Multifractal Method Research Articles

Multifractality in Humanitarian Applications: A Case Study of Internally Displaced Persons/Refugee Camps

The coordination of humanitarian relief is always difficult due to a lack of data required for management and planning. Remote sensing imagery can be an important source of information about the in-situ situation, notably, no-access areas. Scenarios include situation awareness after natural disasters or military conflicts, damage assessment, or monitoring camp structure, either as a one-off exercise or on an ongoing basis. In this article, we propose a multifractal approach to automating information extraction about internally displaced persons/refugee camps and discuss its potential and limitations. Our case study uses multifractal features to determine the extent of camps in Kenya and Sudan. The results show that the method can be usefully applied to camp growing analysis and help to make a rough, but rapid estimation of their extent. Our multifractal method appears to be a reasonable step forward on the road between manual mapping and (not yet developed) fully automated, highly accurate processing.

Epileptic seizures regularities, revealed from encephalograms time series by nonlinear mechanics methods

Recurrence quantification analysis and multifractal method were used for investigation of the encephalograms time series of the children with epileptic seizures. It has been shown that epileptic seizures are accompanied by an increase in determinism of the brain electric process due to neuron activity synchronization. This behavior is typical for nonlinear systems catastrophes of different nature

Multi‐fractal detrended fluctuation half‐spectrum analysis of HRV

Electrocardiogram is a comprehensive reflection of cardiac electrical activity on the surface of the body. It is of great research value to analyse ECG signals and diagnose heart diseases. Since the heartbeat process is a chaotic system, this study proposes a multi-fractal detrended fluctuation half-spectrum analysis method of short-term heart rate variability-y (HRV) signals from the perspective of non-linear dynamics, for patients with normal heart rate, sporadic heart rate variation and supraventricular rate, the heart rate mutation signal of patients with heart rate variation is calculated, and the statistical value distribution of multi-fractal parameter α has a certain change trend with the change of disease condition. The classification accuracy can reach >75% through box plot analysis.

Ship target detection and segmentation method based on multi‐fractal analysis

A novel automatic ship target detection and segmentation approach based on modified multi-fractal method in synthetic aperture radar (SAR) image is proposed in this study. Firstly, singular power spectrum (SPS) is developed to two-dimensional SPS (2D-SPS), and applied to feature extraction and SAR target detection. Secondly, 2D Holder exponent and 2D multi-fractal spectrum are jointly used to ship target segmentation in SAR image. The algorithm steps and detection flow is here conceptually assessed, analytically derived, numerically verified, and also tested on actual SAR images. Results indicate that the proposed method has significant advantages in ship target detection and segmentation in SAR image.

Geostatistical and multi-fractal modeling of geological and geophysical characteristics in Ghalandar Skarn-Porphyry Cu Deposit, Iran

This work aims at figuring out the spatial relationships between the geophysical and geological models in a case study pertaining to copper-sulfide mineralization through an integrated 3D analysis of favorable target. The Ghalandar Skarn-Porphyry Cu Deposit, which is located in NW Iran, is selected for this research work. Three geophysical surveys of direct current electrical resistivity and induced polarization tomography along with magnetometry are performed to construct the physical properties of electrical resistivity, chargeability, and magnetic susceptibility, respectively. Inverse modeling and geostatistical interpolation are utilized to generate the physical 3D models. A 3D model of Cu grade is generated using ordinary kriging; however, the indicator kriging method is run to design a 3D model of rock types through incorporating the drilling results. Block models of geophysical and geological characteristics are cast in a similar 3D mesh to investigate their relationships in copper mineralization. A concentration-volume multi-fractal method is utilized to divide each model into its sub-sets, where the most productive portions in association with Cu-bearing mineralization are distinguished. Note that sub-sets of geophysical models are spatially matched with geological models of Cu grade and rock types. The zones with low electrical resistivity, high chargeability, and low magnetic susceptibility correspond to the main source of Cu mineralization in a dominated skarn rock type setting.

Black silicon – correlation between microstructure and Raman scattering

Abstract Black silicon layers were formed on silicon substrate by the surface structure chemical transfer method and by anodic etching method. Properties of microstructure of formed layers were experimentally studied by the electron microscopy methods (TEM) and characterized by statistical, Fourier and multifractal methods. Theoretical structures with defined fractal properties and surface roughness were generated and their microstructure properties were evaluated. Obtained results were used for the explanation of the real structure development during the forming procedure. By using of this approach, we study the correlation of roughness and fractality with optical properties. Black silicon layers were also investigated by using of Raman scattering method. Optimized theoretical model describing the 1st order of black Si Raman scattering profile was constructed and used for evaluation of the biaxial tensile stress introduced during etching procedure.

Multifractal detrended cross-correlation analysis of the relation between price and volume in European carbon futures markets

Recently, with the development of European carbon futures markets, the complex dynamic relationship between price and volume has become the focus of energy research area. In our research, we applied multifractal detrended cross-correlation analysis method (MF-DCCA) to investigate the cross-correlation and multifractality between price and volume in European carbon futures markets. Firstly, the analysis of basic statistics show that the European carbon futures market has obvious non-normal, peak-heavy-tailed distribution, which does not satisfy the normal distribution hypothesis in the efficient market. Secondly, through multifractal detrended cross-correlation analysis, we conclude that trading price and volume relationship in European carbon futures markets has a non-linear dependence. Thirdly, we find the existence of power-law cross-correlations between price and volume in European carbon futures markets. Finally, the relationship between price and volume is non-linear. After random rearrangement and phase treatment of correlation coefficients, the long-range correlation of small fluctuations and large fluctuations and the fat tail distribution of series are the reasons for the formation of multifractality.

Improved Multifractal Fusion Method to Blend SMOS Sea Surface Salinity Based on Semiparametric Weight Function

AbstractThe multifractal fusion method has proved to be an effective algorithm to mitigate the noise of the sea surface salinity (SSS) of Soil Moisture Ocean Salinity (SMOS) mission. However, the traditional nonparametric weight function used in this method is unable to fully capture the dynamic evolution of the oceanic environment. Considering the multiscale, nonuniform, anisotropic, and flow-dependent nature of the ocean, a prototype with the so-called flexible circle (FLC) weight function or flexible ellipse (FLE) weight function with a set of predefined parameters is proposed in this paper. The improved weight functions could draw dynamic information from the sea surface temperature, Rossby radius of deformation, and surface geostrophic flow to improve the quality of the remotely sensed SSS. The validation against the in situ data indicates that the improved weight functions perform better than the traditional one with a reduced root-mean-square (RMS) and standard deviation (STD) of the differences with respect to EN 4.2.0 profiles (from 0.50 and 0.46 to 0.42 and 0.38 for FLC and 0.39 and 0.36 for FLE in the global ocean). In particular, the FLE scheme could highlight the variation of the strong currents without affecting the computational efficiency. Furthermore, this paper discusses the influences of the error distribution on the fusion results and underlines the importance of error-based adaptions for further improvements.

Delimiting geochemical anomalies in the exploration of covered deposits with multifractal methods and using stream sediment data from the Iberian Pyrite Belt, Southwest Iberia

The Iberian Pyrite Belt (IPB) is a world-class province of Upper Paleozoic age for volcanic- and shale-hosted massive sulfides that includes 5 of the 10 supergiant (>150 Mt) deposits of their kind in the World. The volcano-sedimentary sequences hosting the ores are locally exposed, commonly covered by flysch-type successions of variable thickness (the Baixo Alentejo Flysch Group, BAFG) as a result of primary (sedimentary) features and superimposed effects due to Variscan folding and tectonic stacking. In addition and when affected by late-developed geomorphic depressions, the exhumed Paleozoic basement is also capped by Meso-Cenozoic sediments. The total thickness of cover sequences on top of the litho-stratigraphic sequence of interest can reach 100′s of meters, creating problems for mineral exploration.A large stream sediment (multi-element) geochemical database for the IPB was done in the 90′s during a wide-ranging regional survey. This database was reassessed by means of multifractal methods in view of their faculty to enhance the detection of faint geochemical anomalies and provide geochemical indicators to deep buried ore systems. Several anomalies were detected, namely for Cu, Zn, and Co along NW-SE to WNW-ESE trends. The main anomalous geochemical trends identified include zones that: (i) extend along the Mira Formation of BAFG, opening for the first time two non-traditional areas for further exploration studies; (ii) run along the boundary between the Mira and Mértola Formations of BAFG, probably related to different geochemical signatures and late-developed, fault-controlled, epigenetic ore systems; and (iii) overlap windows of volcanic-sedimentary sequences within the Cenozoic detrital cover and the site below which the deep buried deposit of Lagoa Salgada is located.

Chaos based novel non-invasive approach to assess intracranial pressure in the brain using ECG and arterial BP data

The study explores the applicability of a multifractal cross-correlated method to assess the degree of correlation of intracranial pressure (ICP) signals with ECG and ABP in traumatic brain injury (TBI) patients. Multifractal width (w) and cross-correlation exponent (γx ) estimated using multifractal detrended cross-correlation (MFDXA) method describe interesting features about the complexity of ICP signals of traumatic brain injury (TBI) patients. The degree of multifractality (w) of ECG signals reveal that even without going into complicated ICP signal measurement which is not only invasive but costly procedure, the trauma suffered due to brain injury can easily be assessed. With the help of two quantitative parameters w and γx appropriate software can easily be developed and coupled with ECG system.

Analysis of Carotid Plaque Using Multifractal method in Ultrasound Images.

This paper analyses the non-linear parameters of the plaque in carotid B-mode ultrasound images. In this work an attempt has been made to differentiate textural features of the symptomatic and asymptomatic plaque by multifractal method. The fractal dimension represents irregularity and hence the fractal nature. The interwoven sets of singularities are characterized by its own scaling behaviour, quantitatively represented as the fractal dimension, explains the multifractal behavior. The multifractal characteristics are plotted using a multifractal formalism proposed by Halsey. Certain multifractal features are extracted namely bandwidth (BW) of the spectrum and singularity exponent peak (SXPpeak). These extracted features have coefficient of variation in the range of 0.1 to 0.3; hence they have lower inter-subject variability. This analysis could aid in the study of the symptomatic and asymptomatic plaques.

Application of fractal analysis methods to images obtained by crystallization modified by an additive

The fractal and multifractal methods are now widely used for analysis and classification of digital images having complex structure. We present the results of the application of such methods to the images of crystallograms obtained by crystallization with additives. This technique was developed for studying images of blood crystals, and now finds increasing use in the analysis of medicines, checking food and soil quality. In this work we study images of crystallograms of various milk dilutions and crystallograms obtained with bean leave extracts. The results show that the proposed mathematical methods seem to be rather perspective both in comparing images of different classes and in obtaining classifying signs.

Using multi-fractal and joint multi-fractal methods to characterize spatial variability of reconstructed soil properties in an opencast coal-mine dump in the Loess area of China

Opencast coal mining activities have significant effects on the reconstructed soil properties, including its physical, chemical and biological properties. To assess the effects of opencast mining and associated dumping activities on reconstructed soil properties, the spatial variability of soil properties, i.e., soil particle size distribution (PSD), organic matter (OM) and total nitrogen (TN), in an inner dump after dumping and before reclamation in Antaibao opencast coal-mine of Shanxi Pingshuo was analyzed using multi-fractal and joint multi-fractal methods. Soil samples at 78 sampling sites at the depths of 0–20 cm and 20–40 cm were collected using an auger. The singularity spectra of soil properties in the study area were analyzed, and three multi-fractal parameters (i.e., D1, D2 and ∆α) were calculated. The singularity spectra and multi-fractal parameters can quantitatively characterize the spatial variability of soil PSD, OM and TN. The variability of soil PSD, OM and TN at the depth of 20–40 cm was higher than that at the depth of 0–20 cm. At same soil depth, the sand and clay contents showed relatively high spatial variability, and the silt content, OM and TN exhibited relatively low spatial variability. The spatial variability in sand content, OM and TN between 0 and 20 cm and 20–40 cm depths had a relatively high correlation, while the clay and silt contents at the depths of 0–20 cm and 20–40 cm showed a relatively low correlation. Multi-fractal and joint multi-fractal methods exhibited more advantages in characterizing spatial variability of reconstructed soil properties. This study provided a theoretical basis for the soil sampling optimization of in the opencast coal-mine dump in the Loess area of China.

Chemical Compositions and Distribution Characteristics of Cements in Longmaxi Formation Shales, Southwest China

Shale gas resources have been regarded as a viable energy source, and it is of great significance to characterize the shale composition of different cements, such as quartz and dolomite. In this research, chemical analysis and the multifractal method have been used to study the mineral compositions and petrophysical structures of cements in shale samples from the Longmaxi Formation, China. X-ray diffraction, electron microprobe, field emission scanning electron microscopy, cathodoluminescence microscopy and C-O isotope analyses confirmed that cements in the Longmaxi Formation shales are mainly composed of Fe-bearing dolomite and quartz. Fe-bearing dolomite cements concentrate around dolomite as annuli, filling micron-sized inorganic primary pores. Quartz cements in the form of nanoparicles fill primary inter-crystalline pores among clay minerals. Theoretical calculation shows that the Fe-bearing dolomite cements formed slightly earlier than the quartz cements, but both were related to diagenetic illitization of smectite. Moreover, multifractal analysis reveals that the quartz cements are more irregularly distributed in pores than the Fe-bearing dolomite cements. These results suggest that the plugging effect of the quartz cements on the primary inoraganic pore structures is the dominant factor resulting in low interconnected porosity of shales, which are unfavorable for the enrichment of shale gas.

Multifractal detrended fluctuation analysis on air traffic flow time series: A single airport case

Exploring multifractal characteristics of air traffic flow time series is helpful in understanding the self-similarity and the correlations embedded in the time series, and thus in obtaining insights into the evolution mechanism and the regular patterns of the air traffic flows, which may help to develop effective air traffic flow management measures. With the multifractal detrended fluctuation analysis method, we identify that the total, the arrival and the departure air traffic flow time series of Beijing Capital International Airport in the summer season of 2017 are of multifractality on the scales below the corresponding crossovers, and the primary cause of the multifractality is the long-range correlations of small and large fluctuations. Comparisons on the multifractality of the time series indicate that the total and the arrival air traffic flows are of the strongest and the weakest multifractality respectively, and that of departure air traffic flow is in-between. The comparison results also show that the total and the arrival air traffic flows are insensitive to large fluctuations and dominated by small fluctuations, whereas the departure air traffic flow is insensitive to small fluctuations and dominated by large fluctuations. In addition, an investigation of multifractal characteristics of the time series during the thunderstorm season and the non-thunderstorm season reveals that the impact of the thunderstorm season on the total air traffic flow is the strongest, and there are significantly essential differences in the multifractality of the total air traffic flow before and after the thunderstorm season. For the arrival air traffic flow, there is only a difference in the extreme fluctuation rate, whereas there is no difference in essence for the departure air traffic flow, except for some quantity differences.

Multifractal detrended partial cross-correlation analysis on Asian markets

The impact of the recession on global markets motivates us to explore the effect of the stock market on the economic stability of a country. The growth of a stock market depends on various factors such as inter-dependency, globalization, economic policies etc. In this paper, we apply the recently developed multifractal detrended partial cross-correlation analysis method on some Asian markets by assuming NASDAQ composite index commonly influences these markets. For this purpose, we analyze the daily returns of three Asian markets such as China’s Shanghai composite index (SHCOMP), Japan’s Nikkei stock average index (NIKKEI 225), and India’s Bombay stock exchange (BSE) sensex index over a period of 18 years (2000–2018). The analysis was performed between all bivariate time series after removing the external influence factor and the results indicate that there exists long-range partial cross-correlation behavior and multifractal nature. For comparison, we also performed analysis without removal of external influence factor i.e. multifractal detrended cross-correlation analysis. From the calculated Hurst scaling exponents, we found that the NASDAQ composite index shows significant influence on these analyzed Asian stock markets.

Cross Correlation Singularity Power Spectrum Theory and Application in Radar Target Detection Within Sea Clutters

The cross correlation power spectrum of multiple signal sequences in the singularity domain is studied in this paper. With theoretical derivation and quantitative analysis, the cross correlation singularity power spectrum (CSPS) distribution theory is proposed. Developed from correlation function (CF), spectrum CF (SCF), singularity power spectrum (SPS), and multifractal cross correlation analysis, the CSPS can be applied for the correlation analysis of multiple fractal time series. In this paper, the CSPS is rigorously derived based on SPS and SCF, and is verified with classical multifractal time series. Furthermore, a target detection method based on the proposed CSPS method is also proposed. The proposed methodology is tested on sea clutters, both with and without target, from the Ice Multiparameter Imaging X-Band radar data set. The simulation results indicate that the target detection based on CSPS performs better than conventional multifractal spectrum methods, and can achieve almost 100% detection probability of detecting low-observable targets within sea clutters.

Multifractal Detrended Cross-Correlation Analysis of Electricity and Carbon Markets in China

With the development of carbon market, the complex dynamic relationship between electricity and carbon market has become the focus of energy research area. In this paper, we applied a new developed multifractal detrended cross-correlation analysis method to investigate the cross-correlation and multifractality between electricity and carbon markets. We analyze the daily return of electricity and carbon prices over a period of 6 years to do the research. The results show that, firstly, we find that there is a strong negative correlation between domestic carbon price and electricity price and a significant cross-correlation between the return series of electricity and carbon markets. Secondly, through multifractal detrended fluctuation analysis, it is proven that there are obvious multifractal characteristics in the return series of electricity and carbon markets, and the results of traditional linear analysis are unreliable. We also find that, based on multifractal detrended cross-correlation analysis, the law cross-correlation between electricity and carbon markets exists significantly. The long-range correlation of small fluctuations and large fluctuations and the fat tail distribution of return series are the reasons for the formation of multifractality.

Intraday downward/upward multifractality and long memory in Bitcoin and Ethereum markets: An asymmetric multifractal detrended fluctuation analysis

This study examines high-frequency asymmetric multifractality, long memory, and weak-form efficiency for two major cryptocurrencies, namely, Bitcoin (BTC) and Ethereum (ETH), using the asymmetric multifractal detrended fluctuation analysis method to consider different market patterns. Our results show evidence of structural breaks and asymmetric multifractality. Moreover, the multifractality gap between the uptrend and downtrend is small when the time scale is small, and it increases as the time scale increases. The BTC market is more inefficient than ETH. The inefficiency is more (less) accentuated when the market follows a downward (upward) movement. The efficiency level varies based on each subperiod.

Asymmetric risk transmission effect of cross-listing stocks between mainland and Hong Kong stock markets based on MF-DCCA method

With the implementation of “Shanghai–Hong Kong Stock Connect” and “Shenzhen–Hong Kong Stock Connect,” the mainland and Hong Kong stock markets are becoming more closely linked. Based on the A+H cross-listed A-share and H-share market indices, this study employs asymmetric multifractal cross-correlation methods to analyze the asymmetric cross-correlation between the A-share and H-share markets from diverse perspectives of different ups and downs and various conduction directions with 79 sample stocks from January 1 a=2004 to May 26, 2017. Empirical results show that the A+H shares have long memory in different trends, which is stronger in the downward trend of stock price. It indicates that regardless of which market with A+H shares showing a downward trend are on, driving the future on the local market and the corresponding cross-listed market show a downward trend is easier than driving the rising trend. In addition, a bidirectional risk conduction effect exists between A and H shares, and the A-share market has a strong transmission effect on the H-share market.