Multifractal Method Research Articles

Combining global and local scaling methods to detect soil pore space

Computed tomography (CT) images provide very useful information to characterize the pore space structure in different porous materials. Being a non-destructive technique, it allows preserving the real characteristics of the analysed samples. Moreover, providing 3 dimensional (3D) information, it also allows making more exhaustive studies on other characteristics of pore space such as connectivity or pore geometry. However, this technique also has drawbacks such as the insertion of errors and distortions from CT reconstruction that could drive to low contrast greyscale images hindering the detection of solid-void interface.The Singularity-CA (S-CA) binarization method has already proven to be an efficient method when detecting the pore space in CT soil images. This method is primarily based on the existence of self-similar properties in the singularity value (SV) spatial distribution. These self-similar properties derive to thresholds in the singularity map which are used to delimit the pore space. This method detects the medium and large pore sizes with an excellent fit. However, due to its high sensibility to low fluctuations in the greyscale images, some small pores are incorrectly detected and therefore its amount is overestimated.This work attempts to solve this drawback introducing a new approach named the Combining Singularity-CA (CS-CA) method. This improved method is based on the combination of a global thresholding method, the Maximum Entropy method, and the S-CA method. The CS-CA methodology is fully automatic and did not require human adjustment. A set of soil synthetic images with porosities from 3% to 25% were used to validate the new approach. These images were constructed using the Truncated Multifractal (TM) method, especially useful to simulate CT soil images. Based on parameters such as porosity, relative error and misclassification error, CS-CA method gave better pore detection than the S-CA and the Maximum Entropy method applied individually to the images.Finally, it was compared the two main methods used to detect slope-change points in plots with several linear segments: the linear regression (LR) method and the wavelet transform modulus maxima (WTMM) method. These methods are very important in the S-CA methodology since they are responsible of defining the threshold in the binarization step. The WTMM method proved to be easier to deal with when using by the new CS-CA method.

Price–volume multifractal analysis of the Moroccan stock market

In this paper, we analyzed price–volume multifractal cross-correlations of Moroccan Stock Exchange. We chose the period from January 1st 2000 to January 20th 2017 to investigate the multifractal behavior of price change and volume change series. Then, we used multifractal detrended cross-correlations analysis method (MF-DCCA) and multifractal detrended fluctuation analysis (MF-DFA) to analyze the series. We computed bivariate generalized Hurst exponent, Rényi exponent and spectrum of singularity for each pair of indices to measure quantitatively cross-correlations. Furthermore, we used detrended cross-correlations coefficient (DCCA) and cross-correlation test (Q(m)) to analyze cross-correlation quantitatively and qualitatively. By analyzing results, we found existence of price–volume multifractal cross-correlations. The spectrum width has a strong multifractal cross-correlation. We remarked that volume change series is anti-persistent when we analyzed the generalized Hurst exponent for all moments q. The cross-correlation test showed the presence of a significant cross-correlation. However, DCCA coefficient had a small positive value, which means that the level of correlation is not very significant. Finally, we analyzed sources of multifractality and their degree of contribution in the series.

The Time-Scaling Analysis of Chinese Stock Market Based on Fractal Methods

Detrended fluctuation analysis (DFA), detrended cross-correlation fluctuation analysis (DCCA) and multifractal methods are applied to the time-scaling properties analysis of daily closing prices of Shanghai composite index (SHCI) and Shenzhen composite index (SZCI) of Chinese stock market. The results show that these index series are characterised by long-term memory, multifractal scaling and power-law behavior, and these characteristics have obvious differences between the SHCI and SZCI. The comparison results suggest that heterogeneity (disordered state) characterises of the SHCI dynamics owing to the wider fluctuation of the power of impulsion and suppression outside to stock, and other related factors in Shanghai stock market than that of SZCI in Shenzhen stock market. Furthermore, we investigated the frequency-size distribution of SHCI and SZCI series. This work can be helpful to improvement of modelling of Chinese stock market.

Analysis of normal human retinal vascular network architecture using multifractal geometry.

To apply the multifractal analysis method as a quantitative approach to a comprehensive description of the microvascular network architecture of the normal human retina. Fifty volunteers were enrolled in this study in the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and January 2014. A set of 100 segmented and skeletonised human retinal images, corresponding to normal states of the retina were studied. An automatic unsupervised method for retinal vessel segmentation was applied before multifractal analysis. The multifractal analysis of digital retinal images was made with computer algorithms, applying the standard box-counting method. Statistical analyses were performed using the GraphPad InStat software. The architecture of normal human retinal microvascular network was able to be described using the multifractal geometry. The average of generalized dimensions (Dq ) for q=0, 1, 2, the width of the multifractal spectrum (Δα=αmax - αmin ) and the spectrum arms' heights difference (|Δf|) of the normal images were expressed as mean±standard deviation (SD): for segmented versions, D0 =1.7014±0.0057; D1 =1.6507±0.0058; D2 =1.5772±0.0059; Δα=0.92441±0.0085; |Δf|= 0.1453±0.0051; for skeletonised versions, D0 =1.6303±0.0051; D1 =1.6012±0.0059; D2 =1.5531±0.0058; Δα=0.65032±0.0162; |Δf|= 0.0238±0.0161. The average of generalized dimensions (Dq ) for q=0, 1, 2, the width of the multifractal spectrum (Δα) and the spectrum arms' heights difference (|Δf|) of the segmented versions was slightly greater than the skeletonised versions. The multifractal analysis of fundus photographs may be used as a quantitative parameter for the evaluation of the complex three-dimensional structure of the retinal microvasculature as a potential marker for early detection of topological changes associated with retinal diseases.

Geometric and fractal analysis of dynamic cracking patterns subjected to wetting-drying cycles

Desiccation cracks detrimentally affect the hydraulic and mechanical properties of soils. Soil cracking dynamic subjected to wetting-drying (W-D) cycles aids in the understanding of soil shrinking-swelling mechanism and the regularity of water movement in unsaturated soils with variable solid phase. In the present paper, we experimentally investigate the geometric and fractal features of dynamic cracking patterns with the soils undergoing four W-D cycles. Crack patterns were recorded by a digital camera mounted above the soil specimen. Digital image processing and morphological algorithms were employed to yield the geometric and fractal indexes that depict crack patterns (incl. crack ratio, length, mean width, similarity degree, box-counting fractal and multifractal dimensions). The analysis of dynamic crack patterns suggest an inconsistency exhibited in the crack morphology towards the drying (DR) process and backwards the wetting (WT) process, which could be possibly attributed to the different suction stress between soil particles during moisture desorption and sorption, closely analogous to the hysteresis phenomenon of soil-water characteristic curve. The gradient of crack propagating and dissolving related to water content reaches different peak values during an individual W-D cycle. High repeatability is found across the stabilized crack patterns in the four drying cycles and the cracking degree reaches equilibrium with the increasing number of W-D cycles. The crack patterns in the subsequent drying processes are significantly dominated by the first cracking. The similarity degree between the first and respectively the second, third, fourth crack patterns decreases with the increasing W-D cycles, suggesting the soil cracked in the previously cracking vicinity, and saturated wetting/submerge hardly healed the failure zones that caused the repeatable cracking. Attempt of fractal and multifractal methods applied to the crack networks corroborates the applicability of the scaling method to the crack patterns. The analysis indicates that soil cracks progress in hierarchical networks in the first drying cycle, while the gradational crack segments nearly simultaneously emerged or vanished in the subsequent drying or wetting cycles. The study reveals the discrepancy of crack morphology between the crack formation and dissolution during W-D cycles, and demonstrates the repeatability degree and hierarchy of cracks across W-D cycles, which would aid in quantifying physical processes in cracked soil shrinking/swelling and modelling of water transport in unsaturated soils of dynamic structure.

Fourier transform infrared spectroscopy microscopic imaging classification based on multifractal methods.

Multifractal analysis (MFA) based on generalized concepts of fractals has been applied to biological tissues composed of complex structures. In this paper, a new MFA methodology based on the neighborhood spatial correlation (NSC) is proposed for an extracting texture feature. NSC is used to extract spatial features, and the obtained spatial features are combined with spectral features of characteristic absorption peaks (CAPs) to promote more feature information. This spatial-spectral structure is used as a feature to differentiate cholesterol from Fourier transform infrared spectroscopy microscopic imaging of a rabbit artery by a support vector machine classifier. The dataset was collected between 4000 and 720 cm-1 on rabbit arteries as research objects. The experimental results show that the accuracy of the proposed spatial-spectral structure is higher than that of other multivariate analysis methods (PCA and 2DPCA). The NSC method, compared to the bottom interface method, new bottom interface method, variance method multi-weight method, and neighborhood spatial correlation method, could effectively reduce the influence of speckle noise, and the convergence rate of the weight factor q is not increased.

Dynamical behaviors of multifractal strengths in meteorological factors

In this paper, we study the multifractal properties of cloud observation time-series data in Daegwanryung, Korea, containing two meteorological factors, the effective radius of a cloud droplet and the average particle size of a raindrop. We simulate and analyze the generalized Hurst exponent, the Renyi exponent, the spectrum, and the multifractal strength by using the multifractal detrended fluctuation analysis method. The result obtained may have a useful and effective influences on determining the observation location.

Multifractal manifold for rotating machinery fault diagnosis based on detrended fluctuation analysis

The vibration signals of rotating machinery in fault conditions are non-stationary and nonlinear. For the non-stationary and nonlinear characteristics of fault vibration signals, a novel multifractal manifold (MFM) method based on detrended fluctuation analysis (DFA) is proposed. The proposed method consists of three steps. Firstly, calculate the multifractal fluctuation functions of signal series with an appropriate polynomial order, according to multifractal DFA method. Secondly, construct multifractal feature vector for each signal sample to reveal the nonlinear characteristics in different scales. Finally, implement manifold learning to reduce the dimension of multifractal feature vectors. The obtained low-dimensional MFM features can reveal the differences of signal samples from different fault patterns effectively, which are benefit for automatic pattern recognition and multiple fault diagnosis. The recognition performance of the proposed MFM method is verified by fault experiments of gearbox and rolling element bearing, which demonstrates the superiority of MFM method in rotating machinery fault diagnosis compared to other DFA-based methods.

Generalized Multi-fractal Method for Soft Fault Feature Extraction of Non-linear Analog Circuits Based on EMD

The soft fault signal of non-linear analog circuits is non-linear and unstable. Its inherent characteristics cannot be analyzed by the traditional linear method. Therefore, a new method based on the empirical mode decomposition (EMD) and a generalized multi-fractal dimension algorithm is proposed for soft fault signal feature extraction of non-linear analog circuits. The EMD decomposed the circuits soft fault signal into several intrinsic mode functions (IMF). The EMD then modified a traditional G-P correlation dimension algorithm to simply and reliably use it for calculating the generalized multi-fractal dimensions of the fault signals. According to fractal theory, the generalized fractal dimensions of the original fault signals and IMF can be obtained for an appropriate weight factor q. The principle IMF component can then be filtered using the cross-correlation coefficient criterion. The obtained generalized multi-fractal dimension of the principle IMF component was continually input in the supporting vector machine as a characteristic vector to diagnose the soft fault of the non-linear analog circuits. The feasibility and validity of this proposed method were demonstrated by experiments.

The application of the multifractal cross-correlation analysis methods in radar target detection within sea clutter

Abstract Many complex systems generate multifractal time series which are long-range cross-correlated. This paper introduces three multifractal cross-correlation analysis methods, such as multifractal cross-correlation analysis based on the partition function approach (MFXPF), multifractal detrended cross-correlation analysis (MFDCCA) methods based on detrended fluctuation analysis (MFXDFA) and detrended moving average analysis (MFXDMA), which only consider one moment order. We do comparative analysis of the artificial time series (binomial multiplicative cascades and Cantor sets with different probabilities) by these methods. Then we do a feasibility test of the fixed threshold target detection within sea clutter by applying the multifractal cross-correlation analysis methods to the IPIX radar sea clutter data. The results show that it is feasible to use the method of the fixed threshold based on the multifractal feature parameter Δ f ( α ) by the MFXPF and MFXDFA-1 methods. At last, we give the main conclusions and provide a valuable reference on how to choose the multifractal algorithms, the detection parameters and the target detection methods within sea clutter in practice.

Fractal and multifractal analysis of different hydraulic flow units based on micro-CT images

Pore structure is one of the key factors affecting the transport behavior in porous media, which is mainly determined by petrophysical facies. To accurately characterize the diversity of pore structure, hydraulic flow units (HFU) based on the modified flow zone indicator data are subdivided, and experiments of mercury injection and routine core analysis are conducted for 601 sandstone samples in this study. Five samples selected from different HFU (porosity is approximately equal, however permeability varies widely) are chosen to conduct micro-computerized tomography test. Then, fractal and multifractal methods are used to characterize complex pore space of chosen samples based on the number-area model, radius-volume model and number-radius model respectively, in which the effect of image scale on the fractal dimension is investigated. Each HFU has similar box dimension and generalized fractal dimension, but has significant difference in multifractal spectrum. Specially, the different values of singularity exponent in the multifractal spectrum have a good relationship with the experimentally measured permeability. There are clear inflections points on the fractal curves from mercury injection curve. Based on the value of maximum pore throat radius, only one segment (located in the middle) can be used to calculate the fractal dimension. Fractal dimension obtained by pore network model does not satisfy the traditional relationship. Correspondingly, fractal and multifractal characteristics pore structure of different hydraulic flow units are analyzed in detail.

Asymmetric joint multifractal analysis in Chinese stock markets

In this paper, the asymmetric joint multifractal analysis method based on statistical physics is proposed to explore the asymmetric correlation between daily returns and trading volumes in Chinese stock markets. The result shows asymmetric multifractal correlations exist between return and trading volume in Chinese stock markets. Moreover, when the stock indexes are upward, the fluctuations of returns are always weaker than when they are downward, whether the trading volumes are more or less.

Specific features of proton interaction with transistor structures having a 2D AlGaN/GaN channel

It has been shown that the interaction of 1 MeV protons at doses of (0.5–2) × 1014 cm–2 with transistor structures having a 2D AlGaN/GaN channel (AlGaN/GaN HEMTs) is accompanied not only by the generation of point defects, but also by the formation of local regions with a disordered nanomaterial. The degree of disorder of the nanomaterial was evaluated by multifractal analysis methods. An increase in the degree of disorder of the nanomaterial, manifested the most clearly at a proton dose of 2 × 1014 cm–2, leads to several-fold changes in the mobility and electron density in the 2D channel of HEMT structures. In this case, the transistors show a decrease in the source–drain current and an order-of-magnitude increase in the gate leakage current. In HEMT structures having an enhanced disorder of the nanomaterial prior to exposure to protons, proton irradiation results in suppression of the 2D conductivity in the channel and failure of the transistors, even at a dose of 1 × 1014 cm–2.

Separation of a geochemical anomaly from background by fractal and U-statistic methods, a case study: Khooni district, Central Iran

Separation of geochemical anomalies from background are one of the important steps in mineral exploration. The Khooni mineral district (Central Iran) has complex geochemical surface expression due to a complex geological background. This region was chosen as a study area for recognition of the spatial distribution of geochemical elements and separating anomalies from background using stream sediment geochemical data. In the past decades, geochemical anomalies have been identified by means of various methods. Some of these separation methods include: statistical analysis methods, spatial statistical methods and fractal and multi-fractal methods. In this article, two efficient methods, i.e. U-statistics and the fractal concentration-area for separation and detection of anomalous areas of the background were used. The U spatial statistic method is a weighted mean, which considers sampling point positions and their spatial relation in the estimation of anomaly location. Also, fractal and multi-fractal models have also been applied to separate anomalies from background values. In this paper, the concentration–area model (C–A) was suggested to separate the anomaly of background. For this purpose, about 256 stream sediment samples were collected and analyzed. Then anomaly maps of elements were generated based on U spatial statistics and the C-A fractal methods for Au, As and Sb elements. According to obtained results, the U-statistics method performed better than C-A method. Because the comparisons of the known deposits and occurrences against the anomalous area created using thresholds from U-statistics and C-A method show that the spatial U-statistics method hits all of 3 known deposits and occurrences, the C-A fractal method hits 1 and fails 2. In addition, the results showed that these methods with regard to spatial distribution and variability within neighboring samples, in addition to concentration value frequency distributions and correlation coefficients, have more accurate results than the traditional approaches.

Multifractal analysis in soil properties: Spatial signal versus mass distribution

The spatial variability of soil properties is a constant expected factor that must be considered in soil studies. This variability is composed of “functional” variations and random fluctuations or noise. Multifractal formalism is suitable for variables with self-similar distributions on a spatial domain. Multifractal analysis can provide insight into the spatial variability of soil parameters. In soil science, it has been quite popular to characterize the scaling property of a variable measured along a transect as a mass distribution of a statistical measure on a length domain of the studied transect. The analysed variable is divided into a number of self-similar segments, and the partition function and mass function are estimated. Based on these estimations, the multifractal spectrum (MFS) is calculated. Another technique that can be applied focuses on the variations of a measure by analysing the absolute differences in the soil property values at different scales, such as the Generalized Structure Function (GSF) and the Universal Multifractal Model (UMM). The aim of this study was to compare both types of multifractal methods on a set of soil physical properties measured on a common 1024m transect across arable fields at Silsoe in Bedfordshire, East-Central England. The studied properties were total porosity (Porosity), gravimetric water content (GWC) and nitrous oxide flux (N2O flux). The results showed that when using both methods, the N2O flux exhibits a distinctive multifractal character, and weak multifractal characters are detected in the GWC and Porosity cases. Additionally, several parameters were calculated and discussed.Finally, the relationship between the mass exponent function (τ(q)) and the GSF (ζ(q)) found in the literature, was positively verified for the three variables. On the contrary, the relationship between ζ(q) and the scaling exponent function based on UMM (K(q)) showed discrepancies in N2O flux and GWC for q values higher than 3. This is the first time that these comparisons have been made on soil property data.

Temporal and spatial characteristics of extreme precipitation events in the Midwest of Jilin Province based on multifractal detrended fluctuation analysis method and copula functions

Environmental changes have brought about significant changes and challenges to water resources and management in the world; these include increasing climate variability, land use change, intensive agriculture, and rapid urbanization and industrial development, especially much more frequency extreme precipitation events. All of which greatly affect water resource and the development of social economy. In this study, we take extreme precipitation events in the Midwest of Jilin Province as an example; daily precipitation data during 1960–2014 are used. The threshold of extreme precipitation events is defined by multifractal detrended fluctuation analysis (MF-DFA) method. Extreme precipitation (EP), extreme precipitation ratio (EPR), and intensity of extreme precipitation (EPI) are selected as the extreme precipitation indicators, and then the Kolmogorov-Smirnov (K-S) test is employed to determine the optimal probability distribution function of extreme precipitation indicators. On this basis, copulas connect nonparametric estimation method and the Akaike Information Criterion (AIC) method is adopted to determine the bivariate copula function. Finally, we analyze the characteristics of single variable extremum and bivariate joint probability distribution of the extreme precipitation events. The results show that the threshold of extreme precipitation events in semi-arid areas is far less than that in subhumid areas. The extreme precipitation frequency shows a significant decline while the extreme precipitation intensity shows a trend of growth; there are significant differences in spatiotemporal of extreme precipitation events. The spatial variation trend of the joint return period gets shorter from the west to the east. The spatial distribution of co-occurrence return period takes on contrary changes and it is longer than the joint return period.

Dynamical mechanism in aero-engine gas path system using minimum spanning tree and detrended cross-correlation analysis

Abstract Identifying the mutual interaction in aero-engine gas path system is a crucial problem that facilitates the understanding of emerging structures in complex system. By employing the multiscale multifractal detrended cross-correlation analysis method to aero-engine gas path system, the cross-correlation characteristics between gas path system parameters are established. Further, we apply multiscale multifractal detrended cross-correlation distance matrix and minimum spanning tree to investigate the mutual interactions of gas path variables. The results can infer that the low-spool rotor speed (N1) and engine pressure ratio (EPR) are main gas path parameters. The application of proposed method contributes to promote our understanding of the internal mechanisms and structures of aero-engine dynamics.

Multifractal characterization of mechanical vibration signals through improved empirical mode decomposition-based detrended fluctuation analysis

A novel multifractal detrended fluctuation analysis based on improved empirical mode decomposition for the non-linear and non-stationary vibration signal of machinery is proposed. As the intrinsic mode functions selection and Kolmogorov–Smirnov test are utilized in the detrending procedure, the present approach is quite available for contaminated data sets. The intrinsic mode functions selection is employed to deal with the undesired intrinsic mode functions named pseudocomponents, and the two-sample Kolmogorov–Smirnov test works on each intrinsic mode function and Gaussian noise to detect the noise-like intrinsic mode functions. The proposed method is adaptive to the signal and weakens the effect of noise, which makes this approach work well for vibration signals collected from poor working conditions. We assess the performance of the proposed procedure through the classic multiplicative cascading process. For the pure simulation signal, our results agree with the theoretical results, and for the contaminated time series, the proposed method outperforms the traditional multifractal detrended fluctuation analysis methods. In addition, we analyze the vibration signals of rolling bearing with different fault types, and the presence of multifractality is confirmed.

Comparative Multifractal Analysis of Dynamic Infrared Thermograms and X-Ray Mammograms Enlightens Changes in the Environment of Malignant Tumors.

There is growing evidence that the microenvironment surrounding a tumor plays a special role in cancer development and cancer therapeutic resistance. Tumors arise from the dysregulation and alteration of both the malignant cells and their environment. By providing tumor-repressing signals, the microenvironment can impose and sustain normal tissue architecture. Once tissue homeostasis is lost, the altered microenvironment can create a niche favoring the tumorigenic transformation process. A major challenge in early breast cancer diagnosis is thus to show that these physiological and architectural alterations can be detected with currently used screening techniques. In a recent study, we used a 1D wavelet-based multi-scale method to analyze breast skin temperature temporal fluctuations collected with an IR thermography camera in patients with breast cancer. This study reveals that the multifractal complexity of temperature fluctuations superimposed on cardiogenic and vasomotor perfusion oscillations observed in healthy breasts is lost in malignant tumor foci in cancerous breasts. Here we use a 2D wavelet-based multifractal method to analyze the spatial fluctuations of breast density in the X-ray mammograms of the same panel of patients. As compared to the long-range correlations and anti-correlations in roughness fluctuations, respectively observed in dense and fatty breast areas, some significant change in the nature of breast density fluctuations with some clear loss of correlations is detected in the neighborhood of malignant tumors. This attests to some architectural disorganization that may deeply affect heat transfer and related thermomechanics in breast tissues, corroborating the change to homogeneous monofractal temperature fluctuations recorded in cancerous breasts with the IR camera. These results open new perspectives in computer-aided methods to assist in early breast cancer diagnosis.

Change Detection Using Global and Local Multifractal Description

In this letter, we apply the multifractal formalism to land cover change detection on very high spatial resolution data. Specifically, multifractal spectra are determined and, with modifications, are used as an initial general indicator of change on the subsets of IKONOS and Pleiades images. Next, we calculate Holder exponents for each pixel in the images and use them to generate a change mask. Our analysis shows that Holder exponents enable a detailed evaluation of changes in land cover. A comparison with change detection based solely on panchromatic images shows that the multifractal description method has significant advantages as it reduces the number of false positives. In addition, we show that our change detection results are comparable with other multiscale techniques.