Multifractal Method Research Articles

FRACTAL ANALYSIS OF PORE MICROSTRUCTURES OF JUNCUS EFFUSUS

It is found that the pore microstructures of Juncus effusus (JE), a yearly renewable and high-yield cellulose fiber with unique three-dimensional network structure and interconnected channels, exhibit the fractal characters. The cross-sectional images are measured by SEM and the fractal dimensions are calculated by the box-counting method. In order to reveal more details of JE, the multi-fractal spectrum method is employed to analyze its local features, which further shows the fractal structures. The results indicate that the fractal characteristics are obvious although JE is complex and disordered in structure. The present fractal study may have the potential and significance on fractal analysis of mechanical properties and transport properties of these natural fibers.

Multifractal analysis of model fractal and multifractal signals

One of the topical directions of modern fractal radio physics is the multifractal analysis of signals and processes of various origins. A set of deterministic and stochastic models of monofractal and multifractal signals and processes in the time domain is proposed. New multifractal signal characteristics, namely, the coefficient of asymmetry of the multifractal spectrum function, the relative width of the multifractal spectrum and the dimension of the multifractal support, are introduced, the necessity of their use is demonstrated on examples. Using Wavelet Transform Modulus Maxima Method and Multi-Fractal Detrended Fluctuation Analysis Method, a detailed multifractal analysis of model signals is performed. The features of multifractal analysis of monofractal, multifractal and non-fractal signals are established, the appropriate recommendations for practitioners are formulated. Convenient formats for presenting analysis results have been developed. It was found that during the transition of the multifractal signal to the monofractal regime, the function of the multifractal spectrum of the physical fractal does not collapse into a point, as it should happen in theory for a mathematical fractal. Threshold values of multifractal characteristics, which are indicators of the appearance of the monofractal, are proposed. It has been shown that multifractal analysis of non-fractal signals leads to the appearance of multifractal spectra with anomalous values of multifractal characteristics. The correction function method is modified for the methods of multifractal analysis of signals and processes. It is proved that its usage makes it possible to reduce the deviation of the obtained estimate of the generalized Hurst exponent from the true known value of the Hölder exponent of the analyzed signal from 5 – 90% to 3 – 8%.

Asymmetric Multifractal Analysis of the Chinese Energy Futures and Energy Stock Markets under the Impact of COVID-19

The volatility and cross-correlations of the energy market and the stock market during the COVID-19 pandemic have been paid close attention by scholars and investors. In this paper, we use the asymmetric multifractal analysis methods to study the fluctuation characteristics, market risks and cross-correlations of the Chinese energy futures market (EFM) and two energy stock markets before and after the COVID-19 outbreak, while the return series of Shanghai fuel oil futures, CSI Energy Index and CSI Mainland New Energy Theme Index are considered. The empirical evidences indicate that the auto- and cross-correlations of the three markets have the asymmetric multifractality, and that the multifractality of the cross-correlations is mainly caused by the fat-tailed distribution of the original series. After the COVID-19 outbreak, the risks of both the traditional energy stock market in the uptrend and the entire new energy stock market become larger, while those of the entire EFM become smaller. In addition, the COVID-19 pandemic has increased the multifractality of the cross-correlations between the energy futures and energy stock markets when the EFM is in downward trend.

Comprehensive Characterization Integrating Static and Dynamic Data for Dynamic Fractures in Ultra-Low Permeability Reservoirs: A Case Study of the Chang 6 Reservoir of the Triassic Yanchang Formation in the Ordos Basin, China

The generation of dynamic fractures during the process of water injection in ultra-low permeability reservoirs aggravates the heterogeneity of the reservoir, resulting in a rapid rise of water cut and directional flooding of the producers, which affects waterflood sweep efficiency and recovery. A dynamic fracture, as the geological feature of ultra-low permeability reservoirs, has a complex genetic mechanism and is difficult to characterize. Taking the Chang 6 reservoir of the Triassic Yanchang Formation in the Ordos Basin of central China as an example, this paper presents the characterization method and workflow of dynamic fractures. On the one hand, through the analysis of triaxial-compression rock-mechanic experiments and the mineral composition of the core, we evaluated rock brittleness in order to identify the lithology that can easily generate new fractures. On the other hand, beginning with the ancient tectonic stress field and combining the fracture characteristics of core and geological outcrop, the multi-fractal method and the probabilistic neural network were applied to identify the natural fractures and to quantitatively predict the intensity of natural fractures. Based on the rock brittleness evaluation and the natural fracture feature, the intensity of dynamic fractures was characterized by integrating the analysis of the bottom hole pressure, fracture pressure, and production response characteristics. A dynamic fracture is a “double-edged sword” during the waterflooding development of ultra-low permeability reservoirs. The premature activation and generation of dynamic fractures could lead to a worse development status. Nevertheless, the rational control and utilization of dynamic fractures play a positive role in improving oil recovery. Dynamic fractures are of great significance to the optimization and adjustment of well patterns for ultra-low permeability reservoirs. This can provide a reference for similar reservoirs.

Novel hybrid multi-head self-attention and multifractal algorithm for non-stationary time series prediction

Traditional time series prediction methods have shown their outstanding capabilities in time series prediction. However, due to essential differences in volatility characteristics among diverse types of non-stationary multivariate time series (NSMTS), it is difficult for traditional methods to maintain robust prediction performance. This study proposes a novel dynamic recurrent neural network to achieve stable and robust prediction performance. First, a multifractal gated recurrent unit (MF-GRU) based on the multifractal method is proposed to extract volatility characteristics. Meanwhile, to strengthen the parameters of the historical hidden layer state that has a more significant impact on the output, a self-attention mechanism is introduced into the MF-GRU, leading to a multifractal gated recurrent unit multi-head self-attention model. The efficiency of the proposed model was verified on public datasets. The experimental results show that the proposed model outperforms the traditional methods, such as long short-term memory (LSTM), the gated recurrent unit (GRU), and the minimal gated unit (MGU). etc.

Implication of multifractal analysis for quantitative evaluation of mineral resources in the Central Kunlun area, Xinjiang, China

Extensive research has been conducted to evaluate mineral resource potential based on geochemical data, but this work is still challenging due to the existence of multiple evaluation solutions based on different methods. In this paper, we combine the multifractal analysis method with typical multivariate statistical methods to analyse the spatial characteristics of geochemical stream sediment data, aiming to quantitatively study the ore-forming potential of the elements in the Central Kunlun area of Xinjiang, China. An R -type cluster analysis, Pearson correlation analysis, and principal component analysis are used to explore the correlations among the 12 target elements. The multifractal model is constructed using the method of moments to analyse the spatial distribution patterns of the elements, and corresponding multifractal parameters are extracted to quantitatively describe their ore-forming strengths in the study area. The results show that Co, V, Ti, Fe 2 O 3 , MgO and Cu compose a group of elements closely related to the regional geological background, while Pb, Zn, Bi, Sn, Au and Ba are potential metallogenic elements with relatively high ore-forming strengths and favourable ore-forming potential. Multifractal theory further validates and evaluates the favourable ore-forming element group obtained through conventional geochemical multivariate statistical methods, thus providing a new idea for small-scale geochemical prospecting. Thematic collection: This article is part of the Applications of Innovations in Geochemical Data Analysis collection available at: https://www.lyellcollection.org/cc/applications-of-innovations-in-geochemical-data-analysis

Interplay multifractal dynamics among metal commodities and US-EPU

This research employs the Multifractal Detrended Fluctuations Analysis (MF-DFA) and the Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) to study the components of the observed multifractality considering eight metals commodities time series (Gold, Silver, Platinum, Palladium, Copper, Lead, Nickle and tin) and economic policy uncertainty (US-EPU) index time series. For both multifractal methods, we analyze the generalized Hurst exponent h(q) and the Rényi exponent τ(q), and evaluate their statistical properties, which allow us to examine separately the small-scale contributing (primarily via the negative moments q) and the large scale (via the positive moments q). Also, we calculate the multifractal spectrum f(α) and perform a fourth-degree polynomial regression fit to evaluate the complexity parameters that describe the degree of multifractality, considering the volatility time series and the volatility time series pairs. We find that these volatility time-series (α0>0.5) and these volatility time series pairs (αxy(0)>0.5) are characterized by overall persistent behavior. We discover that Gold seems more related to the other commodities than (US-EPU) index, which is relevant for the different market agents, mainly investors, which could retrieve important information from these results, for example, for diversification purposes.

The Impact of Price Support Policies and Public Emergencies on Agricultural Futures Markets — A Multifractal Cross-Correlations Analysis between China and the US

This paper investigates the presence and asymmetry of cross-correlations between agricultural futures markets in China and the US as well as the impact of price support policies and public emergencies (Sino–US trade conflict and COVID-19 pandemic) on the cross-correlations by the multifractal methods. The results show that the fluctuation characteristics and conduction directions of cross-correlations are asymmetric. The price fluctuations of soybean and corn futures in China are easier to be affected by the US soybean and corn futures. We find that the cross-correlations are multifractal under different price support policies and pubic emergencies. The price support policies with greater interventions on soybean and corn prices have aggravated the complexity of cross-correlations between the two futures markets in China and the US. The soybean and corn futures in China are hardly correlated to the US futures under the dual effect of the Sino–US trade conflict and the COVID-19 pandemic. The Sino–US trade conflict strengthens the complexity of cross-correlation for soybean futures and weakens it for corn futures, while the COVID-19 pandemic enhances the complexity of cross-correlations for soybean and corn futures. In addition, the fat-tailed probability distributions in different price support policy and public emergency periods have a dominant influence on the multifractality of cross-correlations.

The correlation between external magnetic field and multifractal characteristics of PLD deposited DLC films

The present study explores the application of the multifractal method in examining the surface characteristics of the diamond-like carbon (DLC) thin films deposited onto substrates of silicon through a magnetically-assisted pulsed laser deposition (MAPLD) method at room temperature (RT). Multifractal spectra f(α), obtained from atomic force microscopy (AFM) analysis, is adopted to investigate the surface morphology of the produced samples. The relationship between the applied magnetic field and multifractal characteristics of the obtained samples is examined as well. The findings attest to the apparent effect of the magnetic field on the quality of surface morphology and the multifractal spectrum width. Our results reveal that the root-mean-square (RMS) and average surface roughness are enhanced by applying and intensifying the magnetic field. Furthermore, the findings suggest that both the non-uniformity of the height distributions and the singularity spectrum width of the samples fabricated in this study increases with applying and intensifying the magnetic field. Further, Multifractal spectra f(α) demonstrate the multifractality nature of all the obtained samples, including conventionally and magnetically processed DLC thin films. Besides, the multifractality strength of thin films rises with increasing the magnetic field.

Fractal and multifractal spatiotemporal patterns of land surface temperatures in a coastal city

ABSTRACT In order to discover the fractal and multifractal nature of land surface temperature (LST) spatiotemporal patterns, this study computed the LST in thecoastal city, Xiamen, southeastern China using Landsat TM/OLI/TIRS images from 1994 to 2015, and introduced two-dimensional box-counting and two-dimensional multifractal box-counting methods to quantitatively characterize the LST pattern. Results suggest that increasing the binarization threshold decreased the box-counting dimensions with values from 1.86 to 1.65 in 1994, 1.67 in 2000, 1.56 in 2004, 1.57 in 2010, and 1.56 in 2015, respectively. The two-dimensional multifractal approach employs a set of intertwined nonfractal subsets over different spatial scales to describe the LST patterns. LST pattern possesses a increased multifractality and the similar multifractal spectra of left-hook shapes throughout the study period, suggesting the increasing trends of the spatial heterogeneity in LST distribution. The probability of a given pixel having a high LST value is consistently high in the study area, as indicated by the positive ratio between the regions in which the probability measure appears most concentrated and those in which it is most sparse. Rapid urbanization and the large-scale urban land surface changes in the coastal city determine the variation in the multifractal parameters.

Discrimination of Microseismic Events in Coal Mine Using Multifractal Method and Moment Tensor Inversion

Discrimination of various microseismic (MS) events induced by blasting and mining in coal mines is significant for the evaluation and forecasting of rock bursts. In this paper, multifractal and moment tensor inversion methods were used to investigate the waveform characteristics and focal mechanisms of different MS events in a more quantitative way. The multifractal spectrum calculation results indicate that the three types of MS waveform have different distribution ranges in the multifractal parameters of ∆α and Δf(α). The results show that the blasting schemes also have a great influence on MS waveform characteristics. Consequently, the multifractal parameters of ∆α and Δf(α) can be used to discriminate different MS events. Further, the focal mechanisms of MS events were calculated by seismic moment tensor inversion. The results show that an explosion is not the dominant mechanism of deep-hole blasting MS events, and the CLVD and DC components account for an important proportion, indicating that some additional processes occur during blasting. Moreover, the coal-rock fracture MS events are characterized by compression implosion or compression/shear implosion mixed focal mechanisms, while the overburden movement MS events are tensile explosion or tensile/shear explosion mixed focal mechanisms. The focal mechanisms and nodal plane parameters have close relationships with the inducing factors and occurrence processes of MS events.

Geochemical Fractal Characteristics of Deep-Sea REE-Rich Sediments in the Western Pacific

At present, the challenge for geochemical prospecting of deep-sea, rare, earth-rich sediments is the selection of exploration sites. Because of the unpredictability of offshore operations, the distribution and selection of survey line stations face great challenges. In this paper, we study the fact that the concentration distribution of geochemistry in space may be a part of a special complex process, which is called the multifractal. It requires a large number of indexes to characterize its scale characteristics. Based on the multifractal spectrum, 38 geochemical indices in the Pigafetta Basin region of the western Pacific Ocean are divided into three categories. The three indexes are distinguished by the multifractal parameters. The results of multifractals are in good agreement with those of the cluster analysis and principal component analysis. In addition, on the basis of principal component analysis, we further used the multifractal filtering method to extract element anomalies and delineate element enrichment regions. The first principal component clearly represents the potential element enrichment area. The spatial analysis technique and multifractal method proposed in the paper provide a new idea for geochemical prospecting of deep-sea, rare, earth-rich sediments.

Multifractal properties of oil and gas indices in the Songnan Low Uplift of the Qiongdongnan Basin deepwater area

Recently, a high-quality reservoir was discovered in the Songnan Low Uplift in the eastern deepwater area of the Qiongdongnan Basin, northern South China Sea. To establish whether this area has good exploration prospects, this paper uses the multifractal moment method to analyse 2689 data points acquired from the basin in the SE of the southern low bulge. It is pointed out that the fractal behaviours of different types of geochemical indices are often distinguished by multifractal spectra of different shapes, therefore multifractal spectrum function analysis is used in this paper. These data were considered within the context of 13 oil and gas indices describing the multifractal spectrum function morphological characteristics. The results show that these indices can be divided into three types according to the multifractal spectrum. Indices with a strong multifractal spectrum function in the Songnan Low Uplift have the characteristics of a wide and continuous multifractal spectrum function, forming a right-skewed arc shape, while the other indices show weak or single-fractal characteristics. The results are in good agreement with principal component analysis results, where the spectral function of each principal component shows that the gas source of the C1, C2, C3, nC4, nC5, and heavy hydrocarbon indices in the Songnan Low Bulge has multifractal characteristics. The weight of these indices with strong multifractal characteristics shows that the principal components C1, C2 and C3, and heavy hydrocarbons, are important indices for delineating hydrocarbon prospects in the Songnan Low Uplift. Multifractal and spatial analysis techniques provide new ideas for index selection and comprehensive information extraction for oil and gas exploration. Thematic collection: This article is part of the Applications of Innovations in Geochemical Data Analysis collection available at: https://www.lyellcollection.org/cc/applications-of-innovations-in-geochemical-data-analysis

Elimination of Image Saturation Effects on Multifractal Statistics Using the 2D WTMM Method.

Imaging artifacts such as image saturation can restrict the computational analysis of medical images. Multifractal analyses are typically restricted to self-affine, everywhere singular, surfaces. Image saturation regions in these rough surfaces rob them of these core properties, and their exclusion decreases the statistical power of clinical analyses. By adapting the powerful 2D Wavelet Transform Modulus Maxima (WTMM) multifractal method, we developed a strategy where the image can be partitioned according to its localized response to saturated regions. By eliminating the contribution from those saturated regions to the partition function calculations, we show that the estimation of the multifractal statistics can be correctly calculated even with image saturation levels up to 20% (where 20% is the number of saturated pixels over the total number of pixels in the image).

Correlation Analysis of Network Big Data and Film Time-Series Data Based on Machine Learning Algorithm

To expand the application of machine learning in movie data, in order to explore the correlation between network big data and film time-series data, based on the machine learning algorithm, the correlation and multifractal characteristics of happiness index (HI) and film box office (BO) were studied and described by introducing multifractal crossover method. On this basis, some indicators are introduced to optimize the neural network model so that the optimization model can describe and predict the box office and other related information well. The results show that the critical values of the happiness index and box office show a linear change trend with the increase of freedom, and the corresponding change curves of the happiness index and box office show obvious nonlinear characteristics, which can be divided into slow increase stage, steady increase stage, and approximately gentle stage. With the increase of iteration parameter q value, the change trend of the long-term and short-term curves of the generalized Hurst function is basically the same, and the difference between the two is getting smaller and smaller, while the difference between the two curves is getting bigger and bigger with the increase of q value of Renyi function. The changing trend of the dynamic Hurst index in the sliding window period all shows that it first rises rapidly to a certain value, then fluctuates rapidly with the increase of time, then drops rapidly to a constant value, and finally continues to show repeated small range fluctuation. Under the influence of time-series parameter α, the original sequence changes the most, the replacement sequence changes the medium, and the corresponding rearrangement sequence changes the least. The overall distribution of box office prediction data conforms to the characteristics of linear variation. The prediction index of the optimized HI-LSTM (Happiness Index-Long term short term memory neural network) model is higher in the box office, indicating that the model has better performance in describing and predicting the box office. This study can provide a theoretical basis for the correlation study of network big data and film data.

Molecular cancer classification on microarrays gene expression data using wavelet‐based deep convolutional neural network

AbstractMicroarray data analysis is a most promising and difficult process due to the complex nature of data. It includes higher dimensionality, several unbalanced classes, a smaller sample size, the presence of noise, and a higher variation of feature values. This has resulted in a decrease in classification accuracy as well as an overfitting problem. This work proposed an efficient and hybrid deep learning technique for molecular cancer classification using expression data to solve these limitations. The different steps in the proposed work are preprocessing, clustering, extraction, selection, and classification. The input data is preprocessed using a scalable range adaptive bilateral filter. Then clustering is done with the help of an improved binomial clustering approach. After that, the data is extracted with the help of the multifractal Brownian motion method. Then the important features are selected with the help of an improved cuckoo search optimization algorithm. Finally, the data classification is performed using a wavelet‐based deep convolutional neural network. This work is validated with the help of five publically available datasets using the PYTHON platform. The different performance measures considered here are accuracy, precision, recall, and F‐measure. The classification accuracy obtained is 98.36%, 98.12%, 98.55%, 97.70%, and 95.30% for ovarian, breast, colon, leukemia, and prostate cancer datasets. The overall result showed that the suggested technique is better than the existing methods.

On the fractal self-organization of the financial time series

Time series of five financial indexes daily returns were analyzed by means of multifractal and recurrence quantification analysis (RQA) methods. It is shown that a financial crisis in 2008 year is accompanied with the increase in determinism and fractal self-organization. Such regularity is noted as analogous to other nonlinear systems behavior in catastrophic situations. At the same time, the global Hürst coefficient is minimal during the crises instead of maximum for physical systems.

Spatial variability of one-parameter model of soil water characteristic curve at field scale in black soil region of northeast China

The soil water characteristic curve (SWCC) is an important parameter for simulating soil water movement and solute transport, and has obvious spatial variability. The models fitting the SWCC generally contain two or more parameters, which makes the spatial variability of the soil water characteristic curve difficult to express. This paper established a one-parameter model of the soil water characteristic curve based on the Gardner model and analyzed its spatial variability with multifractal and joint multifractal methods. Parameter B in the one-parameter model had moderate variation. The local information leading to spatial variability of parameter B in the top soil layer (0–15 cm) and deep soil layer (15–20 cm) was its low values and high values, respectively. At the single scale, spatial variability of parameter B was mainly caused by the bulk density and clay content in the 0–5 cm soil layer, and by the bulk density and sand content in the 5–10, 10–15, and 15–20 cm soil layers. At the multiscale, the most obvious two factors that led to spatial variability of parameter B were the bulk density and silt content in the 0–5 cm soil layer, bulk density and sand content in the 5–10 cm soil layer, and bulk density and clay content in the 10–15 and 15–20 cm soil layers, respectively. The relationships between parameter B and soil properties had scale dependence.

Surface characterization of NiO thin films deposited by RF-magnetron sputtering at different thickness: Statistical and multifractal approach.

Fractal and multifractal are the most important processes and concepts in describing and examining surface morphology, and for this reason, these concepts are an important approach for analyzing the properties and surface geometry of thin films. In this article, multifractal analysis was performed on images, prepared using atomic force microscopy (AFM), of the surface morphology of nickel oxide thin films deposited by RF-Magnetron sputtering at different thicknesses on the glass substrate. The effect of thickness on the surface properties of the layers was studied by applying multifractal and statistical methods on AFM images. The results obtained from the multifractal spectrum show that the surface of the nickel oxide thin films deposited at different thicknesses are multifractal. The multifractal analysis demonstrated that multifractality and complexity of the surface of nickel oxide thin films changes and decrease with thicknesses. We also used statistical parameters to better examine AFM images to study the effects of layers thickness on the deposited NiO thin films. The results indicated that the statistical parameters are a function of the layer's thickness of NiO thin films. Hence, the isotropic properties and functional parameters changed with changing surface thickness. RESEARCH HIGHLIGHTS: Multifractal analysis was applied to the AFM images to study the surface morphology of NiO thin films. The multifractal nature of the surface of NiO thin films is observed. The layers have become more isotropic with increasing thickness. The results illustrated that deposition masses occurred more at the highest sites on the surface. Multifractality of the surface of the sample decreased with increasing layer thickness.

Examining the efficiency and herding behavior of commodity markets using multifractal detrended fluctuation analysis. Empirical evidence from energy, agriculture, and metal markets

We examine the efficiency and herding behavior between four energy markets (Crude, Brent, Gasoline, and Natural gas), five agriculture markets (Corn, Wheat, Soybeans, Cocoa, and Coffee), and three metal markets (Gold, Silver, and Copper) using multifractal detrended fluctuation method. In addition to the full sample, the analysis also uses two most critical sub-periods of GFC 2007-09 and COVID-19 to reveal hidden features and herding patterns of the commodity markets. The results indicate existence of strong multifractality patterns and resultant inefficiency with varied degrees in the commodity markets. We find highest inefficiency in the copper market, followed by silver market. While cocoa remains most efficient commodity market during the entire period of study. In addition, energy markets remain more efficient compared to the rest of the two groups. Furthermore, the results confirm that crises events impact the efficiency of commodity markets significantly, and persistence behavior with a trace of herding is found in some of the markets. Whereas, silver and gold serve as useful hedging instruments during the crises. Our study is useful for risk management, regulatory supervision, and effective decision making for investors related to investments in the commodity markets.