Multifractal Characteristics Research Articles

Cavitation identification in a hydraulic bulb turbine based on vibration and pressure fluctuation measurements

Cavitation can lead to reduced efficiency, increased vibration and noise in hydraulic turbines. Accurate and rapid identification of cavitation states in a turbine is important. In this work, a test system has been constructed for a hydraulic bulb turbine, to synchronously obtain cavitation images on runner blades, vibrations on both the runner hub and the runner chamber, and pressure fluctuations on the draft tube of the turbine at different cavitation conditions. The multifractal characteristics of the collected signals obtained by using the multifractal detrended fluctuation analysis method have been analysed and compared in detail. It is found that the multifractal characteristics of the vibrations and pressure fluctuations depend strongly on the cavitation state. The multifractal strength of the pressure fluctuations in the draft tube is stronger than that of the vibrations on the runner, induced by cavitation. Furthermore, some multifractal parameters of vibrations and pressure fluctuations have been extracted and used for cavitation state identification in the turbine, which is important for safe operation of a turbine.

South Africa's Public Debt: Long-term Dependence, Structural Breaks and Multifractality

This paper aims to analyse the evolution of public debt in South Africa using new and original methods. The case of South Africa has been little studied in the literature on debt because the level of debt in this country remains reasonable. Moreover, the use of non-standard methods allows for a fine-grained analysis of the public debt time series and, consequently, to draw unprecedented conclusions. Using the Multifractal Detrended Fluctuation Analysis (MF-DFA) method borrowed from solid state physics and medicine, we conclude that South Africa's debt has a multifractal character, which originates from the long memory effect. Thus, South Africa's public debt is unsustainable. The inefficiency of this market has been particularly exacerbated by the various shocks throughout the sample period.

Multifractal characteristics of neuronal activity of the <i> globus pallidus </i> in patients with dystonia

Dystonia is a movement disorder characterized by involuntary muscle contractions resulting in anomalous postures, often accompanied by dystonic tremors. Among the indicative dystonic postures is an involuntary tonic tilt or rotation of the head. Currently, the etiology and pathogenesis of the disease remain theories. Currently, only low-frequency (theta-alpha) oscillations in the globus pallidus are regarded as the possible biomarker of pathological activity in dystonia. In many patients with dystonia, the absence of rhythmic neuronal activity in the basal ganglia renders this biomarker unusable for surgical treatment of the disease. Additionally, the mechanisms underlying the rearrangement of temporary patterns of neural activity that result in pathological symptoms remain unclear. We propose that the appearance of these pathological rhythms is associated with the reduction of dynamic complexity in the pattern of neural activity within the globus pallidus. We aim to use multifractal analysis in this study to evaluate changes in the globus pallidus neural activity pattern’s dynamic complexity and their correlation with the clinical manifestations of dystonia. The study examined the electrical activity of the globus pallidus in 23 patients receiving deep brain stimulation (DBS) treatment. We used the microelectrode registration technique and local field potential (LFP) registration during intraoperative neuron monitoring to precisely identify the specific locations of the external and internal segments of the globus pallidus (GPe and GPi). In four patients, an additional postoperative recording of local field potentials (LFP) was conducted while performing DBS and vibrational impact on the neck muscles. A neurologist evaluated the patients to determine the severity of the disease using the Burke-Fahn-Mardsen Dystonia Rating Scale (BFMDRS). The LEAD DBS software package was employed for postoperative reconstruction of the DBS stimulating electrode positions and for simulating the optimal stimulation zone, selected by the neurologist (i.e., volume of tissue activated, VTA). Subsequently, we identified the region that presented the most significant variation in multifractal spectrum parameters following DBS to understand its correlation with the optimal VTA. A total of 745 local potential records were analyzed, with 433 in GPe and 312 in GPi. The severity of dystonia was found to be correlated with the parameters of the multifractal spectrum (rho: –0.5630882, p=0.005149 for GPe and rho: –0.5180609, p=0.01133 for GPi). Moreover, as the dystonia severity increased along the BFMDRS scale, the multifractal spectrum became narrower and more asymmetric. In post-surgery recordings, we observed changes in the shape of the multifractal spectrum during DBS stimulation or vibration. In particular, when exposed, the multifractal spectrum significantly widened, and its symmetry was restored. We identified the region with the greatest alteration in the width of the multifractal spectrum following DBS and discovered that this region considerably intersected or fully matched the neurologist’s chosen zone for optimal DBS stimulation. We demonstrated that multifractal analysis can serve as an ancillary approach to evaluate the extent of proprioceptive feedback impairment and as a biomarker of dystonia. The noteworthy and substantial correlation with dystonia severity underscores the considerable potential of multifractal spectrum features as a biomarker of dystonia. Additionally, the overlapping areas with the most significant DBS impact and the widest variation in the multifractal spectrum in DBS can be used as a tool for identifying the most favorable region for DBS stimulation.

Extreme Risk Measurement of Carbon Market Considering Multifractal Characteristics

Extreme Risk Measurement of Carbon Market Considering Multifractal Characteristics

Fractal modeling of oil and gas geochemical data in the Taiwan Strait Basin

This study investigated the fractal characteristics of oil and gas indicators in the Taiwan Strait Basin using multifractal methods. The results revealed that these indicators can be categorized into five groups through cluster analysis, principal component analysis, and multifractal analysis. Acidolysis hydrocarbon indicators exhibited strong multifractal characteristics, while other indicators showed weak multifractal characteristics. The high values of Polycyclic aromatic hydrocarbons (F) indicators were mainly found in the southwest direction of the study area, whereas the high values of total aromatic hydrocarbons and their derivatives (UV) indicators were distributed in the eastern part. Erosion carbonate (ΔC) indicators were predominantly located in the western part, while the distribution of acid extractable hydrocarbons (A) and pyrolytic hydrocarbons (P) indicators was sporadic. The distribution of Erosion carbonate (ΔC) indicators in the Jinjiang Depression resembled that in the Jiulongjiang Depression, but the distribution of other indicators varied. The S-A method was employed to identify oil and gas anomaly target areas and predict and evaluate favorable prospect zones in the basin. This study provides a theoretical foundation for future oil and gas exploration and evaluation in the Taiwan Strait.

Groundwater level complexity analysis based on multifractal characteristics: a case study in Baotu Spring Basin, China

Groundwater level complexity analysis based on multifractal characteristics: a case study in Baotu Spring Basin, China

Multifractal Characteristics of Smooth Blasting Overbreak in Extra-Long Hard Rock Tunnel

With the development of infrastructure construction in mountainous areas, the number of new extra-long tunnels is increasing. However, these tunnels often face the challenge of complex and variable surrounding rock grades, resulting in a large number of overbreak and underbreak due to the untimely adjustment of smooth blasting parameters. This study focuses on the optimization of the peripheral hole charging structure and blasting parameters for extra-long hard rock tunnels, aiming to improve the effectiveness of smooth blasting technology. The results of this study demonstrate a significant improvement in the effect of smooth blasting after implementing bidirectional polymerization blasting in the tunnel. A comparison between the bidirectional shaped charge and spaced decoupled charge blasting reveals that the former yields better results. To obtain accurate data on the tunnel section profile during excavation, a laser cross-section meter is used for measurement. Furthermore, this study quantitatively compares the optimization effect of smooth blasting parameters. The multifractal characteristics of the tunnel profile overbreak point sequences are analyzed under different smooth blasting schemes using the multifractal detrended fluctuation analysis (MF-DFA) method. It is found that both the spaced decoupled charge and the bidirectional shaped charge blasting exhibit multifractal features in the overbreak measurement point sequences. The calculation results of the multifractal features of the tunnel profile under different smooth blasting plans are in line with the actual situation.

End-to-end programmable computing systems

Recent technological advances have contributed to the rapid increase in algorithmic complexity of applications, ranging from signal processing to autonomous systems. To control this complexity and endow heterogeneous computing systems with autonomous programming and optimization capabilities, we propose a unified, end-to-end, programmable graph representation learning (PGL) framework that mines the complexity of high-level programs down to low-level virtual machine intermediate representation, extracts specific computational patterns, and predicts which code segments run best on a core in heterogeneous hardware. PGL extracts multifractal features from code graphs and exploits graph representation learning strategies for automatic parallelization and correct assignment to heterogeneous processors. The comprehensive evaluation of PGL on existing and emerging complex software demonstrates a 6.42x and 2.02x speedup compared to thread-based execution and state-of-the-art techniques, respectively. Our PGL framework leads to higher processing efficiency, which is crucial for future AI and high-performance computing applications such as autonomous vehicles and machine vision.

Time-Frequency Response of Acoustic Emission and Its Multi-Fractal Analysis for Rocks with Different Brittleness under Uniaxial Compression

The occurrence of rock burst hazards is closely related to the brittleness of rocks. Current research has paid less attention to the in-depth relationship between rock brittleness and acoustic emission (AE) signal characteristics and precursor information caused by rock fracture. Therefore, in order to further improve the accuracy of the AE monitoring of rockburst hazards, uniaxial compression tests were carried out and AE were monitored for rocks with different brittleness (yellow sandstone, white sandstone, marble, and limestone) in this paper. The relationship between the mechanical properties and the time-frequency characteristics of the AE was analyzed. In addition, the multi-fractal theory was introduced to further deconstruct and mine the AE signals, and the multi-fractal characteristics of AE from rocks with different brittleness were investigated. The results show that the stronger the brittleness of the rock, the higher the main frequency and main frequency amplitude of the AE. Brittleness is positively correlated with the multi-fractal parameter Δα (uniformity of data distribution) and negatively correlated with Δf (frequency difference between large and small data). In addition, the dynamics of Δα and Δf provide new indicators for AE monitoring of rock stability, and their abrupt changes can be regarded as precursors of failure. The weaker the brittleness of the rock, the earlier the failure precursor is and the more significant it is. This has potential engineering application value, which can help identify rockburst precursors and take timely protective measures to ensure engineering safety.

Dynamics of green and conventional bond markets: Evidence from the generalized chaos analysis

In this study, we conduct a comparative analysis of the nonlinear dynamics of the time series data for green and conventional bond indices spanning the period from 2014 to 2023. Our research builds upon the existing limited studies conducted in this domain. To examine the inherent characteristics of the empirical data-generating process, we utilize a generalized chaos analysis framework. The findings indicate that both series exhibit a combination of quasi-periodic cycles and deterministic (chaotic) dynamics. Additionally, we employ reconstruction techniques to visualize the strange (fractal) system attractors for both series. Furthermore, we quantify and compare the dynamics that emerge from the data by employing recurrence quantification analysis and multifractal analysis with power-law distributional coherence tests. Consequently, we identify that both bond series exhibit multifractal characteristics. Finally, we conduct a comparative analysis of the distinct dynamical differences observed between the bond index series. Based on our results, we draw concluding remarks regarding the future evolution of the green bond market and discuss the potential implications of our findings in the context of climate change.

Expecting the Unexpected: Entropy and Multifractal Systems in Finance.

Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point's position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor's point of view, a rise in entropy is a signal of abnormal and possibly negative returns. This means he has to expect the unexpected and prepare for it. To explore this, we analyse the New York Stock Exchange (NYSE) U.S. Index as well as its constituents. Through this examination, we assess their multifractal characteristics and identify market conditions (bearish/bullish markets) using entropy, an effective method for recognizing fluctuating fractal markets. Our findings challenge conventional beliefs by demonstrating that price declines lead to increased entropy, contrary to some studies in the literature that suggest that reduced entropy in market crises implies more determinism. Instead, we propose that bear markets are likely to exhibit higher entropy, indicating a greater chance of unexpected extreme events. Moreover, our study reveals a power-law behaviour and indicates the absence of variance.

Multifractal long-range dependence pattern of functional magnetic resonance imaging in the human brain at rest.

Long-range dependence is a prevalent phenomenon in various biological systems that characterizes the long-memory effect of temporal fluctuations. While recent research suggests that functional magnetic resonance imaging signal has fractal property, it remains unknown about the multifractal long-range dependence pattern of resting-state functional magnetic resonance imaging signals. The current study adopted the multifractal detrended fluctuation analysis on highly sampled resting-state functional magnetic resonance imaging scans to investigate long-range dependence profile associated with the whole-brain voxels as specific functional networks. Our findings revealed the long-range dependence's multifractal properties. Moreover, long-term persistent fluctuations are found for all stations with stronger persistency in whole-brain regions. Subsets with large fluctuations contribute more to the multifractal spectrum in the whole brain. Additionally, we found that the preprocessing with band-pass filtering provided significantly higher reliability for estimating long-range dependence. Our validation analysis confirmed that the optimal pipeline of long-range dependence analysis should include band-pass filtering and removal of daily temporal dependence. Furthermore, multifractal long-range dependence characteristics in healthy control and schizophrenia are different significantly. This work has provided an analytical pipeline for the multifractal long-range dependence in the resting-state functional magnetic resonance imaging signal. The findings suggest differential long-memory effects in the intrinsic functional networks, which may offer a neural marker finding for understanding brain function and pathology.

Nanopore Structure and Multifractal Characteristics of Continental Shale Oil Reservoir: A Case Study from Ziliujing Shales in the Sichuan Basin

Thermal maturity of the shales from the Ziliujing Formation of the Jurassic age in the Sichuan Basin is in the hydrocarbon generation window, which makes it a candidate for shale oil and gas development. The meso- and macropore characteristics and heterogeneity of shales are important factors affecting the occurrence and development of oil and gas. However, the meso- and macropores of the Ziliujing shales have not been systematically studied. Thus, the mineral compositions and total organic carbon (TOC) of samples from this formation, as well as its pore structure, are analyzed by low-temperature N2 adsorption technique. Moreover, the heterogeneity of the pores was determined by multifractal analysis. The results show that the Ziliujing shales can be classified into three types according to the distributions of mineral compositions of carbonate and mixed and argillaceous shales. Results revealed that the smallest meso- and macropore volume (PV), the smallest specific surface area (SSA), and the largest average pore diameter (APD) occur in the carbonate shales. However, the largest PV and SSA and the smallest APD are observed in the argillaceous shales. The porosity of carbonate shales is mainly concentrated between 5 nm and 30 nm. Compared with carbonate shales, the porosity with pore sizes less than 30 nm of mixed and argillaceous shales shows a rapid increase. Furthermore, inorganic minerals are the main factors affecting the pore distributions, while TOC shows a weak effect. Herein, clay minerals significantly increase the mesopore volume and the pore number with a size of less than 30 nm. The Dq-q curves reveal that the meso- and macropore distributions of Ziliujing shales show multifractal behavior, but the multifractal characteristics of pores of various shales are distinctly different. The information dimension D1, the Hurst exponent H, and the width of the right side D0–D10 are key indicators to distinguish the local variations within the pore structure of different types of shales. The carbonate shales have the largest multifractal spectra width and the smallest D1 and H, while the opposite trend is found for the argillaceous shales. Clay minerals reduce the heterogeneity of the meso- and macropore distributions and increase the pore connectivity. Nevertheless, the carbonate minerals exhibit a reverse trend. Finally, it was found that TOC does not impact pore complexity as much. Collectively, this study supports our understanding of the occurrence of shale oil within various reservoir facies, thereby providing a guideline for future explorations in the Ziliujing Formation of the Jurassic age in the Sichuan Basin.

A Joint Multifractal Approach to Solar Wind Turbulence

Previous studies have shown that solar wind, a plasma medium with turbulent dynamics, exhibits anomalous scaling features, i.e., intermittency, in the inertial domain. This intermittent nature has primarily been investigated through the study of the scaling features of the structure functions of single quantities. We use a novel approach based on joint multifractal analysis in this study to simultaneously investigate the scaling characteristics of both the magnetic field and the plasma velocity in solar wind turbulence. Specifically, we focus on the joint multifractal behavior of magnetic and velocity field fluctuations in both fast and slow solar wind streams observed by the ESA-Ulysses satellite, with the goal of identifying any differences in their joint multifractal characteristics.

Multifractal approach for a biological porous media: Human dentin case.

A multifractal characterization to human dentin porosity is made. Micrographs of human dentin samples gotten from scanning electron microscopy (SEM) were studied in order to characterize porosity. We got the generalized dimensions (multifractal moments) for pore space and matrix skeleton on gray scale and binary images for samples of both gender. For the case, we found that superficial porosity is linked to mass fractal dimension in an approximately linear relationship as for binary images. In addition, probability density distribution (PDD) for pore diameters is found a normal PDD type. Other quantities of interest like mean, standard deviation, maximum and minimum pore diameters, voit ratios, and percentage of chemical composition are reported. RESEARCH HIGHLIGHTS: SEM sample, images and procedure for multifractal analysis are detailed. Micro patterns in grayscale are multifractal and in binary scale are monofractals. Superficial porosity is relate to mass fractal dimension approximately linearly. From the porosity values, voit ratios are determined. Pores diameters obey a normal PDD.

Study of pore-throat structure characteristics and fluid mobility of Chang 7 tight sandstone reservoir in Jiyuan area, Ordos Basin

Abstract Quantitative studies of the pore-throat structure (PTS) characteristics of tight sandstone reservoirs and their effects on fluid mobility were proposed to accurately evaluate reservoir quality and predict sweet spots for tight oil exploration. This study conducted high-pressure mercury injection (HPMI) and nuclear magnetic resonance (NMR) experiments on 14 tight sandstone samples from the Chang 7 member of the Yanchang Formation in the Jiyuan area of the Ordos Basin. The HPMI was combined with the piecewise fitting method to transform the NMR movable fluid transverse relaxation time (T 2) spectrum and quantitatively characterize the PTS characteristics and the full pore-throat size distribution (PSD). Then, movable fluid effective porosity (MFEP) was proposed to quantitatively evaluate the fluid mobility of tight sandstone reservoirs and systematically elucidate its main controlling factors. The results showed that the PTS could be divided into four types (I, II, III, and IV), which showed gradual decreases in average pore-throat radius (R a), continuous increases in the total fractal dimension (D t), and successive deterioration of reservoir fluid mobility and percolation capacity. Moreover, the full PSD (0.001–10 μm) showed unimodal and multi-fractal characteristics. According to the Swanson parameter (r apex), the reservoir space types can be divided into small and large pore-throat and the corresponding fractal dimension has a relationship where D 1 < D 2. Large pore-throat had higher permeability contribution and pore-throat heterogeneity but a lower development degree and MFEP than small pore-throat, which had a relatively uniform and regular PSD and represented the primary location of movable fluids. Moreover, the development degree and heterogeneity of small pore throat controlled the flowability of reservoir fluids. MFEP can overcome the constraints of tiny throats and clay minerals on movable fluid, quantify the movable fluid content occupying the effective reservoir space, and accurately evaluate the reservoir fluid mobility. The combination and development of various pore-throat sizes and types in tight sandstone reservoirs results in different PTS characteristics, whereas differences in the mineral composition and content of reservoirs aggravate PTS heterogeneity, which is the main factor controlling the fluid mobility.

Multifractal Characteristics of Uranium Grade Distribution and Spatial Regularities in a Sandstone-Type Uranium Deposit in Xinjiang, China

Based on an analysis of the spatial distribution of uranium grade in 338 boreholes of a uranium deposit in Xinjiang, the enrichment and spatial variation of uranium ore in two stopes of the deposit are discussed using multifractal theory. The distribution characteristics of the uranium ore of the two stopes are studied by multifractal parameters: the scaling exponent of mass τ(q), the scaling exponent α(q) of each sub-set and its corresponding fractal dimension f(α), the fractal dimension D0 and information dimension D1. The differences of uranium distribution in the two stopes can be quantified well by using multifractal spectrum and multifractal parameters such as Δα, Δf and R. After a comprehensive multifractal distribution analysis, 10 m × 10 m is defined as a fence unit, and the window sizes ε=3,6,9⋯,45 are set; the singularity exponents α of the two stopes are calculated by using this element concentration–area method. The results show that the multifractal theory and model can organically combine spatial structure information, scale change information and anisotropy information to obtain low-grade and weak mineral resources information and can effectively distinguish complex and superimposed anomalies. This will provide a basis for the local concentration and spatial variation rules of uranium distribution and the design of the parameters of the leaching uranium mining well site.

Study on the evolution of microscopic pore structure of sandstone under freeze-thaw cycles

In cold regions, rocks undergo freeze-thaw (FT) cycles, leading to damage caused by the evolution of pores. This phenomenon significantly influences the rock mass stability. A comprehensive and accurate study of FT damage evolution in rock pores was conducted in this research. The microscopic morphology and pore size distribution (PSD) of sandstone samples subjected to FT cycles were measured utilizing scanning electron microscopy (SEM), nitrogen adsorption/desorption (NAD) method, and mercury intrusion porosimetry (MIP). Multifractal analysis was applied to explore the evolution of PSD. A gradual weakening of the cementation degree of mineral particles under FT cycles was observed based on SEM results, resulting in the formation of larger connected pores. A decrease of 37.3% in the number of pores was recorded after 40 cycles. According to NAD and MIP results, a greater effect on mesopores and large pores were found, with a minimal impact on micropores and transition pores. Multifractal characteristics were identified in the PSD of sandstone samples with different FT cycles. Initially, the connectivity of sandstone pores was found to decrease and then increase gradually after 10 cycles. Uneven PSD was observed to concentrate stress among pores, exacerbate frost heave damage, and drive the PSD towards a more uniform direction. It was concluded that the non-uniform degree of PSD does not consistently increase with FT cycles. The findings of this study can serve as a reference for further research on rock FT damage mechanisms.

Fractal Analysis of S&P 500 Sector Indexes

In this study multifractal properties of S&P 500 sector indexes are investigated with Multifractal Detrended Fluctuation Analysis (MF-DFA). The MF-DFA is a signal processing technique that is used to describe the multifractal properties of a time series data. It is an extension of Detrended Fluctuation Analysis (DFA), which is a widely utilized method for estimating the scaling behavior of a time series. Main idea behind MF-DFA is to decompose a time series into multiple scales using a coarse-graining procedure, and then to estimate the scaling behavior of each scale using DFA. This gives a set of scaling exponents that describe the multifractal features of the time series. Our MF-DFA results indicates the presence of multifractality in all S&P 500 sector indexes. Since these indexes are multifractal, we can conclude that they possess properties such as scaling variability, nonlinear dynamics, self-similarity, long-range dependence, multiscale correlations and nonstationary.

Multi-objective optimization of surface morphology using fractal and multi-fractal analysis for dry milling of AISI 4340

Machining of difficult-to-cut metals is typically associated with inconsistent machining quality, high machining costs, and significant environmental pollution. This work proposes a novel method that utilizes fractal and multi-fractal features of the machined surface morphology to evaluate surface quality in high-speed dry milling of 4340 steel. The results demonstrate that the fractal dimension can be used as an evaluation index for machined surface defects. It accurately characterizes the contour information of machined surface defects, including voids, tears, and material side flow. A smaller fractal dimension indicates fewer surface defects and represents a higher-quality machined surface. However, the surface defects cannot be described using the traditional evaluation method based solely on surface roughness (Ra). Moreover, multi-objective optimization of high-speed milling of 4340 steel is achieved by using the combined indices Ra and fractal features as responses. The Ra generated using the optimized parameters can reach 97 nm, which represents a reduction of 4.9 % compared to the non-optimization state. And typical defects on machined surfaces virtually disappear. Meanwhile, the machining cost has been reduced by 2.52 % and the machining efficiency has increased by 31.45 %. The results demonstrate that the proposed surface quality evaluation method is possible for practical engineering applications to achieve high-quality, high-efficiency and low-cost high-speed dry milling of high-strength steels.