The graph-based random walk model of fractal diffusion is limited in its use to the connected sets and does not allow for direct fractal dimension estimation based on observed data. We discuss a task ...of directly obtaining accurate fractal dimension estimates and propose butterfly diffusion as an alternative approach using an explicit relation between walk and fractal dimensions. The validity of the presented approach is evaluated and statistical properties of the dimension estimates are presented. Through experiments on self-similar sets, we demonstrate the effectiveness of this approach in producing unbiased dimension estimates, offering a promising tool for fractal analysis and Monte Carlo simulations. The estimate of fractal dimension can be also created from spectral dimension, but this approach is less general and less accurate.
•Butterfly diffusion as a resampled random walk.•Introduced simple relation between walk and fractal dimensions.•Novel data-driven dimension estimation procedure.•Statistical verification using Monte Carlo simulations on self-similar structures.
Over the past 40 years, from its classical application in the characterization of geometrical objects, fractal analysis has been progressively applied to study time series in several different ...disciplines. In neuroscience, starting from identifying the fractal properties of neuronal and brain architecture, attention has shifted to evaluating brain signals in the time domain. Classical linear methods applied to analyzing neurophysiological signals can lead to classifying irregular components as noise, with a potential loss of information. Thus, characterizing fractal properties, namely, self-similarity, scale invariance, and fractal dimension (FD), can provide relevant information on these signals in physiological and pathological conditions. Several methods have been proposed to estimate the fractal properties of these neurophysiological signals. However, the effects of signal characteristics (e.g., its stationarity) and other signal parameters, such as sampling frequency, amplitude, and noise level, have partially been tested. In this chapter, we first outline the main properties of fractals in the domain of space (fractal geometry) and time (fractal time series). Then, after providing an overview of the available methods to estimate the FD, we test them on synthetic time series (STS) with different sampling frequencies, signal amplitudes, and noise levels. Finally, we describe and discuss the performances of each method and the effect of signal parameters on the accuracy of FD estimation.
The fractal dimension (FD) is a classical nonlinear dynamic index that can effectively reflect the dynamic transformation of a signal. However, FD can only reflect signal information of a single ...scale in the whole frequency band. To solve this problem, we combine refined composite multi-scale processing with FD and propose the refined composite multi-scale FD (RCMFD), which can reflect the information of signals at a multi-scale. Furthermore, hierarchical RCMFD (HRCMFD) is proposed by introducing hierarchical analysis, which successfully represents the multi-scale information of signals in each sub-frequency band. Moreover, two ship-radiated noise (SRN) multi-feature extraction methods based on RCMFD and HRCMFD are proposed. The simulation results indicate that RCMFD and HRCMFD can effectively discriminate different simulated signals. The experimental results show that the proposed two-feature extraction methods are more effective for distinguishing six types of SRN than other feature-extraction methods. The HRCMFD-based multi-feature extraction method has the best performance, and the recognition rate reaches 99.7% under the combination of five features.
In cold regions, hydraulic concrete structures are exposed to harsh environmental conditions that greatly impact their durability and safety. The role of aggregates in influencing concrete ...performance is paramount. This study aims to investigate how the fractal distribution and size of aggregates affect the frost resistance of hydraulic concrete, with air void characteristics, mass loss rate, Elastic Modulus of dynamic, and strength serving as primary references. Two experiment series, fractal distribution influence and aggregate size influence of concrete frost resistance, were conducted to analyze the frost resistance of hydraulic concrete with different aggregates. Based on the fractal theory, four grading fractal distributions were utilized to fractalize aggregate gradation, aiming to investigate the influence of aggregate fractal dimension on the frost resistance of hydraulic concrete. Simultaneously, concrete specimens of fully graded aggregate and wet-screened aggregate were also constructed to simultaneous freeze-thaw cycle tests, the impact of aggregate size on concrete's frost resistance performance was also studied by analyzing the air void structure in the frost-resistant concrete specimens. Results indicate that the specimens exhibit varying frost resistance performance under different aggregate gradation fractals, ranking from strongest to weakest as follows: D=2.5, D=2.3, D=2.1, D=2.7, which is attributed to the differences of air void spacing factors. Increasing aggregate size leads to greater internal structural heterogeneity in concrete, resulting in an increase in defects and weak points. This causes microcracks to develop more rapidly in the transition zone of fully graded concrete during freeze-thaw cycles, leading to more severe damage. The findings of this study can serve as a fundamental scientific basis for the designing and constructing of hydraulic concrete structures in cold areas.
•Frost resistance of hydraulic concrete declines in order of fractal dimensions: D=2.5, D=2.3, D=2.1, D=2.7.•The wet-screened concrete with smaller coarse aggregate sizes demonstrates superior frost resistance to fully graded concrete.•The varying frost resistance performance of concrete specimens is attributed to differences in the air void structures.
•A hybrid intelligent approach for forecasting COVID-19 time series combining fractal theory and fuzzy logic is presented.•The fractal dimension is used to measure the complexity of the dynamics in ...the time series of the countries in the world.•Fuzzy Logic is used to represent the uncertainty in the process of making a forecast.•The hybrid approach consists on a fuzzy model formed by a set of fuzzy rules that uses as input values the linear and nonlinear fractal dimensions of the time series.•The outputs are the forecast for the countries based on the COVID-19 time series of confirmed cases and deaths.
We describe in this paper a hybrid intelligent approach for forecasting COVID-19 time series combining fractal theory and fuzzy logic. The mathematical concept of the fractal dimension is used to measure the complexity of the dynamics in the time series of the countries in the world. Fuzzy Logic is used to represent the uncertainty in the process of making a forecast. The hybrid approach consists on a fuzzy model formed by a set of fuzzy rules that use as input values the linear and nonlinear fractal dimensions of the time series and as outputs the forecast for the countries based on the COVID-19 time series of confirmed cases and deaths. The main contribution is the proposed hybrid approach combining the fractal dimension and fuzzy logic for enabling an efficient and accurate forecasting of COVID-19 time series. Publicly available data sets of 10 countries in the world have been used to build the fuzzy model with time series in a fixed period. After that, other periods of time were used to verify the effectiveness of the proposed approach for the forecasted values of the 10 countries. Forecasting windows of 10 and 30 days ahead were used to test the proposed approach. Forecasting average accuracy is 98%, which can be considered good considering the complexity of the COVID problem. The proposed approach can help people in charge of decision making to fight the pandemic can use the information of a short window to decide immediate actions and also the longer window (like 30 days) can be beneficial in long term decisions.
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•Irregular magnetite is broken from initial size to target size using drop hammer.•Multistage crushing test model based on fractal theory is first proposed.•The change rule of unit ...energy with fractal dimension and product size is revealed.•The model between unit energy, product size and fractal dimension is constructed.
To study the theoretical relationship between the ore fragmentation energy and the particle size, the multistage crushing test model of magnetite was constructed with the fractal geometry to simulate the general process of magnetite crushing from the initial particle size to the target particle size, and the theoretical relationship between the fragmentation energy and fragment size and fractal dimension was established. Results show that the multistage crushing of magnetite has fractal characteristics, the rate of particle size reduction in the multistage crushing process decreases, and the fractal dimension increases gradually to be a stable value. The cumulative unit fragmentation energy is affected by the fragment size and fractal dimension, which is defined as the ratio of the cumulative fragmentation energy to ore mass. With the increase in the median particle size of the fragments (d50), the cumulative unit fragmentation energy decreases in the form of exponential function. With the increase in the fractal dimension of the fragment size, the logarithm of the cumulative unit fragmentation energy increases linearly. The smaller the fragment size is, the faster the cumulative unit fragmentation energy changed with the fractal dimension. The larger the fractal dimension of the fragments is, the more significant change in the cumulative unit fragmentation energy with the fragment size is. The mathematical relationship between cumulative unit fragmentation energy, fragment particle size, and fractal dimension is determined.
Owing to the gradual depletion of easily accessible mineable resources, deep mining has now become standard practice. In deep shale gas reservoirs, a substantial level of the total shale gas resource ...is obtained from adsorbed methane. Moreover, methane adsorption in deep shale gas reserves has been defined as supercritical adsorption. The effect of high temperature and high pressure on the adsorption capacity of methane has become more critical. Shale possesses an unmistakably multi-scale pore structure, whose adsorption mechanisms differ in pores of different sizes. Therefore, it was considered necessary to understand precisely the supercritical adsorption behavior of shale gas under different adsorption mechanisms. To this end, a supercritical weighted adsorption model for shale gas, that analyzed micropore-filling with a mesoporous monolayer surface was developed. Based on this model, this research analyzed the adsorption and thermodynamic characteristics of shale gas at different temperature and pressure levels. The effects of the weighting factors on shale’s adsorption properties, pore fractal dimensions and thermodynamic parameters have also been discussed. The study’s results revealed a significant negative correlation between methane adsorption and temperature. Furthermore, there was a consistent increase in the isosteric heat of adsorption and a corresponding decrease in the standard entropy of adsorption under increasing adsorption. Absolute adsorption tended to increase following an increased weighting factor. Upon an analysis of shale's pore structure characteristics, a positive correlation between the fractal dimension and the weighting factor was discovered. With regard to the thermodynamic parameters, the isosteric heat of adsorption rose as the weighting factor increased, while the standard entropy of adsorption tended to decrease. The research findings have established a theoretical basis for evaluating deep shale gas reserves.
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•A supercritical weighted adsorption model for shale gas considering different adsorption mechanisms was constructed.•The key role of weighting factors in determining the adsorption properties and fractal dimension, among others, is revealed.•The supercritical adsorption evolution of shale gas under different adsorption mechanisms and temperatures is described.
The fractal property is one of the most important properties in complex networks. It describes the power law relationship between characteristics of the box and the box size. There are numerous ...research studies focusing on the fractal property in networks through different dimensions. In order to study the problems across various disciplines, fractal dimension and local dimension are proposed to study network and node properties respectively. In this review paper, various network covering algorithms, which form the basis for obtaining fractal dimension are being reviewed. The different dimensions used to describe the fractal property of networks and their applications are then discussed. Through these studies, we emphasize that the fractal property is an important tool for understanding network characteristics. In the last section, we give our conclusion and discuss possible future directions for fractal dimension research.
•Network covering algorithms with different features are discussed.•Fractal dimension which describes the fractal property of network is surveyed.•The local dimensions are also systematically surveyed.•Experiments involving real-world problems are given.
•Three fractal parameters (dimension, lacunarity and succolarity) were employed to characterize the microstructure of reservoir rocks.•Twelve 3D CT images of sandstone reservoir rocks with different ...permeabilities were used to estimate the three fractal parameters.•The fractal parameters, especially succolarity, identified differences in core micro-structure as related to differences in permeability.
An explicit relationship between seepage properties and microscopic structure of porous media is being sought for the development of oil and gas resources. An effective method to accurately characterize and quantify the microscopic structure of porous media is a key issue. Fractal geometry can take advantage of several physically-based parameters to analyze microstructures of reservoir rocks. In this study, three fractal structural parameters, fractal dimension, lacunarity and succolarity, were employed to characterize scale-invariant complexity, heterogeneity, and anisotropy of rock microstructures, respectively. Twelve three-dimensional digital cores of sandstone reservoir rocks were used to evaluate permeability in terms of fractal dimension, lacunarity and succolarity. The parameters were utilized to quantitatively characterize differences in core micro-structure and predict their effects on permeability. Due to the confounding influence of porosity, the fractal dimension was not an accurate predictor of the variation in permeability on its own. Instead, the results reveal that lacunarity and succolarity were better able to predict differences in structure and permeability. Succolarity, in particular, showed an exponential relationship with permeability, yielding a coefficient of determination of 0.940. Using a combination of fractal structural parameters in the place of pore-size distribution, can provide a better explanation of the relationship between fluid flow, and the heterogeneous structure and anisotropic physical properties of reservoir rocks.
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