Multifractal Model Research Articles

Multifractal Characterization and Modeling of Blood Pressure Signals

In this paper, a multi-fractal analysis on a diastolic blood pressure signal is conducted. The signal is measured in a time span of circa one day through the multifractal detrended fluctuation analysis framework. The analysis is performed on asymptotic timescales where complex regulating mechanisms play a fundamental role in the blood pressure stability. Given a suitable frequency range and after removing non-stationarities, the blood pressure signal shows interesting scaling properties and a pronounced multifractality imputed to long-range correlations. Finally, a binomial multiplicative model is investigated showing how the analyzed signal can be described by a concise multifractal model with only two parameters.

Review on the Application of Low-Field Nuclear Magnetic Resonance Technology in Coalbed Methane Production Simulation.

Low-field nuclear magnetic resonance has become one of the main methods to characterize static parameters and dynamic changes in unconventional reservoirs. The research focus of this paper is process simulation of coalbed methane (CBM) production. The dynamic variation of pore volume with different pore sizes during pressure drop, methane desorption–diffusion process, and methane–water interaction during migration is discussed. Moreover, the calculation principles of NMR single and multifractal models are systematically described, and the applicability of NMR fractal models within different research contexts is discussed. Four aspects need urgent attention in the application of this technology in CBM production: (1) overburden NMR technology has limitations in characterizing the stress sensitivity of shale and high-rank coal reservoirs with micropores developed, and we should aim to enable an accurate description of micropore pore stress sensitivity; (2) dynamic NMR physical simulation of reservoir gas and water production based on in-situ and actual geological development conditions should become one of the key aspects of follow-up research; (3) low-temperature freeze–thaw NMR technology, as a new pore–fracture characterization method, needs to be further applied in characterizing the distribution characteristics of pores and fractures; and (4) NMR fractal model should be used as the main theoretical method to expand the simulation results. The applicability of different fractal models in characterizing pore–fracture structure (static) and CBM production process (dynamic) needs to be clarified.

Multifractal test for nonlinearity of interactions across scales in time series.

The creativity and emergence of biological and psychological behavior tend to be nonlinear, and correspondingly, biological and psychological measures contain degrees of irregularity. The linear model might fail to reduce these measurements to a sum of independent random factors (yielding a stable mean for the measurement), implying nonlinear changes over time. The present work reviews some of the concepts implicated in nonlinear changes over time and details the mathematical steps involved in their identification. It introduces multifractality as a mathematical framework helpful in determining whether and to what degree the measured series exhibits nonlinear changes over time. These mathematical steps include multifractal analysis and surrogate data production for resolving when multifractality entails nonlinear changes over time. Ultimately, when measurements fail to fit the structures of the traditional linear model, multifractal modeling allows for making those nonlinear excursions explicit, that is, to come up with a quantitative estimate of how strongly events may interact across timescales. This estimate may serve some interests as merely a potentially statistically significant indicator of independence failing to hold, but we suspect that this estimate might serve more generally as a predictor of perceptuomotor or cognitive performance.

From rough to multifractal volatility: The log S-fBM model

We introduce a family of random measures MH,T(dt), namely log S-fBM, such that, for H>0, MH,T(dt)=eωH,T(t)dt where ωH,T(t) is a Gaussian process that can be considered as a stationary version of an H-fractional Brownian motion. Moreover, when H→0, one has MH,T(dt)→M˜T(dt) (in the weak sense) where M˜T(dt) is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal (H=0) and rough volatility (0<H<1/2) models. The main properties of the log S-fBM are discussed and their estimation issues are addressed. We notably show that the direct estimation of H from the scaling properties of ln(MH,T([t,t+τ])), at fixed τ, can lead to strongly over-estimating the value of H. We propose a better GMM estimation method which is shown to be valid in the high-frequency asymptotic regime. When applied to a large set of empirical volatility data, we observe that stock indices have values around H=0.1 while individual stocks are characterized by values of H that can be very close to 0 and thus well described by a MRM. We also bring evidence that unlike the log-volatility variance ν2 whose estimation appears to be poorly reliable (though used widely in the rough volatility literature), the estimation of the so-called ”intermittency coefficient” λ2, which is the product of ν2 and the Hurst exponent H, appears to be far more reliable leading to values that seem to be universal for respectively all individual stocks and all stock indices.

Spectral analysis of multifractional LRD functional time series

Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, multifractionally integrated functional autoregressive moving averages processes can be introduced in this framework. The convergence to zero in the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved. Under a Gaussian scenario, a weak-consistent parametric estimator of the long-memory operator is then obtained by minimizing, in the norm of bounded linear operators, a divergence information functional loss. The results derived allow, in particular, to develop inference from the discrete sampling of the Gaussian solution to fractional and multifractional pseudodifferential models introduced in Anh et al. (Fract Calc Appl Anal 19(5):1161-1199, 2016; 19(6):1434–1459, 2016) and Kelbert (Adv Appl Probab 37(1):1–25, 2005).

Scaling of Acceleration Statistics in High Reynolds Number Turbulence

The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as an extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our Letter highlights the need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.

Exploiting fractal features to determine fatigue crack growth rates of metallic materials

An alternative approach to the classical fracture mechanics equation to quantify the fatigue crack growth rate is proposed and examined. The method exploits the fractal features of the propagating fatigue crack. The reference crack growth response of AISI 410 martensitic stainless steel is established using compact tension C(T) specimens. Fractal analysis of microscopic images along the edge length of the crack establishes the fractal dimensions, dF of the fatigue crack. The box-counting algorithm is developed for the fractal analysis based on optical images of the crack at 100X magnification and the resolution of 1090 pixels/mm. Results show that the crack initially exhibits a Euclidean nature (dF ≈ 1). The fractal dimension increases steadily with increasing crack length in the Paris crack growth rate region with 1.05 < dF < 1.24. The corresponding stress intensity factor range varies between 18 ≤ ΔKI ≤ 40 MPa√m. The fractal dimension, dF correlates linearly with the normalized stress intensity factor range, ΔKKIC within the Paris crack growth region. This enables the multifractal fatigue crack growth rate equation to be established in terms of the fractal dimension, Paris coefficient and exponent, and fatigue fracture properties of the material. The fatigue crack growth rate of the material could then be determined using the multifractal fatigue crack growth rate model without requiring the geometry factor of the crack.

Developments in Quantitative Assessment and Modeling of Mineral Resource Potential: An Overview

The special issue entitled “Developments in Quantitative Assessment and Modeling of Mineral Resource Potential” is composed of 17 papers that cover a diverse range of approaches to mineral resource assessment, including mainly multivariate statistical analysis, fractal and multifractal modeling, geostatistical modeling, machine learning, mathematical morphology and 3D mineral exploration. This introductory article first provides an overview of the developments and existing methods in quantitative assessment of mineral resources. Then, brief introductions of each of the 17 papers are provided. These papers can be grouped into three themes: (1) multifractal theory for mineral resource assessment; (2) machine learning for mineral prospectivity mapping; and (3) GIS-based 3D modeling for mineral exploration. The 17 papers either proposed novel methods or demonstrated innovative applications of existing methods.

Infectious Inflammatory Processes and the Role of Bioactive Agent Released from Imino-Chitosan Derivatives Experimental and Theoretical Aspects.

The paper focuses on the development of a multifractal theoretical model for explaining drug release dynamics (drug release laws and drug release mechanisms of cellular and channel-type) through scale transitions in scale space correlated with experimental data. The mathematical model has been developed for a hydrogel system prepared from chitosan and an antimicrobial aldehyde via covalent imine bonds. The reversible nature of the imine linkage points for a progressive release of the antimicrobial aldehyde is controlled by the reaction equilibrium shifting to the reagents, which in turn is triggered by aldehyde consumption in the inhibition of the microbial growth. The development of the mathematical model considers the release dynamic of the aldehyde in the scale space. Because the release behavior is dictated by the intrinsic properties of the polymer–drug complex system, they were explained in scale space, showing that various drug release dynamics laws can be associated with scale transitions. Moreover, the functionality of a Schrödinger-type differential equation in the same scale space reveals drug release mechanisms of channels and cellular types. These mechanisms are conditioned by the intensity of the polymer–drug interactions. It was demonstrated that the proposed mathematical model confirmed a prolonged release of the aldehyde, respecting the trend established by in vitro release experiments. At the same time, the properties of the hydrogel recommend its application in patients with intrauterine adhesions (IUAs) complicated by chronic endometritis as an alternative to the traditional antibiotics or antifungals.

Sediment and Radioactivity Transport in the Bohai, Yellow, and East China Seas: A Modeling Study

This paper is concerned with the development of a radionuclide dispersion model for the nuclear power plants in the Bohai, Yellow, and East China seas (BYECS) characterized by high turbidity and multi-scale circulations, focusing on the comparison of dispersion processes of 137Cs depending upon, in particular, the suspended sediment concentration and erosion/sedimentation processes. The simulations were carried out using a multi-fraction sediment transport model embedded in the semi-implicit Eulerian–Lagrangian finite-element coupled wave-circulation model linked with the model of radionuclide transport, which describes the key radionuclide transfer processes in the system of water–multi-fraction sediments. In contrast to the Eulerian models used for hydrodynamics and sediment transport processes, the Lagrangian technique was applied to simulate the transport of radionuclides. The simulation results for total suspended concentration agreed with in situ measurements and the Geostationary Ocean Color Imager data. The results of the simulation of hypothetical releases of 137Cs from four nuclear power plants (NPPs) placed in BYECS essentially differ from the real release of activity in the Pacific Ocean shelf due to the Fukushima Daiichi accident, which took place at the same time and released activity that was similar. The total amount of bottom contamination of 137Cs in releases from the Sanmen, Hanbit, and Hongyanhe NPPs was about 40% of dissolved component, and the total amount of suspended component was about 20% of dissolved component, in contrast with the Fukushima Daiichi accident, where the particulate component was only 2%. The results demonstrate the importance of erosion processes in the budget of 137Cs in shallow areas around the Sanmen and Hanbit NPPs, where strong wind and tidal currents took place.

Langmuir Probe Perturbations during In Situ Monitoring of Pulsed Laser Deposition Plasmas.

The recent advancements in pulsed laser deposition (PLD) control via plasma diagnostics techniques have been positive and raised questions on the limitation of some techniques, such as the Langmuir probe (LP). The particularities of laser-produced plasma can lead to incorrect interpretation of collected electrical signal. In this paper, we explored the limitations of LP as a technique for in situ PLD control by performing investigations on several metallic plasmas, expanding in various Ar atmosphere conditions. Sub-microsecond modulation was seen in the reconstructed IV characteristics attributed to non-equilibrium dynamics of the ejected charges. A perturbative regime was recorded for Ar pressures higher than 2 Pa, where ionic bursts were observed in the electron saturation region. This perturbation was identified as a plasma fireball. A non-linear multifractal model was developed here to explore these new regimes of the LP. The strange attractors characterizing each fireball were reconstructed, and their evolution with the Ar pressure is discussed. Both short- and long-time non-linear behavior were correlated via probe bias, and the pressure effect on the strange attractor’s defining the fireball-like behavior was investigated. A good correlation was noticed between the simulated data and experimental findings.

Hidden Markov Model and multifractal method-based predictive quantization complexity models vis-á-vis the differential prognosis and differentiation of Multiple Sclerosis’ subgroups

Hidden Markov Model (HMM) is a stochastic process where implicit or latent stochastic processes can be inferred indirectly through a sequence of observed states. HMM as a mathematical model for uncertain phenomena is applicable for the description and computation of complex dynamical behaviours enabling the mathematical formulation of neural dynamics across spatial and temporal scales. The human brain with its fractal structure demonstrates complex dynamics and fractals in the brain are characterized by irregularity, singularity and self-similarity in terms of form at different observation levels, making detection difficult as observations in real-time occurrences can be time variant, discrete, continuous or noisy. Multiple Sclerosis (MS) is an autoimmune degenerative disease with time and space related dissemination, leading to neuronal apoptosis, coupled with some subtle features that could be overlooked by physicians. This study, through the proposed integrated approach with multi-source complex spatial data, aims to attain accurate prediction, diagnosis and prognosis of MS subgroups by HMM with Viterbi algorithm and Forward–Backward algorithm as the dynamic and efficient products of knowledge-based and Artificial Intelligence (AI)-based systems within the framework of precision medicine. Multifractal Bayesian method (MFM) accordingly applied to identify and eliminate “insignificant” irregularities while maintaining “significant” singularities. An efficient modelling of HMM is proposed to diagnose and predict the course of MS while using MFM method. Unlike the methods employed in previous studies, our proposed integrated novel method encompasses the subsequent approaches based on reliable MS dataset (X) collected: (i) MFM method was applied (X) to MS dataset to characterize the irregular, self-similar and significant attributes, thus, attributes with “insignificant” irregularities were eliminated and “significant” singularities were maintained. MFM-MS dataset (Xˆ) was generated. (ii) The continuous values in the MS dataset (X) and MFM-MS dataset (Xˆ) were converted into discrete values through vector quantization method of the HMM (iii) Through transitional matrices, different observation matrices were computed from the both datasets. (v) Computational complexity has been computed for both datasets. (vi) The results of the HMM models based on observation matrices obtained from both datasets were compared. In terms of the integrated HMM model proposed and the MS dataset handled, no earlier work exists in the literature. The experimental results demonstrate the applicability and accuracy of our novel proposed integrated method, HMM and Multifractal (HMM-MFM) method, for the application to the MS dataset (X). Compared with conventional methods, our novel method has achieved more superiority regarding extracting subtle and hidden details, which are significant for distinguishing different dynamic and complex systems including engineering and other related applied sciences. Thus, we have aimed at pointing a new frontier by providing a novel alternative mathematical model to facilitate the critical decision-making, management and prediction processes among the related areas in chaotic, dynamic complex systems with intricate and transient states.

The Relationship between Fe Mineralization and Magnetic Basement Faults using Multifractal Modeling in the Esfordi and Behabad Areas (BMD), Central Iran

AbstractMultifractal modeling is a mathematical method for the separation of a high potential mineralized background from a non‐mineralized background. The Concentration‐Distance to Fault structures (C‐DF) fractal model and the distribution of the known iron (Fe) deposits/mines seen in the Esfordi and Behabad 1:100,000 sheets from the Bafq region of central Iran are used to distinguish Fe mineralization based on their distance to magnetic basement structures and surface faults, separately, using airborne geophysical data and field surveys. Application of the C‐DF fractal model for the classification of Fe mineralizations in the Esfordi and Behabad areas reveals that the main ones show a correlation with their distance from magnetic basement structures. Accordingly, the distances of Fe mineralizations with grades of Fe higher than 55% (43% &lt; Fe ≤ 60%) are located at a distance of less than 1 km, whereas for surfacial faults with grades of 43% ≤ Fe ≤ 60%, the distances are 3162&lt; DF ≤ 4365 m from the faults. Thus, there is a positive relationship between Fe mineralization and magnetic basement structures. Also, the proximity evidence of Precambrian high‐grade Fe mineralization related to magnetic basement structures indicates syn‐rifting tectonic events. Finally, this C‐DF fractal model can be used for exploration of magmatic and hydrothermal ore deposits.

Universality and Intermittency of Pair Dispersion in Turbulence.

Turbulence can disperse a concentrated parcel of pollutants at a rate over nine orders of magnitude higher than its purely diffusive counterpart. One intriguing signature of turbulent dispersion is its superdiffusive scaling. However, the universality of this scaling law is still in question. By leveraging a new laboratory facility, particle pairs with small initial separations can be tracked over four decades of separation in time and five decades of separation in squared displacement, thereby observing the full range of dispersion scaling laws. The results show that the classical Richardson cubic scaling will be reached not for an initial separation asymptotically close to zero but at a critical value, and this value does not appear to depend on the Reynolds number, providing an effective way to study universal dispersion dynamics. Additionally, the results agree well with the prediction based on the multifractal model and may help reconcile different reported scaling laws from laboratory experiments and field studies.

Spatial Downscaling Model Combined with the Geographically Weighted Regression and Multifractal Models for Monthly GPM/IMERG Precipitation in Hubei Province, China

High spatial resolution (1 km or finer) precipitation data fields are crucial for understanding the Earth’s water and energy cycles at the regional scale for applications. The spatial resolution of the Global Precipitation Measurement (GPM) mission (IMERG) satellite precipitation products is 0.1° (latitude) × 0.1° (longitude), which is too coarse for regional-scale analysis. This study combined the Geographically Weighted Regression (GWR) and the Multifractal Random Cascade (MFRC) model to downscale monthly GPM/IMERG precipitation products from 0.1° × 0.1° (approximately 11 km × 11 km) to 1 km in Hubei Province, China. This work’s results indicate the following: (1) The original GPM product can accurately express the precipitation in the study area, which highly correlates with the site data from 2015 to 2017 (R2 = 0.79) and overall presents the phenomenon of overestimation. (2) The GWR model maintains the precipitation field’s overall accuracy and smoothness, with even improvements in accuracy for specific months. In contrast, the MFRC model causes a slight decrease in the overall accuracy of the precipitation field but performs better in reducing the bias. (3) The GWR-MF combined with the GWR and MFRC model improves the observation accuracy of the downscaling results and reduces the bias value by introducing the MFRC to correct the deviation of GWR. The conclusion and analysis of this paper can provide a meaningful experience for 1 km high-resolution data to support related applications.

The Spatiotemporal Evolution of Rainfall Extremes in a Changing Climate: A CONUS‐Wide Assessment Based on Multifractal Scaling Arguments

AbstractGiven the rapidly changing climate, accurate spatiotemporal information on the evolution of extreme rainfall events is required for flood risk assessment and the design of resilient infrastructure. Consequently, various research efforts have focused on investigating the appropriateness of various parametric and non‐parametric approaches in modeling the observed changes in the frequency of extreme rainfall over time. Yet, the assumption of stationarity, or the change of model parameters when accounting for nonstationary rainfall, may magnify estimation uncertainty of rain rates associated with low exceedance probabilities. Moreover, the use of climate model results may yield inconclusive outcomes, given the existence of epistemic uncertainties in the frequency of extreme events developing on smaller spatial scales or over complex terrain. Herein, we employ a parametric approach based on multifractal scaling arguments, along with high‐resolution (4‐km) hourly precipitation estimates covering a 40‐year period over CONUS, to derive Intensity‐Duration‐Frequency curves and investigate the spatiotemporal evolution of extreme rainfall over a wide range of characteristic temporal scales and exceedance probability levels. Considering the robustness of the multifractal models even when fitted to short rainfall records, we uniquely apply the framework to sequential 10‐year segments of data, where the rainfall process can be reasonably assumed stationary. The obtained results reveal that existing infrastructure may be severely impacted by the intensification of precipitation extremes due to climate change, with the observed trends being significantly influenced by the topography and rainfall climatology of each region, while depending on the averaging durations and return periods of interest.

FRACTAL CHARACTERISTICS OF NETWORK TRAFFIC AND ITS CORRELATION WITH NETWORK SECURITY

Based on the analysis of the self-similarity of network traffic, a network anomaly detection technology is proposed by combining with the fuzzy logic so as to explore the fractal characteristics of network traffic. The concepts of network traffic and network security are introduced. Then, a network traffic model of network traffic is proposed based on the fractal theory and wavelet analysis. Finally, a distributed denial of service (DDoS) that attacks the monitoring and intensity judgment method is put forward based on the fuzzy logic theory. The results show that the autocorrelation function of the multifractal wavelet model constructed based on the local Hurst exponent (LHE) can reach a mean square error (MSE) of [Formula: see text], which proves that the network traffic model proposed can reduce the impact of the non-stationary characteristics of the network traffic on the modeling accuracy. The network security detection method proposed can monitor the DDoS attacks and can accurately judge the attack intensity in real time. The research in this study provides an important reference for the scientific operation of the network.

Scaling Laws for Partially Developed Turbulence

We formulate multifractal models for velocity differences and gradients which describe the full range of length scales in turbulent flow, namely: laminar, dissipation, inertial, and stirring ranges. The models subsume existing models of inertial range turbulence. In the localized ranges of length scales in which the turbulence is only partially developed, we propose multifractal scaling laws with scaling exponents modified from their inertial range values. In local regions, even within a fully developed turbulent flow, the turbulence is not isotropic nor scale invariant due to the influence of larger turbulent structures (or their absence). For this reason, turbulence that is not fully developed is an important issue which inertial range study can not address. In the ranges of partially developed turbulence, the flow can be far from universal, so that standard inertial range turbulence scaling models become inapplicable. The model proposed here serves as a replacement. Details of the fitting of the parameters for the τp and ζp models in the dissipation range are discussed. Some of the behavior of ζp for larger p is unexplained. The theories are verified by comparing to high resolution simulation data.

A correspondence between the multifractal model of turbulence and the Navier-Stokes equations.

The multifractal model of turbulence (MFM) and the three-dimensional Navier-Stokes equations are blended together by applying the probabilistic scaling arguments of the former to a hierarchy of weak solutions of the latter. This process imposes a lower bound on both the multifractal spectrum [Formula: see text], which appears naturally in the Large Deviation formulation of the MFM, and on [Formula: see text] the standard scaling parameter. These bounds respectively take the form: (i) [Formula: see text], which is consistent with Kolmogorov's four-fifths law ; and (ii) [Formula: see text]. The latter is significant as it prevents solutions from approaching the Navier-Stokes singular set of Caffarelli, Kohn and Nirenberg. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Generalization of Higuchi’s fractal dimension for multifractal analysis of time series with limited length

There exist several methodologies for the multifractal characterization of nonstationary time series. However, when applied to sequences of limited length, these methods often tend to overestimate the actual multifractal properties. To address this aspect, we introduce here a generalization of Higuchi’s estimator of the fractal dimension as a new way to characterize the multifractal spectrum of univariate time series or sequences of relatively short length. This multifractal Higuchi dimension analysis (MF-HDA) method considers the order-q moments of the partition function provided by the length of the time series graph at different levels of subsampling. The results obtained for different types of stochastic processes, a classical multifractal model, and various real-world examples of word length series from fictional texts demonstrate that MF-HDA provides a reliable estimate of the multifractal spectrum already for moderate time series lengths. Practical advantages as well as disadvantages of the new approach as compared to other state-of-the-art methods of multifractal analysis are discussed, highlighting the particular potentials of MF-HDA to distinguish mono- from multifractal dynamics based on relatively short sequences.