Multifractal Model Research Articles

Fractal Geometry in High-Frequency Trading: Modeling Market Microstructure and Price Dynamics

This theoretical article delves into the intricate world of high-frequency trading (HFT) without empirical testing of real-world data, focusing on the incorporation of fractal geometry principles to enhance our understanding of market microstructure and price dynamics. In the introduction, we outline the significance of this research in the context of modern financial markets and lay out the objectives of our theoretical analysis. The article then takes an in-depth dive into fractal geometry fundamentals, illuminating its core concepts and its relevance within financial markets. Subsequently, the article explores the landscape of high-frequency trading, offering an overview of this dynamic domain and how fractal geometry can be incorporated into trading models. The section on modeling market microstructure presents theoretical approaches to understanding order flow dynamics, including novel derivations and equations. It then transitions into fractal-based approaches for analyzing the complexities of market microstructure, providing both an original perspective and numbered equations. Moreover, this article investigates the theoretical modeling of price dynamics, underscoring the pivotal role of fractal geometry in enriching these models. The discussion revolves around the fundamental autoregressive models and multifractal models, and it elucidates how fractal geometry principles, such as the Hurst exponent, come into play. We explore the self- similarity of price dynamics, fractal dimensions, and how these aspects can be integrated into high-frequency trading strategies. Overall, this article offers a comprehensive theoretical exploration of fractal geometry's implications in the realm of high-frequency trading, providing valuable insights for both researchers and practitioners seeking to fathom the complexities of market microstructure and price dynamics. The incorporation of fractal principles into financial models fosters a deeper understanding of self-similarity and complexity within financial markets, even in the absence of empirical data.

Heterogeneity of Pore and Fracture Structure in Coal Reservoirs by Using High-Pressure Mercury Intrusion and Removal Curve

The pore structure determines the desorption, diffusion and migration of coalbed methane, and the heterogeneity of the pore structure seriously restricts the diffusion and seepage process and productivity of coalbed methane. Therefore, this paper takes eight coal samples in the Linxing area as the research target and uses the high-pressure mercury injection test to describe the pore structure distribution. On this basis, three kinds of single and multifractal models are used to calculate the progressive mercury removal curve, and the correlation analysis is carried out to determine the physical significance of the mercury removal fractal dimension. Finally, the relationship between the fractal dimension of the mercury curve and the pore structure parameters is defined, and the applicability of fractal models in characterizing pore structure heterogeneity is discussed. The conclusions of this paper are as follows. (1) Samples can be divided into two categories according to porosity and mercury removal efficiency. Among them, the mercury removal efficiency of sample 1–3 is higher than 35%, and porosity is less than 9.5%, while those of sample 4–8 are the opposite. The seepage pore volume percentage of sample 1–3 is 35–60%, which is higher than that in sample 4–8. (2) The difference of the samples’ fractal dimension calculated with the Menger and Sierpinski models is small, indicating that the pore structure distribution heterogeneity of the two types is similar. The multifractal model shows that the adsorption pore and macro-pore heterogeneity of sample 4–8 are stronger than those of sample 1–3, and the pore distribution heterogeneity is controlled by the low value of pore volume. (3) The results of the two single fractal calculations show that the pore structure distribution heterogeneity of sample 4–8 is stronger than that of sample 1–3. The multifractal model calculation shows that the adsorption pore distribution heterogeneity of sample 4–8 is stronger, and the low value of pore volume controls the pore distribution heterogeneity. (4) The mercury fractals based on the Menger model can reflect the adsorption pore distribution and macro-pore distribution heterogeneity, while the Sierpinski model can only reflect the adsorption pore distribution heterogeneity at the mercury inlet stage.

The Effect of Thermal Maturity on the Pore Structure Heterogeneity of Xiamaling Shale by Multifractal Analysis Theory: A Case from Pyrolysis Simulation Experiments

Shale oil and gas, as source-reservoir-type resources, result from organic matter hydrocarbon generation, diagenesis, and nanoscale pore during the evolution processes, which are essential aspects of shale gas enrichment and reservoir formation. To investigate the impact of diagenetic hydrocarbons on shale pore heterogeneity, a thermal simulation of hydrocarbon formation was conducted on immature shale from the Middle Proterozoic Xiamaling Formation in the Zhangjiakou area, covering stages from mature to overmature. Nuclear magnetic resonance (NMR) instruments analyzed the microstructure of the thermally simulated samples, and the multifractal model quantitatively assessed pore development and heterogeneity in the experimental samples. The results reveal that the quartz and clay mineral contents show alternating trends with increasing temperature. Organic matter dissolution intensifies while unstable mineral content decreases, promoting clay mineral content development. Pyrolysis intensity influences Total Organic Carbon (TOC), which reduces as hydrocarbons are generated and released during simulation. Porosity exhibits a decreasing–increasing–decreasing trend during thermal evolution, peaking at high maturity. At maturity, hydrocarbon generation obstructs pore space, resulting in higher levels of bound fluid porosity than those of movable fluid porosity. Conversely, high maturity leads to many organic matter micropores, elevating movable fluid porosity and facilitating seepage. Shale pore heterogeneity significantly increases before 450 °C due to the dissolution of pores and the generation of liquid and gas hydrocarbons. In the highly overmature stage, pore heterogeneity tends to increase slowly, correlated with the generation of numerous micro- and nano-organic matter pores.

Prediction of Au-Associated Minerals in Eastern Thailand Based on Stream Sediment Geochemical Data Analysis by S-A Multifractal Model

Conducted within the scope of geochemical exploration in eastern Thailand, this study aims to detect geochemical anomalies and potential mineral deposits. The objective was to interpret intricate spatial dispersion patterns and concentration levels of deposit pathfinder elements, specifically arsenic (As), copper (Cu), and zinc (Zn), using a comprehensive array of stream sediment geochemistry data. Methodologies involved integrating multifractal properties and traditional statistics, facilitated by the GeoDAS and ArcGIS platforms as instrumental analytical tools. In total, 5376 stream sediment samples were collected and evaluated, leading to the development of an in-depth geochemical map. The results indicated distinct geological units marked by substantially elevated average values of the aforementioned elements. Identification of geochemical anomalies was achieved through the spatial distribution method and the subsequent application of the spectrum-area (S-A) multifractal model. An intriguing link was found between high As concentrations and gold deposits in the area, suggesting As as a viable pathfinder element for gold mineralization. The anomaly maps, generated from the stream sediment data, spotlighted potential zones of interest, offering valuable guidance for future mineral exploration and geological inquiries. Nonetheless, it is vital to recognize that the increased values noted in these maps may be influenced by regional geological factors, emphasizing the necessity for a diverse set of analytical methods for accurate interpretation. This study’s significance lies in its pioneering use of the S-A multifractal model in geochemical data analysis. This innovative approach has deepened our comprehension of geochemical dispersion patterns and improved the precision of mineral exploration.

Применение теории катастроф для математического моделирования оползневого процесса на вогнутых склонах горных территорий

Introduction. Analysis of existing theoretical and experimental studies has shown that the model approach is the main method of landslide research. The existing mathematical model of landslides does not meet the requirements of the necessary adequacy. Materials and methods. This study uses a mathematical modeling methodology based on catastrophe theory. Results and discussion. To solve this actual problem, in this article authors developed a mathematical model of the landslide process on the concave slopes of mountainous territories. The developed model contains two components. The first of them is a mathematical model of the stress field in the volume of rocks located inside the slope section. This model uses the framework of fractal and multifractal modeling methods developed by the authors. The results of this model research are final expressions for calculating the stress field used rock pressure and bending stress as the external stress field. The superposition of the field induced by these external stresses gives the stress field in the volume of rocks located inside the slope section. Analysis of program implementation of this model showed that there are two areas in the slope section: compressive and tensile stresses adjacent to each other. At the boundary between these areas, there is a discontinuity of the stress field. A displacement surface passes along this boundary, forming a potentially landslide body. Moreover, it was found that a potentially landslide body on a slope is in a state of local and global instabilities. A potentially landslide body tends to occupy the position of the minimum potential energy. Local instability is expressed as the tendency of movement to a stable equilibrium without changing its location in the rock mass. The tendency of landslide body to move down the slope is a demonstration of global instability. The second mathematical model describes the realization of local instability that leads to the formation of a landslide body. Conclusion. According to the model analysis, it was found that the implementation of instability leads to the formation of a landslide body. At the same time, according to this analysis a landslide body can take up three stable equilibrium positions, allowing it to stay on the slope without global instability. Suggestions for practical application and direction of future research. The research results can be used to predict landslides on concave slopes of mountainous territories and to develop new mathematical models allowing to make the analysis of concave slopes of mountainous territories taking into account fracturing.

Size effect on tensile strength of interface between coarse aggregate and mortar under freeze-thaw cycle

Through the cleavage tensile test and freeze-thaw cycle experiment, the variation of tensile strength of mortar and interfacial bond-splitting tensile strength of coarse aggregate-mortar interface with different specimen sizes under freeze-thaw cycle was studied. The results indicate that the splitting tensile strength of the mortar specimen and the interfacial bond-splitting tensile strength of the coarse aggregate-interface specimen have a significant size effect, and the splitting tensile strength of the coarse aggregate-mortar interfacial specimen is significantly lower than that of the mortar specimen. At the same time, it is found that the statistical size effect model, the multifractal size effect model, and the fracture mechanics size effect model all can well describe the size effect of the split tensile strength of mortar specimens. Finally, considering the splitting tensile strength of standard mortar and the number of freeze-thaw cycles, a two-parameter size effect calculation model for the interfacial bond-splitting tensile strength of the coarse aggregate-mortar interface was established, basing on the Bažant-Xi statistical theory size effect correction formula, which was verified to be universal by experimental data.

On observational signatures of multi-fractional theory

We study the multi-fractional theory with q-derivatives, where the multi-fractional measure is considered to be in the time direction. The evolution of power spectra and also the expansion history of the universe are investigated in the q-derivatives theory. According to the matter power spectra diagrams, the structure growth would increase in the multi-fractional model, expressing incompatibility with low redshift measurements of large scale structures. Furthermore, concerning the diagrams of Hubble parameter evolution, there is a reduction in the value of Hubble constant which conflicts with local cosmological constraints. Thus, primary numerical investigations imply that q-derivatives theory has no potential to relieve observational tensions. We also explore the multi-fractional model with current observational data, principally Planck 2018, weak lensing, supernovae, baryon acoustic oscillations (BAO), and redshift-space distortions (RSD) measurements. Numerical analysis reveals that the degeneracy between multi-fractional parameters makes them remain unconstrained under observations. Furthermore, observational constraints on H0 and σ8, detect no significant departure from standard model of cosmology.

Driving forces of hydrological health and multifractal response of fish habitat in regulated rivers

Climate change and anthropogenic activities are major influences on the hydrological cycle, further altering river hydrological health. However, the characteristics of the forces in driving the variations of hydrological health at long-short time scales (annual, seasonal, monthly), as well as the potential impacts of these variations on aquatic habitats, remain unclear. In this study, the flow threshold method was introduced to identify the inherent characteristics of river hydrological health degree (RHD) evolution in the upper reaches of the Yangtze River (URYR) through the extreme-point symmetric modal decomposition (ESMD) method and range of variation approach (RVA). The RHD under unregulated conditions was reconstructed to quantify the impacts of anthropogenic activities and climate change. Subsequently, a multifractal model was proposed to establish the relationship between RHD and habitat-weighted usage area (WUA) during the spawning period of the Four Famous Major Carps, aiming to analyze the response mechanisms of habitat conditions to RHD fluctuations. The results showed that the RHD in the URYR exhibited degradation characteristics, experiencing a moderate change with a value of 0.44. Climate change was identified as the dominant factor causing the annual-scale decline in RHD, with an average impact weight of 62.9%. At the annual scale, Anthropogenic activities exacerbate (−3.4), counteract (20.1), and counteract (20.5) the adverse climatic impacts at Yichang, Cuntan, and Zhutuo stations, respectively. Additionally, the effect of human activities during the flood season is slight, with the most favorable and unfavorable impacts occurring in December (50.7) at the Zhutuo station and in October (−27.2) at the Yichang station. Under the influence of driving forces, the multifractal correlation of the RHD-WUA system tended to homogenized as the time window increased, indicating the presence of potential nonlinear dependence, asymmetric fractal characteristics, and positive-to-negative persistence transitions. Therefore, modeling river health considering fish habitat cannot be limited to linear paradigms. The findings provide valuable insights for the management and restoration of aquatic ecosystems.

A novel system for multivariate analysis of discontinuities in fractured rock masses based on manifold learning and fractal models

The engineering-geological characteristics of fractured rock masses depend on the discontinuity properties, which can be analyzed by the stereonet-based system. However, this system faces challenges in describing orientations, which are spherical data, and it cannot incorporate other scalar parameters (e.g., trace length, aperture, etc.) that affect engineering practice. Therefore, a novel system based on manifold learning is proposed to analyze the distribution patterns of orientation data and characterize multiple discontinuity properties. The proposed system uses manifold learning to explore and visualize the manifold structure of orientation data by a two-dimensional kernel density estimation (2D-KDE) technique. It also integrates other parameters into the manifold structure to capture multiple properties. To quantify the manifold structure and reveal the real fractal behaviors of orientation data, a detailed analysis process including monofractal and multifractal models is proposed. The effectiveness of the proposed system is verified by comparison with the stereonet-based system in several aspects. The equivalent manifold structure is also proposed to enhance the engineering applicability of the system, and one of its applications is demonstrated. Finally, the proposed system is applied to a significant road project and further developed by unmanned aerial vehicle (UAV) photogrammetry. The proposed system provides new insights for research on fractured rock masses and has great potential for solving multi-parameter problems in engineering geology.

Effect of litho-facies on nano-pore structure of continental shale in shuinan formation of Jiaolai Basin

Continental shale lithofacies has an important influence on nano-pore structure? However, nano-pore distribution heterogeneity (NDH) in continental shale lithofacies is not well documented. Continental shale was collected from the Shuinan Formation in the Jiaolai Basin, and the shale lithofacies of all samples are identified based on the mineralogical composition and TOC content. Nano-pore size distributions are characterized by LPN2 and CO2GA tests. Then, micro-pore (0.3∼2 nm) and meso-pore (2∼100 nm) distribution heterogeneity of different facies are discussed by single and multi-fractal model. Finally, the pore structure mechanism in shale lithofacies is clarified, and the results are as follows. The lithofacies of all shale samples are classified into three types, (A) calcareous shale lithofacies is characterized by the higher siliceous minerals and highest TOC content; (B) Calcareous/siliceous mixed shale lithofacies is characterized by middle TOC content; (C) Siliceous shale lithofacies is characterized by higher clay minerals and lowest TOC content. Micro- and meso-pore volume and A shale lithofacies NDH is much lower than that of B and C, which indicates that the lithofacies type has a great influence on the nano-pore structure. Felsic mineral and TOC content has a weak negative correlation with nano-pore structure, and there is a linear relationship between clay mineral content and nano-pore volume, indicating that clay mineral content is the main factor that can affect nano-pore structure. Moreover, fractal characteristics of the nitrogen desorption curve have been proved, and there are obvious differences in the fractal values of adsorption and desorption curves among those different lithofacies. Single fractal dimension D1 derived from LPN2 and CO2GA tests has a linear correlation with the multifractal parameter D-10-D0, which shows that a lower pore volume of pore diameter determines NDH variation. In conclusion, siliceous rich shale can provide a larger pore volume and specific surface area, and it acts as the dominant lithofacies for shale gas drainage in the study area.

Ergodic characterization of nonergodic anomalous diffusion processes

Anomalous diffusion in various complex systems abounds in nature and spans multiple space and time scales. Canonical characterization techniques that rely upon mean squared displacement break down for nonergodic processes, making it challenging to characterize anomalous diffusion from an individual time-series measurement. Nonergodicity reigns when the time-averaged mean square displacement differs from the ensemble-averaged mean squared displacement even in the limit of long measurement series. In these cases, the typical theoretical results for ensemble averages cannot be used to understand and interpret data acquired from time averages. The difficulty then lies in obtaining statistical descriptors of the measured diffusion process that are not nonergodic. We show that linear descriptors such as the standard deviation, coefficient of variation, and root mean square break ergodicity in proportion to nonergodicity in the diffusion process. In contrast, time series of descriptors addressing sequential structure and its potential nonlinearity: multifractality change in a time-independent way and fulfill the ergodic assumption, largely independent of the time series' nonergodicity. We show that these findings follow the multiplicative cascades underlying these diffusion processes. Adding fractal and multifractal descriptors to typical linear descriptors would improve the characterization of anomalous diffusion processes. Two particular points bear emphasis here. First, as an appropriate formalism for encoding the nonlinearity that might generate nonergodicity, multifractal modeling offers descriptors that can behave ergodically enough to meet the needs of linear modeling. Second, this capacity to describe nonergodic processes in ergodic terms offers the possibility that multifractal modeling could unify several disparate nonergodic diffusion processes into a common framework.

Characterization method of core pore structure based on truncated Gaussian and its application in shale cores

The core pore structure is mainly described by fractal theory. Conventional single fractal models cannot express the multi-peak distribution characteristics of coal rock, shale, carbonate rock, and other cores. A multifractal model has numerous dimensions and discontinuous functions, making the model parameter selection difficult. Based on the capillary bundle theory and the truncated Gaussian distribution model, a model for characterizing the multi-peak distribution structure of cores is developed in this study. The model's accuracy is verified using the tested NMR experimental data of shale bimodal distribution and the cited coal trimodal distribution. This study further analyzes the influence of tortuosity and tortuosity fractal dimensions on this model. The results show that tortuosity and tortuosity fractal dimensions are conducive to improving the accuracy of the multi-peak pore structure model. The structure model, which can characterize the multi-scale and multi-peak distribution characteristics of shale pore diameter, provides a new concept for describing the pore structure of porous media.

Robustness and sensitivity of some wavelet multifractal models in fractal data modelling

AbstractMany multifractal models such as self‐similar and scaling law types have been proved to be efficient modellers and estimators in many fields such as financial time series where the data hide fractal and multifractal structures, allowing its processing without sophisticated models to be difficult. However, in statistical analysis, a necessary part that should take place for any model and estimator consists in tests of performance such as confidence intervals and generally statistical tests to confirm the adequacy of the model. The present paper provides the consideration of multifractal models based on wavelets and self‐similar type processes to study statistical tests. To test the efficiency, accuracy and robustness of the models, different inferential statistics are introduced, provided with some empirical examples due to the EURO/USD exchange rate time series with a sample covering the period 03/01/2000 to 30/08/2022. Contrarily to existing works, we showed in the present work that quasi‐self‐similar type models are better for many reasons. They indeed guarantee the well fitting of the data dynamics, the nonlinearity in both the model and the multifractal spectrum, the renormalization parameters which may differ from one scale to another and the preservation of the quasi multiplicative structure.

A multifractal model of asset (in)variances

This study extends Mandelbrot’s (2008) multifractal model of asset returns to model realized variances across different time frequencies. In a comparative manner, various degrees of time deformations are explored for implementation of the multiplicative cascade. In doing so, this study focuses on two effects: discontinuity measured by the specific power-law exponent and dependency measured by the Hurst exponent. This study shows that the benchmark model, for which Mandelbrot’s (2008) “cartoon” is the foundation, has some remarkable properties as it is capable of explaining the realized variances for the GBP/USD exchange rate and Bitcoin. Notably, the realized variances for crude oil and the S&P 500 require a more extreme time deformation. The invariance hypothesis is confirmed for all realized variances because the power-law exponents for weekly and monthly data coincide with predictions of the multifractal model. Overall, the novel results derived from the proposed multifractal models suggest that some realized variances of otherwise unrelated asset markets are driven by the same underlying “driving force”—a common multifractal cascade.

Multi-Fractal Weibull Adaptive Model for the Remaining Useful Life Prediction of Electric Vehicle Lithium Batteries.

In this paper, an adaptive remaining useful life prediction model is proposed for electric vehicle lithium batteries. Capacity degradation of the electric car lithium batteries is modeled by the multi-fractal Weibull motion. The varying degree of long-range dependence and the 1/f characteristics in the frequency domain are also analyzed. The age and state-dependent degradation model is derived, with the associated adaptive drift and diffusion coefficients. The adaptive mechanism considers the quantitative relations between the drift and diffusion coefficients. The unit-to-unit variability is considered a random variable. To facilitate the application, the convergence of the RUL prediction model is proved. Replacement of the lithium battery in the electric car is recommended according to the remaining useful life prediction results. The effectiveness of the proposed model is shown in the case study.

Grain size distribution drives microbial communities vertically assemble in nascent lake sediments

Sediment microbes are crucial for maintaining biogeochemical cycles in aquatic ecosystems, yet the influence of sediment geophysical structure on microbial communities remains unclear. In this study, we collected sediment cores from a nascent reservoir in its initial stage of deposition and utilized the multifractal model to comprehensively characterize the heterogeneity of sediment grain size and pore space. Our results demonstrate that both environmental physiochemistry and microbial community structures varied significantly with depth, with the grain size distribution (GSD) being the key driver of sediment microbial diversity, as revealed by the partial least squares path model (PLS-PM) method. GSD can potentially impact microbial communities and biomass by controlling pore space and organic matter. Overall, this study represents the first attempt to apply soil multifractal models into the integrated description of physical structure in sediment. Our findings provide valuable insights into the vertical distribution of microbial communities.

Real-time Multi Fractal Ensemble Analysis CNN Model for Optimizing Brain Tumor Classification and Survival Prediction Using SVM

Classification and Prediction of brain tumors towards survival prediction has been well studied. There exist different schemes around the problem but struggle with poor performance in survival prediction and classification. To overcome the deficiency in classification, a real-time multi-fractal ensemble analysis CNN model (RMFEA-CNN) is presented in this article. The method not just considers basic low-level features like gray, texture, and binary features but also considers Coverage, Mass Index, and Intensity Fraction features. By preprocessing the image with the histogram equalization technique, the image quality has been increased. Further, the above-said features are extracted and trained by generating a multi-fractal ensemble towards various classes using a convolution neural network. The intermediate layers apply a support vector machine toward the classification of an ensemble. The neurons of the intermediate layer apply a support vector machine in estimating Ensemble Centric Coverage Support Measure (ECCSM), Ensemble Centric Mass Support Measure (ECMSM), and Ensemble centric Intensity Support Measure (ECISM) towards various classes. Disease Attraction Weight (DAW), which is measured by the support vector machine using a variety of support metrics, is computed using the estimated values by the method and produced at the output layer. The method carry out disease prediction and estimates survival stage support (SSS) measures to perform survival prediction, as determined by the DAW value. The proposed method improves disease prognostication performance and introduces a lower false ratio.

A wavelet multifractal model for quality of life index measuring during pandemics and crises

AbstractIn the present article, we investigate the impact of the timescale factor on the quality of life index behaviour on specific time intervals characterized by the presence of socio‐economic, political, and/or health severe movements such as pandemics and crises. We essentially aim to show that effectively the quality of life evaluation based on a single index as in the existing studies may be described more adequately by a variable index due to the social, political, economic, and also healthy environment. The variability discovered is expressed by the existence and the estimation of a multi‐index instead of a single one with relatively too many factors. Our focus is mainly on the effect of the COVID‐19 pandemic and crises or crashes on the quality of life. It turns out that the first essays of empirical treatments of such a series bring out a fractal/multifractal aspect. This motivates our main idea reposing on the fractal/multifractal structure of the data to construct a quantitative model based on wavelets combined with change‐point analysis. Our model is applied empirically on a sample corresponding to Saudi Arabia as a case of study during the period from January 1990 to December 2021. The end of this period is strongly affected by the COVID‐19 pandemic. The sample is based on social media conversations and texts discussing and describing the satisfaction with the quality of life. The study confirms effectively that the role of the timescale factor is more described when considering a multi‐index rather than measurement on the whole time interval. Besides, this multi‐index is clearly illustrated by means of the multifractal spectra of the data used.

Lagrangian acceleration and its Eulerian decompositions in fully developed turbulence

We study the properties of various Eulerian contributions to fluid particle acceleration by using well-resolved direct numerical simulations of isotropic turbulence, with the grid resolution as high as $12288^3$ and the Taylor-scale Reynolds number $R_\lambda$ in the range between 140 and 1300. The variance of convective acceleration, when normalized by Kolmogorov scales, increases linearly with $R_\lambda$, consistent with simple theoretical arguments, but very strongly differing from phenomenological predictions of Kolmogorov's hypothesis as well as Eulerian multifractal models. The scaling of the local acceleration is also linear $R_\lambda$ to the leading order, but more complex in detail. The strong cancellation between the local and convective acceleration -- faithful to the random sweeping hypothesis -- results in the variance of the Lagrangian acceleration increasing only as $R_\lambda^{0.25}$, as recently shown by Buaria \& Sreenivasan [Phys. Rev. Lett. 128, 234502 (2022)]. The acceleration variance is dominated by irrotational pressure gradient contributions, whose variance also follows an $R_\lambda^{0.25}$ scaling; the solenoidal viscous contributions are relatively small and follow a $R_\lambda^{0.13}$, consistent with Eulerian multifractal predictions.

A classification-based multifractal analysis method for identifying urban multifractal structures considering geographic mapping

Identifying urban multifractal structures are helpful for understanding urban spatial organization patterns as complex systems. Multifractal analysis is a powerful tool to model multifractal structures. However, due to the use of statistical moments to delineate the multifractal spectrum for multifractal analysis, the great majority of existing studies cannot map urban multifractal structures to geographic space. Lack of geographic mapping makes it difficult to interpret the causes of the anomalous scaling characteristics of urban multifractal structures. For the few mappable multifractal structure modeling methods, they model multifractal structures from a global or local perspective that generates inadequate or redundant scaling characteristics. Here, a classification-based multifractal analysis method (CMFA) was proposed to overcome the shortcomings. It classifies the urban areas into zones according to the density of urban elements and builds up the multi-scaling relationships of urban elements for each zone. The corresponding multifractal structures can be mapped according to the spatial distribution of zones across scales. A case study was conducted to identify the multifractal structures of nighttime light in Beijing, China, to verify the CMFA method. In conclusion, when there are abnormal scaling characteristics reflected by the multifractal spectrum, the multifractal structure maps can diagnose the land use problems leading to disordered spatial organization patterns. Urban planners should focus on such problem land parcels and carry out urban renewal to optimize urban spatial structures.