Multifractal Measures Research Articles

Remarks on the mutual singularity of multifractal measures

In the present work, we study the mutual singularity of multifractal Hausdorff and packing measures which provide a positive answer to Olsen’s questions in a more general framework. Our main results apply to a family of measures supported by the full 5-adic grid of [0, 1], namely the quasi-Bernoulli measures.

The mutual singularity of the relative multifractal measures

Abstract M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

The mutual singularity of multifractal measures for some non-regular Moran fractals

The mutual singularity of multifractal measures for some non-regular Moran fractals

Another example of the mutual singularity of multifractal measures

We propose an example for which the multifractal Hausdorff and packing measures are mutually singular.

Multifractal point processes and the spatial distribution of wildfires in French Mediterranean regions

We introduce a simple and wide class of multifractal spatial point patterns as Cox processes which intensity is multifractal, i.e., the class of Poisson processes with a stochastic intensity corresponding to a random multifractal measure. We then propose a maximum likelihood approach by means of a standard Expectation–Maximization procedure in order to estimate the distribution of these intensities at all scales. This provides, as validated on various numerical examples, a simple framework to estimate the scaling laws and therefore the multifractal properties for this class of spatial point processes. The wildfire distribution gathered in the Prométhée French Mediterranean wildfire database is investigated within this approach that notably allows us to compute the statistical moments associated with the spatial distribution of annual likelihood of fire event occurrence. We show that for each order q, these moments display a well defined scaling behavior with a non-linear spectrum of scaling exponents ζq. From our study, it thus appears that the spatial distribution of the wildfire ignition annual risk can be described by a non-trivial, multifractal singularity spectrum and that this risk cannot be reduced to providing a number of events per km2. Our analysis is confirmed by a direct spatial correlation estimation of the intensity logarithms whose the peculiar slowly decreasing shape corresponds to the hallmark of multifractal cascades. The multifractal features appear to be constant over time and similar over the three regions that are studied.

Superconductivity at the three-dimensional Anderson metal-insulator transition

We study a disordered weakly coupled superconductor around the Anderson transition by solving numerically the Bogoliubov--de Gennes (BdG) equations in a three-dimensional lattice of size up to $20\ifmmode\times\else\texttimes\fi{}20\ifmmode\times\else\texttimes\fi{}20$ in the presence of a random potential. The spatial average of the order parameter is moderately enhanced as disorder approaches the transition but decreases sharply in the insulating region. The spatial distribution of the order parameter is sensitive to the disorder strength: for intermediate disorders below the transition, we already observe a highly asymmetric distribution with an exponential tail. Around the transition, it is well described by a log-normal distribution and a parabolic singularity spectrum. These features are typical of a multifractal measure. We determine quantitatively the critical disorder at which the insulator transition occurs by an analysis of level statistics in the spectral region that contributes to the formation of the order parameter. Interestingly, spectral correlations at the transition are similar to those found in noninteracting disordered systems at the Anderson transition. A percolation analysis suggests that the loss of phase coherence may occur around the critical disorder.

To what extent can multifractal measures provide an accurate model of the porosity of soils?

AbstractOver the last decades, several authors have suggested that multifractal measures, that is, self‐similar measures defined on fractal or non‐fractal objects, could be useful to describe soil properties, to model soil processes, and to deal with their extreme microscale heterogeneity. In this context, a key question relates to the extent to which multifractal measures can indeed fulfill all the expectations they have generated. To address this question, we discuss the possibility of generating a synthetic soil image exhibiting multifractal porosity. To this end, a simple geometrical multifractal model in 2D is developed, which helps us to better understand the concept of multifractality and to generate images. We show that it is possible to generate synthetic binary images over a limited range of scales, but that a pure multifractal model for the distribution of the solid or pore mass cannot be developed due to physical constraints. Moreover, in the generated images mimicking multifractal solid space, a higher degree of multifractality corresponds to a larger porosity, rendering it difficult to tune model parameters to match actual soil properties. In addition, simple statistics relying on power‐law fits appear insufficient to characterize soil architecture even if they may capture some key multiscale indicators of observed spatial heterogeneity. We argue that the same conclusions would be reached in a three‐dimensional space, as well as for grey‐scales images.

Forecasting stock market in high and low volatility periods: a modified multifractal volatility approach

We propose a new multifractal volatility measure (MVA) and accordingly construct new types of multifractal volatility models. We compare these models to some alternatives, and consider long memory features, the benefits of leverage effects, and the presence of jumps. The analysis is applied to the multifractal volatility and realized volatility series of three representative indices, adopting six loss functions as guidance for the model selection. The forecast performance is evaluated and tested using model confident set (MCS) in both high- and low-volatility periods, as well as based on robustness checks on the continuous rank probability score statistics. The results show satisfactory performances for the new multifractal volatility models. Specifically, we find that our new multifractal volatility model significantly improves the one-day-ahead volatility forecasts in the high-volatility period. While in the low-volatility periods, the out-of-sample test results highlight the superiority of the traditional multifractal volatility models in the accuracy of volatility forecasting. These findings will be helpful for market volatility prediction and investment strategies.

Evaluation of the Heterogeneous Spatial Distribution of Population and Stores/Facilities by Multifractal Analysis

The spatial distribution of a population is not homogeneous---some areas attract many residents, while others do not. The spatial distribution of stores and facilities that have been coevolving with that of the population is also heterogeneous. Previous studies have shown that multifractality is a characteristic of the spatial distribution of a population, as well as other quantities associated with the urban system. We found that stores/facilities belonging to some categories also exhibit multifractality in spatial distribution. We quantified the spatial distributions of the population and stores/facilities in each category by multifractal analysis and compared their multifractal properties. Multifractal measures that reflect the heterogeneity of the densities in each location were able to capture additional features that cannot be seen when only the box-counting dimension was observed. Further, high concentrations of stores/facilities in categories relating to professional or commercial businesses were observed, consistent with previous studies on the scaling law, another pattern observed in urban morphology. We discuss the implications of the multifractal properties on the arrangement of locations where stores/facilities are concentrated. We believe that multifractal analysis is a powerful tool for the quantification of spatial distributions and expect that our interpretation of multifractal measures will stimulate further investigations into urban morphology.

On the Hewitt–Stromberg measure of product sets

In this paper, we shall be concerned with evaluation of Hewitt–Stromberg measure of Cartesian product sets by means of the measure of their components. This is done by the construction of new multifractal measures, on the Euclidean space $${\mathbb {R}}^n$$ , in a similar manner to Hewitt–Stromberg measures but using the class of all n-dimensional half-open binary cubes of covering sets in the definition rather than the class of all balls.

A note on the multifractal Hewitt-Stromberg measures in a probability space

In this note, we investigate the multifractal analogues of the Hewitt-Stromberg measures and dimensions in a probability space.

Retinal vessel segmentation using multifractal characterization

This study presents a supervised classification method for the segmentation of retinal vessels using fundus images. This work proposes a novel retinal vasculature segmentation method based on multifractal characterization of the vessels to minimize the noise and enhance the vessels during segmentation. The Holder exponent, a multifractal measure is employed for the first time to segment the retinal vessels. The Holder exponent is used to quantify the local regularity of the vessels. The Holder exponents are computed from the Gabor wavelet responses for the effective segmentation of vessels, which is a novel feature of the method. The Gaussian mixture model (GMM) classifier is used for the classification of pixels. Different multifractal measures used to compute the Holder exponents are evaluated for the output quality. The effectiveness of the method is evaluated using three publicly available datasets for fundus images namely, DRIVE, STARE and CHASE_DB1. The proposed method provides robust segmentation of retinal vessels for both normal and abnormal images (i.e., images with pathologies) at a reasonable segmentation speed and the method is also simple to configure. It achieves an average accuracy and area under the receiver operating characteristic curve of 0.948 and 0.959 respectively on the DRIVE dataset; 0.9542 and 0.9711 respectively on the STARE dataset; 0.9459 and 0.9592 respectively on the CHASE_DB1 dataset; 0.9500 and 0.9623 respectively on the abnormal images.

Controlling the Multifractal Generating Measures of Complex Networks

Mathematical modelling of real complex networks aims to characterize their architecture and decipher their underlying principles. Self-repeating patterns and multifractality exist in many real-world complex systems such as brain, genetic, geoscience, and social networks. To better comprehend the multifractal behavior in the real networks, we propose the weighted multifractal graph model to characterize the spatiotemporal complexity and heterogeneity encoded in the interaction weights. We provide analytical tools to verify the multifractal properties of the proposed model. By varying the parameters in the initial unit square, the model can reproduce a diverse range of multifractal spectrums with different degrees of symmetry, locations, support and shapes. We estimate and investigate the weighted multifractal graph model corresponding to two real-world complex systems, namely (i) the chromosome interactions of yeast cells in quiescence and in exponential growth, and (ii) the brain networks of cognitively healthy people and patients exhibiting late mild cognitive impairment leading to Alzheimer disease. The analysis of recovered models show that the proposed random graph model provides a novel way to understand the self-similar structure of complex networks and to discriminate different network structures. Additionally, by mapping real complex networks onto multifractal generating measures, it allows us to develop new network design and control strategies, such as the minimal control of multifractal measures of real systems under different functioning conditions or states.

On the mutual singularity of multifractal measures

<p style='text-indent:20px;'>The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.</p>

Higher-order statistics based multifractal predictability measures for anisotropic turbulence and the theoretical limits of aviation weather forecasting

Theoretical predictability measures of turbulent atmospheric flows are essential in estimating how realistic the current storm-scale strategic forecast skill expectations are. Atmospheric predictability studies in the past have usually neglected intermittency and anisotropy, which are typical features of atmospheric flows, rendering their application to the storm-scale weather regime ineffective. Furthermore, these studies are frequently limited to second-order statistical measures, which do not contain information about the rarer, more severe, and, therefore, more important (from a forecasting and mitigation perspective) weather events. Here we overcome these rather severe limitations by proposing an analytical expression for the theoretical predictability limits of anisotropic multifractal fields based on higher-order autocorrelation functions. The predictability limits are dependent on the order of statistical moment (q) and are smaller for larger q. Since higher-order statistical measures take into account rarer events, such more extreme phenomena are less predictable. While spatial anisotropy of the fields seems to increase their predictability limits (making them larger than the commonly expected eddy turnover times), the ratio of anisotropic to isotropic predictability limits is independent of q. Our results indicate that reliable storm-scale weather forecasting with around 3 to 5 hours lead time is theoretically possible.

Differentiation of EMCI in sMR images using segmented brainstem multifractal texture measures

In this Letter, an attempt has been made to segment the brainstem structure from structural magnetic resonance (sMR) images and differentiate early mild cognitive impairment (EMCI) from normal and Alzheimer's condition using multifractal texture measures. The images considered from public domain database are preprocessed and brainstem is segmented using fuzzy ‘C’ means-based connected component labelling method. Multifractal spectrum (MS) is evaluated for the segmented brainstem structure using the multifractal detrended moving average technique. Seven MS-based texture measures are extracted and statistically analysed using one-way analysis of variance. Classification algorithms namely, naïve Bayes, decision tree and random forest (RF) are employed to distinguish the EMCI condition. Ten-fold cross-validation approach is executed and its performance is evaluated using accuracy, precision, recall and area under the curve. The results indicate that the segmented brainstem structures possess multifractal characteristics. Of the extracted MS-based texture measures, αmax, α0, αR, B and Δα are found to be statistically significant. RF gives the highest accuracy of 92.3% in distinguishing EMCI from other subject groups. Hence, the proposed approach with brainstem MS-based texture measures in combination with RF can be used as an imaging biomarker for the diagnosis of EMCI.

Multifractal Analysis of Realized Volatilities in Chinese Stock Market

This paper presents a sliding window multifractal detrending moving average (MF-DMA) method for multifractal analysis of time series. Numerical experiments on the binomial multifractal measure show improvements in fitting theoretical values by sliding window MF-DMA method compared with the original MF-DMA algorithm. We try to assess the multifractality displayed by realized volatility series of SSEC (Shanghai (securities) composite index) and SZSEC (Compositional Index of Shenzhen Stock Market) indexes in Chinese stock market via sliding window MF-DMA method, and the scaling exponents obtained are fitted to a Log-normal multifractal model. Empirical analysis shows clear evidences of the existences of multifractal features in realized volatility series of SSEC and SZSEC indexes, the multifractality degree of SSEC index is stronger than that of SZSEC index. The major sources of multifractality exhibited in realized volatility series of both SSEC and SZSEC indexes are long-range correlations of small and large fluctuations, and the fat-tailed distributions have certain effects on multifractality. The Log-normal multifractal model shows great availability to capture the scaling behavior of realized volatility series of real financial data.

Generalized dimensions, large deviations and the distribution of rare events

Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a chaotic dynamical system, generalized dimensions take on a dynamical meaning, as they provide the rate function for the large deviations of the first hitting time, which is the (average) time required to connect any two different regions in phase space. We prove this result rigorously under a set of stringent assumptions. As a consequence, the statistics of hitting times provides new algorithms for the computation of the spectrum of generalized dimensions. Numerical examples, presented along with the theory, suggest that the validity of this technique reaches far beyond the range covered by the theorem. We state our result within the framework of extreme value theory. This approach reveals that hitting times are also linked to dynamical indicators such as stability of the motion and local dimensions of the invariant measure. This suggests that one can use local dynamical indicators from finite time series to gather information on the multifractal spectrum of generalized dimension. We show an application of this technique to experimental data from climate dynamics.

Kernel density approach to error estimation of MF-DFA measures on time series

MF-DFA is a widely used method for the extraction of multifractal patterns that may appear in time series. Current techniques for determining the range of scales employed in the MF-DFA procedure are mainly empirical and in some cases require the researcher to visually select such range, so multifractal measures often differ among experiments, and the estimation of errors in such measures is often disregarded. This problem also represents a serious obstacle for incorporating MF-DFA in automated processes. In this paper, we present a Kernel Density based method that provides metrics for errors on the estimations of the multifractal spectrum obtained by MF-DFA, and gives the probabilities of errors being inside a certain range for each moment considered in the spectrum. Our approach makes use of Kernel Density Estimation in combination with a variant of the Theil–Sen estimator to compute MF-DFA exponents. We tested our technique on simulated and real-world time series. Comparison between this approach and the conventional method applied to deterministic and random multiplicative processes demonstrates robustness to spurious estimations of generalized fractal dimensions in MF-DFA.

Multifractal phenomena and packing dimension

We undertake a general study of multifractal phenomena for functions or measures. We show that the existence of several kinds of multifractal functions or measures can be easily deduced from an abstract statement, leading to new results. This general approach does not work for Fourier or Dirichlet series. Using careful constructions, we extend our results to these cases.