy decreased, the fracture surface peaks tend to dominate, the Hurst exponent increased, the fractal dimension decreased, and its strength retention rate increases, displaying persistence.

Is there a time-varying nexus between stock market liquidity and informational efficiency? – A cross-regional evidence

PurposeThis study aims to measure the degree of volatility comovement between stock market liquidity and informational efficiency across Asia, Europe, North-South America, Africa, and the Pacific Ocean over three decades. In particular, the authors analyze the extent of the time-varying nexus between different aspects of stock market liquidity and multifractal scaling properties of the stock return series across various regions and diversified market conditions. This study further investigates several factors altering the degree of dynamic conditional correlations (DCCs) between the efficiency and liquidity of the domestic stock markets.Design/methodology/approachThe study measures five aspects of stock market liquidity – tightness, depth, breadth, immediacy, and adjusted immediacy. The authors evaluate the multifractal scaling properties of the stock return series to measure the level of stock market efficiency across the regions and diversified market conditions. The study uses the dynamic conditional correlation-multivariate generalized autoregressive conditional heteroscedasticity framework to quantify the degree of volatility comovement between liquidity and efficiency over the period.FindingsThe study finds the presence of stronger volatility comovement between inefficiency and illiquidity due to the price impact characteristics of the stock markets irrespective of different regions and diversified market conditions. The extent of time-variation increased following the shock periods, indicating the significant role of the financial crisis in increasing the volatility comovement between inefficiency and illiquidity. The highest degree of time-varying correlation is observed in the developed stock markets of Northwestern and Northern Europe compared to the regional and emerging counterparts. On the other hand, weak DCCs are observed in the emerging stock markets of Europe.Originality/valueThe output of the present study assists investors in identifying diversification opportunities across the regions. Additionally, the study has significant implications for market regulators, aiding in predicting future troughs and peaks. The prediction, in turn, helps formulate capital market development plans during dynamic economic situations.

Complexity synchronization in living matter: a mini review.

Fractal time series have been argued to be ubiquitous in human physiology and some of the implications of that ubiquity are quite remarkable. One consequence of the omnipresent fractality is complexity synchronization (CS) observed in the interactions among simultaneously recorded physiologic time series discussed herein. This new kind of synchronization has been revealed in the interaction triad of organ-networks (ONs) consisting of the mutually interacting time series generated by the brain (electroencephalograms, EEGs), heart (electrocardiograms, ECGs), and lungs (Respiration). The scaled time series from each member of the triad look nothing like one another and yet they bear a deeply recorded synchronization invisible to the naked eye. The theory of scaling statistics is used to explain the source of the CS observed in the information exchange among these multifractal time series. The multifractal dimension (MFD) of each time series is a measure of the time-dependent complexity of that time series, and it is the matching of the MFD time series that provides the synchronization referred to as CS. The CS is one manifestation of the hypothesis given by a "Law of Multifractal Dimension Synchronization" (LMFDS) which is supported by data. Therefore, the review aspects of this paper are chosen to make the extended range of the LMFDS hypothesis sufficiently reasonable to warrant further empirical testing.

Response of Hydrodynamic Characteristics to Tillage-Induced Microtopography of Rill Erosion Processes under Heavy Rainfalls

Response of Hydrodynamic Characteristics to Tillage-Induced Microtopography of Rill Erosion Processes under Heavy Rainfalls

Application of multifractal analysis for recognition of sleep apnea using electroencephalogram

Sleep apnea is a common sleep disorder that causes an increased risk of mortality and affects quality of life. Identifying cases of sleep apnea is vital to promptly treating the condition. The goal of the work is to develop a software package for automatically detecting episodes of obstructive sleep apnea syndrome. The paper presents an approach based on the study of multifractal properties of the electroencephalogram (EEG) signal for detecting sleep apnea events. The results of tests conducted using the PhysioNet CAP Sleep database indicate excellent quality of detection of sleep apnea against the background of other diseases: using the Hurst index – sensitivity (90.00%) and specificity (91.50%), using the multifractal detrended fluctuation method analysis (by spectrum width) - sensitivity (90.00%) and specificity (90.50%).The use of a sleep apnea episode recognition system ensures timely detection of the disease at an early stage and thereby improves the effectiveness of the treatment of this disease.

Detection of Pipeline Leaks Using Fractal Analysis of Acoustic Signals

In this paper, the possibility of using monofractal and multifractal analysis of acoustic signals of pipelines to detect leaks is considered. An experimental stand has been created to study the fractal characteristics of acoustic signals of pipelines with “slit” type defects. During the experiments, defects of the “slit” type pipeline with dimensions of 2 mm, 8 mm, and 20 mm were modeled. Detrended fluctuation analysis (DFA) and the multifractal detrended fluctuation analysis (MF-DFA) were used. As a result of the experimental studies, it was found that the occurrence of leakage leads to the occurrence of anticorrelated vibrations in a pipeline with multifractal properties. The analyses of acoustic signals by DFA and MF-DFA methods make it possible to reliably determine the leakage. The Hurst exponent and the width of the multifractal spectrum can serve as indicators of the occurrence of leaks in pipelines.

Multifractal characterization of nystagmus eye movements.

In this work, we investigate the multifractal properties of eye movement dynamics of children with infantile nystagmus, particularly the fluctuations of its velocity. The eye movements of three children and one adult with infantile nystagmus were evaluated in a simple task in comparison with 28 children with no ocular pathologies. Four indices emerge from the analysis: the classical Hurst exponent, the singularity strength corresponding to the maximum of the singularity spectrum, the asymmetry of the singularity spectrum, and the multifractal strength, each of which characterizes a particular aspect of eye movement dynamics. Our findings indicate that, when compared to children with no ocular pathologies, patients with infantile nystagmus present lower values of all indices. Except for the multifractal strength, the difference in the remaining indices is statistically significant. To test whether the characterization of patients with infantile nystagmus in terms of multifractality indices allows them to be distinguished from children without ocular pathologies, we performed an unsupervised clustering analysis and classified the subjects using supervised clustering techniques. The results indicate that these indices do, indeed, distinctively characterize the eye movements of patients with infantile nystagmus.

Fractal Analysis of Polarizability in Graphite Deposits: Methodological Integration for Geological Prediction and Exploration Efficiency

Most geophysical and geochemical data are commonly acknowledged to exhibit fractal and multifractal properties, but the fractal characteristics of polarizability have received limited attention from the literature. The present study demonstrates that the polarizability data of the graphite deposits have fractal characteristics and introduces the fractal method for its quantitative analysis to indicate and predict the properties of graphite deposits. The results show that the concentration-area (C-A) method is superior to classical interpolation in anomaly extraction but inferior to the spectrum-area (S-A) method in the coverage region. Because the type of graphite ore is sedimentary-metamorphic in this area, the graphite ore-bodies can be regarded as a special stratum, which is different from most metal deposits, and the anomaly of graphite ore are shown in the background mode of the S-A method. The high values of the background mode effectively indicate the potential areas where the graphite-bearing strata occur, while observing a decrease in the power-law exponent (β) of the background mode as the width of ore-bodies increases. The validity of this conclusion was confirmed based on the vertical profiles of the predicted area, and the uncharted ore vein was thereby identified. Furthermore, it was found that the anomaly mode can serve as a grade indicator of graphite ore rather than delineating the fault. By integrating the background and anomaly modes of the S-A method, we can quantitatively predict and effectively identify high-grade targets from sedimentary deposits containing minerals in future exploration.

Damage evaluation of rock blasting based on multi-fractal study

This study shows the failure mechanism of rock under blasting loads by CT scanning and utilizing fine sandstone as a test specimen. The results indicated that the 3D reconstruction technique can be employed to construct macroscopic fissures in the rock. The complexity of macroscopic fissures in the rock after blasting is quantitatively evaluated using the 3D box-counting dimension. In addition, a multi-fractal method is proposed for comprehensively evaluating macroscopic fissures and microscopic damage of rock after blasting. Compared to the numerical simulation results, the multi-fractal dimension represents the cracks and damage of rock after blasting more precisely than the simple fractal dimension. The blasting experiments of the jointed sandstone demonstrated that the 3D box dimension is suitable for evaluating the 3D macroscopic fractures of the rock after blasting and that it is a straightforward method of macroscopic evaluation. The multi-fractal dimension can evaluate the macroscopic fissures and microscopic damage of various CT scanning rock layers with greater precision.

Complexity synchronization in emergent intelligence

In this work, we use a simple multi-agent-based-model (MABM) of a social network, implementing selfish algorithm (SA) agents, to create an adaptive environment and show, using a modified diffusion entropy analysis (DEA), that the mutual-adaptive interaction between the parts of such a network manifests complexity synchronization (CS). CS has been shown to exist by processing simultaneously measured time series from among organ-networks (ONs) of the brain (neurophysiology), lungs (respiration), and heart (cardiovascular reactivity) and to be explained theoretically as a synchronization of the multifractal dimension (MFD) scaling parameters characterizing each time series. Herein, we find the same kind of CS in the emergent intelligence of groups formed in a self-organized social interaction without macroscopic control but with biased self-interest between two groups of agents playing an anti-coordination game. This computational result strongly suggests the existence of the same CS in real-world social phenomena and in human–machine interactions as that found empirically in ONs.

Pallidal multifractal complexity is a new potential physiomarker of dystonia

ObjectiveLow-frequency 4–12 Hz pallidal oscillations are being considered as potential physiomarkers for dystonia. We suggest investigating the multifractal properties of pallidal activity as an additional marker. MethodsWe employed local field potentials (LFP) recordings from 23 patients with dystonia who were undergoing deep brain stimulation (DBS) surgery to explore the connection between disease severity and the multifractal characteristics of pallidal activity. Furthermore, we performed an analysis of LFP recordings from four patients, following the externalization of DBS lead electrodes, to investigate the impact of DBS and neck muscle vibration on multifractal parameters. ResultsGreater dystonia severity exhibited a correlation with a narrower multifractal spectrum width but higher multifractal spectral asymmetry. Both GPi DBS and muscle vibration in dystonia patients expanded the multifractal spectrum width while restoring multifractal spectral symmetry. Notably, the threshold peak intensities for an increase in multifractal spectrum width substantially overlapped with the optimal volume of tissue activated. A broader multifractal spectrum during DBS corresponded to more favorable clinical outcomes. ConclusionsMultifractal properties of pallidal neuronal activity serve as indicators of neural dysfunction in dystonia. SignificanceThese findings suggest the potential of utilizing multifractal characteristics as predictive factors for the DBS outcome in dystonia.

Investigation on acoustic emission characteristics of fault stick-slip under different lateral pressures

To explore the effect of different stress environments on fault-slip rockbursts. Bidirectional shear friction experiments with different lateral pressures were conducted on precracked syenogranites buried at 800 m. The macroscopic statistical parameters (cumulative number of AE events, magnitude and b value) and local characteristic parameters (amplitude and dominant frequency) of acoustic emission during the stick-slip process under different lateral pressures were investigated. In addition, based on fractal theory, the nonlinear characteristics of AE spectrum were analyzed. On this basis, the microscopic mechanism of fault stick-slip was discussed. The results show that the lateral pressure influences the friction strength of the fault and stick-slip motion characteristics. With increasing lateral pressure, the proportion of transgranular shear fractures increases, which leads to an increase of cumulative number of AE events and magnitude. The periodic decrease in the b value is more significant at high lateral pressure. There is a good correlation between a high-magnitude AE event and a stress drop. The AE frequency with phased response characteristics can be used to effectively identify the evolution of fault stick-slip instability at the laboratory scale. A sharp increase in the amplitude of the dominant frequency can be regarded as one of the precursory features of fault stick-slip instability. The AE frequency spectra have multifractal characteristics, that differ among the different stages. The maximum multifractal dimension and spectral width can reflect the difference in energy released during fault stick-slip motion.

Analysis of the relationship of various pathologies with the degree of multifractality of electrical activity of the brain

The review is devoted to the analysis of the relationship between dynamic changes in patterns of electrical activity of the brain during the occurrence of mental disorders in the form of paranoid schizophrenia and depression and in patterns of brain activity in cardiovascular pathology associated with permanent atrial fibrillation, as well as indicators of multifractality of the studied patterns. To assess these indicators of electroencephalographic patterns, we describe a method of multifractal analysis based on the search for maxima of wavelet coefficient modules, and to isolate the fractal component of the signal in the power spectrum we describe a method of autospectral analysis with irregular resampling. It has been shown that the main differences between the multifractal properties of the electrical activity of the brain in health and in pathology are the different widths of the multifractality spectrum and its location, associated with different types of sequential pattern values. In this regard, the multifractality indicators can serve as informative markers of neuronal disorders and can be included in a set of tests for studying various pathologies.

Multifractal characterization and recognition of animal behavior based on deep wavelet transform

The study conduct an in-depth exploration of the multifractal characteristics of dairy cows behavioral data, aiming to reveal their complexity and representation in behavioral patterns. By means of Multifractal Detrended Fluctuation Analysis (MFDFA) in conjunction with deep wavelet transform, we extract multifractal indices that precisely depict the differences and dynamic changes of cows behavior. Further, we delves into the potential correlations between these multifractal features and the physiological states of dairy cattle, particularly focusing on their potential applications in predicting different physiological conditions. And an assessment model is developed based on these fractal indices and utilize advanced machine learning techniques to evaluate their effectiveness and accuracy in predicting the physiological states of dairy cattle. This study not only unveils the multifractal properties of cattle behavior but also demonstrates how these features can be utilized to predict significant physiological states, providing new scientific bases and methods for dairy cattle health management.

Multifractal dimensions for orthogonal-to-unitary crossover ensemble.

Multifractal analysis is a powerful approach for characterizing ergodic or localized nature of eigenstates in complex quantum systems. In this context, the eigenvectors of random matrices belonging to invariant ensembles naturally serve as models for ergodic states. However, it has been found that the finite-size versions of multifractal dimensions for these eigenvectors converge to unity logarithmically slowly with increasing system size N. In fact, this strong finite-size effect is capable of distinguishing the ergodicity behavior of orthogonal and unitary invariant classes. Motivated by this observation, in this work, we provide semi-analytical expressions for the ensemble-averaged multifractal dimensions associated with eigenvectors in the orthogonal-to-unitary crossover ensemble. Additionally, we explore shifted and scaled variants of multifractal dimensions, which, in contrast to the multifractal dimensions themselves, yield distinct values in the orthogonal and unitary limits as N→∞ and, therefore, may serve as a convenient measure for studying the crossover. We substantiate our results using Monte Carlo simulations of the underlying crossover random matrix model. We then apply our results to analyze the multifractal dimensions in a quantum kicked rotor, a Sinai billiard system, and a correlated spin-chain model in a random field. The orthogonal-to-unitary crossover in these systems is realized by tuning relevant system parameters, and we find that in the crossover regime, the observed finite-dimension multifractal dimensions can be captured very well with our results.

Effect of correlation time of combustion noise on early warning indicators of thermoacoustic instability.

In this paper, we analyze the effects of finite correlation time (noise color) of combustion noise on noise-induced coherence and early warning indicators (EWIs) via numerical and experimental studies. We consider the Rijke tube as a prototypical combustion system and model combustion noise as an additive Ornstein-Uhlenbeck process while varying noise intensity and correlation time. We numerically investigate corresponding effects on coherence resonance and multi-fractal properties of pressure fluctuations. Subsequently, we experimentally validate results and elucidate the influence of noise color and intensity on trends in coherence resonance and multi-fractal measures that can be expected in a practical scenario using an electroacoustic simulator. We find that the coherence factor, which quantifies the relative contribution of coherent oscillations in a noisy signal, increases as the system approaches the thermoacoustic instability-irrespective of the correlation time. It works at most levels of combustion noise (except for too low and too high noise levels). The Hurst exponent reduces as the system approaches thermoacoustic instability only when the correlation time is small. These results have implications on the prediction and monitoring of thermoacoustic instability in practical combustors.