Multifractal Properties Research Articles

Multifractal Analysis of the Influence of Indole-3-Acetic Acid on Fast-Activating Vacuolar (FV) Channels of Beta vulgaris L. Taproot Cells.

In this paper, the multifractal properties of the ion current time series in the fast-activating vacuolar (FV) channels of Beta vulgaris L. taproot cells were investigated. These channels are permeable for only monovalent cations and mediate K+ at very low concentrations of cytosolic Ca2+ and large voltages of either polarity. Using the patch clamp technique, the currents of the FV channels in red beet taproot vacuoles were recorded and analysed by using the multifractal detrended fluctuation analysis (MFDFA) method. The activity of the FV channels depended on the external potential and was sensitive to the auxin. It was also shown that the singularity spectrum of the ion current in the FV channels is non-singular, and the multifractal parameters, i.e., the generalised Hurst exponent and the singularity spectrum, were modified in the presence of IAA. Taking into account the obtained results, it can be suggested that the multifractal properties of fast-activating vacuolar (FV) K+ channels, indicating the existence of long-term memory, should be taken into account in the molecular mechanism of the auxin-induced growth of plant cells.

Coherent forward scattering as a robust probe of multifractality in critical disordered media

We study coherent forward scattering (CFS) in critical disordered systems, whose eigenstates are multifractals. We give general and simple arguments that make it possible to fully characterize the dynamics of the shape and height of the CFS peak. We show that the dynamics is governed by multifractal dimensions D_1D1 and D_2D2, which suggests that CFS could be used as an experimental probe for quantum multifractality. Our predictions are universal and numerically verified in three paradigmatic models of quantum multifractality: Power-law Random Banded Matrices (PRBM), the Ruijsenaars-Schneider ensembles (RS), and the three-dimensional kicked-rotor (3DKR). In the strong multifractal regime, we show analytically that these universal predictions exactly coincide with results from standard perturbation theory applied to the PRBM and RS models.

3D object feature extraction and classification using 3D MF-DFA

In this study, we propose a three-dimensional (3D) object recognition method using multifractal properties. The proposed method is based on a 3D multifractal detrended fluctuation analysis (MF-DFA) using the voxel data of an object. We propose a 3D MF-DFA by extending a conventional MF-DFA, which is widely adopted for analyzing the time series. In the current study, the data processed by the model were changed from the time-series data in the original MF-DFA into the voxels of 3D objects. Thus, the steps in the original model were extended to a 3D processing structure. To voxelize the scatter point data, we apply a modified Allen–Cahn (AC) equation with the Neumann boundary condition to generate the object volume. Various 3D models after voxelization are used for the 3D MF-DFA to calculate the generalized Hurst exponents. The calculated generalized Hurst exponents of different categories show different distributions and are used as the training feature input vector into a multi-classification system, i.e., one-versus-one support vector machines (OvO-SVMs). The computational results show that the proposed method can effectively extract the features of an object. Metrics such as the accuracy, precision, and recall are utilized to measure the efficiency and robustness. When compared with state-of-art algorithms, our empirical tests show that the proposed 3D MF-DFA-OvO-SVM system is superior to most of the methods in terms of classification accuracy. In addition, we also confirm that our proposed model is applicable to object detection in 3D space. An efficient method may be useful for autonomous driving, robot cruises, and AR-based intelligent user interfaces.

Analysis of large-deviation multifractal spectral properties through successive compression for double JPEG detection

Analysis of large-deviation multifractal spectral properties through successive compression for double JPEG detection

Post‐earthquake stiffness loss estimation for reinforced concrete columns using fractal analysis of crack patterns

AbstractThe seismic damage to reinforced concrete (RC) components is conventionally quantified through various damage indices. In this paper, an image‐based procedure is developed for post‐earthquake residual stiffness quantification of RC columns. The proposed scenario‐based methodology is built upon the multifractal indices of the surface crack maps of the seismically damaged RC columns as the complexity measure for the images. An extensive databank from experiments on RC column specimens with a broad range of structural characteristics are collected and employed for generation and validation of the proposed procedure. Based on the availability of the structural information, five closed‐form regression‐aided equations are offered for estimating the stiffness loss in seismically damaged RC columns using predicting parameters among fractal indices, concrete strength, and the aspect ratio of the column. Results reveal that multifractal dimensions of crack maps are largely correlated with the stiffness degradation in RC columns. A sample specimen at various stiffness‐based damage indices is also presented as a case study to evaluate the predicted versus measured damage indices. The updated stiffness values acquired by the proposed equations can be utilized for stability evaluation, system identification, or further analysis of the seismically damaged RC buildings following an earthquake.

Topological quantum criticality of the disordered Chern insulator

We consider the two-dimensional topological Chern insulator in the presence of static disorder. Generic quantum states in this system are Anderson localized. However, topology requires the presence of a subset of critical states, with diverging localization length (the Chern insulator analog of the ‘center of the Landau band states’ of the quantum Hall insulator.) We discuss geometric criteria for the identification of these states at weak disorder, and their extension into the regime of strong disorder by analytical methods. In this way, we chart a critical surface embedded in a phase space spanned by energy, topological control parameter, and disorder strength. Our analytical predictions are supplemented by a numerical analysis of the position of the critical states, and their multifractal properties.

Fractal Characterization of Brass Corrosion in Cavitation Field in Seawater

Cavitation is a physical process that produces complex effects on the machines and components working in conditions where it acts. One effect is the materials-mass loss by corrosion–erosion when components are introduced into fluids under cavitation. The analysis of the damages produced by cavitation is generally performed by using different destructive and non-destructive experimental techniques. Most studies on materials’ behavior in cavitation refer to the erosion–corrosion mechanism, and very few investigate the fissure propagation by fractal methods. None have investigated the fractal characteristics of the sample surface after erosion–corrosion or the multifractal characteristics of materials’ mass variation in time in a cavitation field. Therefore, this research proposes a computational approach to determine the pattern of materials’ damages produced by ultrasound cavitation. The studied material is a brass, introduced in seawater. Fractal and multifractal techniques are applied to the series of the absolute mass loss per surface and the sample’s micrography after corrosion. Such an approach has not been utilized for such a material in similar experimental conditions. This study emphasizes that the box dimension of the series of the absolute mass loss per surface is close to one, and its behaviour is close to a non-/monofractal. It is demonstrated that the material’s surface corrosion is not uniform, and its multifractal character is highlighted by the f(α)− spectrum and the multifractal dimensions, which have the following values: the capacity dimension = 1.5969, the information dimension = 1.49836, and the correlation dimension = 1.4670.

Drug Release from Nanoparticles (Polymeric Nanocapsules and Liposomes) Mimed through a Multifractal Tunnelling-Type Effect.

The present study analyzes (theoretically and experimentally) a drug release process from nanoparticles (polymeric nanocapsules and liposomes). This process is functionalized on the surface with an aptamer. These types of drug release processes can also be included in cream-type formulations. The obtained cream ensures the active targeting of tumor epithelial cells, in the case of skin cancer, because it can be easily administered to the skin by spreading, thus avoiding side effects caused by the toxicity of the drug to healthy cells, increasing both patient compliance and the effectiveness of the treatment. The process of obtaining these formulations is a simple one, easy to use and highly reproductible. The theoretical model, based on the multifractal tunnel effect within the Scale Relativity Theory, considers the system as a complex one. In this model, complexity is replaced with system multifractality, quantified in physical quantities as multifractal dimensions and multifractal functions. The main advantage of this approach consists in the fact that it allows us to obtain information on system behavior at a microscopic level and to evaluate microscopic characteristics of the system, such as intrinsic transparences of the drug molecules, multifractal constants as indicators of the system's complexity, the frequency of interactions within the system and the energy ratio between potential barrier energy and the energy of drug molecules.

Analysis of multifractality of various components of electroencephalograms in mental disorders

The method of finding maxima of the moduli of the wavelet coefficients was used to perform a comparative analysis of the multifractality of various components of electroencephalograms in normal conditions and in mental disorders such as schizophrenia and depression. It is shown that for each type of mental disorders, the position of the singularity spectrum is stable in most areas of the brain, and the main differences between multifractal properties are associated with the alpha components of electroencephalograms. The group of individuals affected with depression is characterized by anticorrelated dynamics of successive values of the alpha components, and the group of individuals with schizophrenia is characterized by a combination of anticorrelated and correlated dynamics, thus, it can be useful for the clinical diagnosis of neuronal disorders.

Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space

We present a novel method for stochastic interpolation of sparsely sampled time signals based on a superstatistical random process generated from a multivariate Gaussian scale mixture. In comparison to other stochastic interpolation methods such as Gaussian process regression, our method possesses strong multifractal properties and is thus applicable to a broad range of real-world time series, e.g. from solar wind or atmospheric turbulence. Furthermore, we provide a sampling algorithm in terms of a mixing procedure that consists of generating a -dimensional field , where each Gaussian component is synthesized with identical underlying noise but different covariance function parameterized by a log-normally distributed parameter ξ. Due to the Gaussianity of each component , we can exploit standard sampling algorithms such as Fourier or wavelet methods and, most importantly, methods to constrain the process on the sparse measurement points. The scale mixture u(t) is then initialized by assigning each point in time t a and therefore a specific value from , where the time-dependent parameter follows a log-normal process with a large correlation time scale compared to the correlation time of . We juxtapose Fourier and wavelet methods and show that a multiwavelet-based hierarchical approximation of the interpolating paths, which produce a sparse covariance structure, provide an adequate method to locally interpolate large and sparse datasets.

Microbial characterization based on multifractal analysis of metagenomes.

The species diversity of microbiomes is a cutting-edge concept in metagenomic research. In this study, we propose a multifractal analysis for metagenomic research. Firstly, we visualized the chaotic game representation (CGR) of simulated metagenomes and real metagenomes. We find that metagenomes are visualized with self-similarity. Then we defined and calculated the multifractal dimension for the visualized plot of simulated and real metagenomes, respectively. By analyzing the Pearson correlation coefficients between the multifractal dimension and the traditional species diversity index, we obtain that the correlation coefficients between the multifractal dimension and the species richness index and Shannon diversity index reached the maximum value when q = 0, 1, and the correlation coefficient between the multifractal dimension and the Simpson diversity index reached the maximum value when q = 5. Finally, we apply our method to real metagenomes of the gut microbiota of 100 infants who are newborn and 4 and 12 months old. The results show that the multifractal dimensions of an infant's gut microbiomes can distinguish age differences. There is self-similarity among the CGRs of WGS of metagenomes, and the multifractal spectrum is an important characteristic for metagenomes. The traditional diversity indicators can be unified under the framework of multifractal analysis. These results coincided with similar results in macrobial ecology. The multifractal spectrum of infants' gut microbiomes are related to the development of the infants.

A nonlinear analysis of cardiovascular diseases using multi-scale analysis and generalized hurst exponent

Congestive heart failure (CHF) and arrhythmia (ARR) are common heart diseases that affect a growing population of patients worldwide. In this work, we employ multi-scale analysis (MSA) to estimate generalized Hurst exponent (GHE) from electrocardiogram (ECG) records under CHF, ARR, and normal sinus rhythm (NSR). As a result, fractal correlations in short and long fluctuations of CHF, ARR, and NSR are measured. Then, a set of six statistical tests are applied to GHE estimates to check how they are different at each time scale between two different ECG conditions. Particularly, the goal is verify if two different ECG conditions can be statistically differentiated by short or by long fluctuations. The battery of statistical tests includes Kolmogorov–Smirnov, Kruskal–Wallis, Wilcoxon rank sum, Student t-test, Ansari–Bradley, and F-test. The results from MSA show evidence that CHF, ARR, and NSR all exhibit multi-fractal properties. Besides, the results from statistical tests revealed that long fluctuations statistically differentiate CHF and ARR, ARR and NSR, and CHF and NSR. Therefore, long fluctuations account most for the characterization of CHF, ARR, and NSR. Our findings are helpful to better understand the mechanics of heart disease and normal heart beats, and also promising to eventually designing computer-aided diagnosis systems for CHF and ARR classification.

The effect of the COVID-19 pandemic on multifractals of price returns and trading volume variations of cryptocurrencies

We investigate the multifractal properties of daily price returns and trading volume variations in 35 cryptocurrencies by using the method of wavelet leaders prior and during the COVID-19 pandemic. The obtained results from the analysis of scaling exponent functions and multifractal spectrums show that, in general, price returns and trading volume variations exhibit multifractal properties prior to the COVID-19 pandemic and that they tend to exhibit monofractal behavior during the pandemic. As a result, the level of multifractality diminished during the COVID-19 for both price returns and trading volume variations. Since complexity in price returns and trading volume variations decreased during the pandemic, cryptocurrencies may offer an interesting investment during times of serious world economic downturns.

Particle-size fractal dimensions and pore structure characteristics of soils of typical vegetation communities in the Kubuqi Desert

The goal of this study was to investigate the soil particle-size distribution (PSD) and pore structure characteristics in the Kubuqi Desert in order to provide basic data for gaining insights into the soil-modifying properties of the local vegetation. Based on laser diffraction analysis, we measured the soil PSD and calculated the single and multi-fractal dimensions of the soils under typical artificial forest and natural grassland vegetation. The diameters, volumes and surface areas of the soil pores were determined via nitrogen adsorption experiments. The relationships between the fractal characteristics and pore structure parameters of the soil were investigated via redundancy analysis and stepwise regression. The clay content was less than 2.0%, and the sand content was greater than 75.0%. There was variability in the PSD and fractal dimension, which was mainly observed for the 0–20 cm, 20–80 cm, and 80–100 cm soil layers. In the 0–80 cm, the fine particle content, single fractal dimension (D), entropy dimension (D1), ratio of D1 to the capacity dimension (D1/D0), and correlation dimension (D2) were smaller for the forest soils than for the grassland soils. The fine particle (clay and silt) content of the 80–100 cm soil layer was approximately 37.8% higher for the Salix psammophila-8a than for the Salix psammophila-3a, and it was approximately 161.4% higher for the Populus popular-35a than for the Pinus sylvestris var. mongolica-8a. The silt content, D1, D1/D0, and D2 were significantly positively correlated with the specific surface area (SSA), total pore volume (TPV), and average pore diameter (APD) of the nanopores (p < 0.05, 0.01, or 0.001), and they were significantly negatively correlated with the percentage of the micropore volume (PMV) (p < 0.05 or 0.01). In the Kubuqi Desert, the fine particle content and fractal dimensions of the soil layer below the root zone of shrub and arbor vegetation increased with increasing stand age, but the trend was reversed in the shallower soil layers. The variability of the soil PSD characteristics was strongly correlated with the variability of the nanopore parameters on the microscopic scale, suggesting that the total pore volume, average pore diameter, and percentage of the micropore volume may be potential indicators of the soil structure and quality.

Pore structure and pore size change for tight sandstone treated with supercritical CO2 fluid

The effects of ScCO2 on pore structure should be investigated in order to properly apply supercritical carbon dioxide (ScCO2) fracturing on tight sandstone. ScCO2 treatment was applied to three sets of tight sandstone samples. The pore structure was characterized using a combination of high-pressure mercury intrusion porosimetry and nuclear magnetic resonance. Furthermore, the pore size distribution’s multi-fractal dimensions were estimated. After SCCO2 treatment, the number of transitional pores decreased significantly, while the number of mesopores and macropores increased. At all scales, the fractal dimension characteristics indicate that pore structure gets more complicated and heterogeneous. Furthermore, due to their larger contact area, large pores have a higher fractal dimension than small pores. The accumulative pore volume of the calcite-rich group increases and a decreasing trend occurs for the calcite-rare group. The content of calcite sharply decreased while clay, feldspar, and plagioclase content have a slight drop after ScCO2 treatment, which affects the alternation of pore structure. Thus, the accumulative porosity volume and pore connectivity do not always increase after ScCO2 exposure because re-precipitation and fine migration can potentially clog pores and channels.

An exact computation for mixed multifractal dimensions of sets and measures

In the present work, we are concerned with the estimation of some mixed variants of multifractal dimensions for a special class of measures characterized by a weak Ahlfors assumption applying mixed multifractal generalizations of Hausdorff and packing measures. Exact computation of such dimensions is shown to be valid for a class of Moran-type measures in some special cases.

Different types of multifractal measures in separable metric spaces and their applications

<abstract><p>The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and the relative packing measures $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.</p></abstract>

On the projections of the multifractal Hewitt-Stromberg dimensions

This paper studies the behavior of the multifractal Hewitt-Stromberg measures and dimensions under projections onto almost all m-dimensional subspaces.

Chaotic Behavior Analysis and Multifractal Feature Learning of Specular Reflection Ionosphere Clutter for Target Detection

Because ionosphere condition is dynamically varying in the Earth’s middle-latitude, the long-range detection performance of operating high frequency surface wave radar (HFSWR) systems is severely restricted by the presence of ionosphere clutter. The development and performance evaluation of target detection under the F-layer specular reflection ionosphere clutter with concentrated power (Fs-concentrated clutter) will be described in this paper. First, this article pays attention to the variable chaotic behavior of Fs-concentrated clutter by using amplitude and phase echo data. Then, this article investigates the multifractal property of the power spectrum between the Fs-concentrated clutter and mixed echo component. The various fractal characteristics of different types of echo component and Doppler scale are analyzed. Next, the singularity intensity correlation function width, the accumulation area of multifractal correlation spectrum and asymmetry parameter are extracted as the learning features. These multifractal feature parameters are fed into the improved AlexNet and Resnet networks for distinguishing the types of echo component. Experimental results using real HFSWR echoes datasets have demonstrated that this proposed method can increase the detection probability by about 20% than the state-of-art detection methods and classical target detection methods under various signal-clutter ratio (SCR) background.

Relationships between Reference Evapotranspiration and Meteorological Variables in the Middle Zone of the Guadalquivir River Valley Explained by Multifractal Detrended Cross-Correlation Analysis

The multifractal relationship between reference evapotranspiration (ET0), computed by the Penmann-Monteith equation (PM), relative humidity (RH) and mean surface temperature (Tmean) was studied in the middle zone of the Guadalquivir River Valley (south Spain) in a previous study. This work extends that study to the average wind speed (U2) and solar radiation (SR), focusing on more recent years. All agro-meteorological variables were analyzed by multifractal detrended cross-correlation analysis (MFCCA) and multifractal detrended fluctuation analysis (MFDFA). The outcomes revealed persistent long-term autocorrelations, with Tmean and RH having the highest persistence (H>0.75). More precise results of multifractal properties than in the previous study were obtained for ET0, Tmean, and RH due to the elimination of trends in the signals. Only medium and large fluctuations in ET0 showed multifractal cross-correlations with its controlling factors, except for U2. Moreover, joint scaling exponents differed from individual exponents. These phenomena contrast with what has been observed in previous cross-correlation studies, revealing that some differences exist in the dynamics of multifractality among the analyzed variables. On the other hand, the Tmean–ET0 relation showed that extreme events in ET0 are mainly ruled by high temperature fluctuations, which match conclusions drawn in the previous study.