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We investigate stochastic properties of the backward continued fraction expansion of irrational numbers in ( 0 , 1 ) . For the mean process associated with a real-valued observable which depends only on the first digit of the expansion, we establish the large deviation principle. For any such observable which is non-negative, we completely determine the set of minimizers of the rate function in terms of a growth rate of the observable. Our method of proof employs the thermodynamic formalism for topological Markov shifts, and a multifractal analysis on Lyapunov exponents for the Renyi map generating the backward continued fraction expansion.

Real time anomaly detection is important to performance and efficiency in many areas. This paper offers a complex method for detecting abnormal telecommunication traffic. The proposed method includes components based on entropy analysis, signature analysis and machine-learning detection of anomalies using fractal and recurrence analysis. Datasets containing realizations of normal traffic and realizations of attacks of different types were used as input data. Traffic containing the attack is considered to be abnormal. The multifractal and recurrence methods are briefly described. The results of multifractal and recurrence analysis of data-set traffic showed significant differences between normal and attacked realizations. Based on the obtained results, machine-learning algorithms are proposed. The classifiers random forest and neural network were used and showed good classification accuracy. The methods signature and entropy analysis are briefly described, and the results of their application showed a high degree of anomaly detection. The complex anomaly detection method combines entropy analysis, signature analysis, and machine-learning using the multifractal and recurrence analysis. The proposed complex method is tested on virtual network and is compared pervious methods. Simulation results showed that the proposed complex method has the highest percentage of detected attacks, the lowest percentage of undetected attacks and lost data.

We examine the complexity of financial returns generated by popular agent-based models through studying multifractal properties of such time series. Specifically, we are interested in the sensitivity of the models to their parameter settings and whether some patterns emerge in the connection between complexity and a specific type of parameter. We find that (i) herding behavior mostly boosts the model complexity as measured by multifractality, (ii) various in-built stabilizing factors increase model complexity, while (iii) the role of the intensity of choice, the number of agents, as well as the chartists’ representation have rather model-specific effects. Finally, the core feature driving the model complexity seems to be the implementation of a switching mechanism governing agents’ interactions. The heterogeneous set of nine analyzed models thus offers some universal concepts that hold across their range. Our results also indicate that complex dynamics are observed not only for the benchmark parameter settings but also for other combinations of parameter values for most models. This opens new avenues for future research and specifically motivates examining the models in more detail by focusing on other dynamic properties in addition to the herein presented multifractality.

The concept of dimension is an important task in geometry. It permits a description of the growth process of objects. It may be seen as an invariant measure characterizing the object. Fractal dimensions are a kind of invariants permitting essentially to describe the irregularity hidden in irregular objects, by providing a suitable growth law. Among fractal geometrical objects, Moran’s types play an important role in explaining many situations, in pure mathematics as the general context of Cantor’s, and in applied physics as a suitable context for studying scaling laws. In the present paper, some non-regular homogeneous Moran measures are investigated, by establishing some new sufficient conditions permitting an explicit computation of the relative multifractal dimensions of the level sets for which the classical formulation does not hold. Besides, the mutual singularity of the relative multifractal measures for the homogeneous Moran case with different multifractal dimensions is investigated. This is very important, as in quasi-all existing situations, the validity of the multifractal formalism passes through the equality of the multifractal Hausdorff dimension with the packing one.

The COVID-19 pandemic creates a worldwide threat to human health, medical practitioners, social structures, and finance sectors. The coronavirus epidemic has a significant impact on people's health, survival, employment, and financial crises; while also having noticeable harmful effects on our environment in a short span of time. In this context, the complexity of the Corona Virus transmission is estimated and analyzed by the measure of non-linearity called the Generalized Fractal Dimensions (GFD) on the chest X-Ray images. Grayscale image is considered as the most important suitable tool in the medical image processing. Particularly, COVID-19 affects the human lungs vigorously within a few days. It is a very challenging task to differentiate the COVID-19 infections from the various respiratory diseases represented in this study. The multifractal dimension measure is calculated for the original, noisy and denoised images to estimate the robustness of COVID-19 and other noticeable diseases. Also the comparison of COVID-19 X-Ray images is performed graphically with the images of healthy and other diseases to state the level of complexity of diseases in terms of GFD curves. In addition, the Mean Absolute Error (MAE) and the Peak Signal-to-Noise Ratio (PSNR) are used to evaluate the performance of the denoising process involved in the proposed comparative analysis of the representative grayscale images.

In this article, noise induced chaos is investigated for a finance system. To characterize chaotic paradigm, period analysis is done with the variation of a parameter and noise strength. Later on, chaos has been quantified by 0–1 tests under the same variation. A phase space analysis is also done to investigate the effect of noise on the system dynamics. However, we have noticed that the respective asymptotic dynamics of the deterministic and noise induced chaos are very similar. To classify chaos between noisy and noise free systems, multifractal analysis is then performed. Though the phase spaces are showing similar trajectories, the multifractal analysis confirms more complex dynamics of the noise induced system in compare to the deterministic model. This investigation is an application of multifractal analysis, in case of quantifying chaos.

This study provides the first evidence of market efficiency of drug indices, especially cannabis and tobacco, which are known in finance as sin markets. The multifractal detrended fluctuation analysis (MFDFA) is employed on the daily data of six cannabis and one tobacco indices in order to measure efficiency by quantifying the intensity of self-similarity. The findings confirm multifractality in all sample series. Interestingly, Dow Jones Tobacco (DJUSTB) Index shows the highest multifractality, demonstrating the lowest efficiency, whereas S&P/TSX Cannabis (SPTXCAN) Index is the most efficient of all the time series under analysis, with the lowest multifractality levels. Only the North American Marijuana (NAMMAR), Cannabis World Index Gross Total Return (CANWLDGR) and DJUSTB show persistent behavior. These findings could be of interest to policymakers and regulators to establish new reforms to improve the efficiency of these markets, as well as for actual and potential investors.

This paper concerns an investigation on the multifractal features of liquidity in China's stock markets based on multifractal detrended fluctuation analysis, and an analysis on multifractal features of market liquidity and fractal degree by determining the generalized Hurst exponent. Also involved in this paper are the identification of the trended fluctuations by tendency entropy dimension and a study on the validity of the correct rate of identification by stochastic correct rate. The results show that the multifractal features of liquidity are obvious in China's stock markets and that the multifractal liquidity degree of the large-cap stock is lower than those of the medium and small-cap stocks. The trend entropy dimension can be used to identify the trend of liquidity effectively, serving as an effective method to identify the fluctuation trend of fractal market.

We investigate gold’s role as a hedge or safe haven against oil price and currency movements across calm and extreme market conditions. For the empirical analysis, we extend the intraday multifractal correlation measure developed by Madani et al. (Bankers, Markets & Investors, 163:2–13, 2020) to consider the dependence for calm and extreme movement periods across different time scales. Interestingly, we employ the rolling window method to examine the time-varying dependence between gold-oil and gold-currency in terms of calm and turmoil market conditions. Based on high frequency (5-min intervals) across the period 2017–2019, our analysis shows three interesting findings. First, gold acts as a weak (strong) hedge for oil (currency) market movements, across all agent types. Second, gold has strong safe-haven capability against extreme currency movements, and against only short time scales of oil price movements. Third, hedging strategies confirm the scale-dependent gold's role in reducing portfolio risk as a hedge or safe haven. Implications for investors, financial institutions, and policymakers are discussed.

Network theory helps us understand, analyze, model, and design various complex systems. Complex networks encode the complex topology and structural interactions of various systems in nature. To mine the multiscale coupling, heterogeneity, and complexity of natural and technological systems, we need expressive and rigorous mathematical tools that can help us understand the growth, topology, dynamics, multiscale structures, and functionalities of complex networks and their interrelationships. Towards this end, we construct the node-based fractal dimension (NFD) and the node-based multifractal analysis (NMFA) framework to reveal the generating rules and quantify the scale-dependent topology and multifractal features of a dynamic complex network. We propose novel indicators for measuring the degree of complexity, heterogeneity, and asymmetry of network structures, as well as the structure distance between networks. This formalism provides new insights on learning the energy and phase transitions in the networked systems and can help us understand the multiple generating mechanisms governing the network evolution.

The two-phase flow patterns (air–water) in horizontal square minichannel (3 × 3 mm) has been analysed. The multifractal analysis of pressure drop fluctuations was used for qualitative assessment of two-phase flow patterns. The results of the complexity analysis using the multifractal spectral width (Δh) are presented. The proposed method allows us to identify the following two-phase flow patterns: micro-bubbles flow, micro- and minibubbles flow, micro- and mini-bubbles with confined bubbles flow, slug flow, stratified flow. The obtained results confirm that this type of analysis can be considered as an alternative way of identification of two-phase flow patterns in the minichannel. The work also focuses on the discussion of the occurrence and identification of bubbles churns in slugs and churns.

To understand the pore structure and heterogeneity of pore size distribution (PSD) is essential for revealing fluid mechanics and evaluating the utilization of unconventional resources. In this study, there are multiple shale examples collected from the Chang 7 section in the Ordos Basin for the investigation was conducted on the basis of various experiments on total organic carbon (TOC), X-ray diffraction (XRD), and nitrogen gas adsorption, through scanning electron microscopy (SEM) and multifractal method. The multifractal characteristic parameters, including the width of singularity spectra ( Δ α ), Hurst exponent ( H ), D 1 / D 0 , and nitrogen gas adsorption, were used to find out about the characteristics of pore development and to quantify the complexity and heterogeneity of pore structure. Depending on the exact mineral composition, the Yanchang Formation of Chang 7 shales is classified into either silty mudstone (SM) or muddy siltstone lithofacies (MS). According to the investigative results, the Chang 7 lacustrine shale features a complex pore system with the pores ranging from 1.5 to 10 nm in diameter. Besides, mesopores contribute significantly to the total pore volume (TPV) and total surface area (TSA). As for TPV and TSA of the SM lithofacies in the samples under investigation, they are nearly 1.09–1.78 and 0.80–1.72 times greater as compared to the MS lithofacies samples. The dominant types of reservoir spaces include organic matter (OM) pore and interparticle pore which are related to inorganic minerals. The value of Δ α is higher for MS lithofacies than for SM lithofacies, indicating a greater heterogeneity of PSD in the MS lithofacies. The pore structure of MS lithofacies is determined mainly by TOC and siliceous mineral content, whereas the influencing factors for SM lithofacies are TOC and clay mineral content. There is a significant relationship between multifractal parameters and pore structure parameters for both SM and MS lithofacies. The TOC of SM and MS lithofacies exhibits a close correlation with Δ α , suggesting that the pores in organic matter are dominated by those nanopores with a complex and heterogeneous pore structure. The rock composition of the lithofacies can affect Δ α to a varying extent, which means that the minerals have an evident impact on the heterogeneity of MS and SM lithofacies.

Stream sediment geochemical survey is an efficient tool for regional prospecting of ore minerals. However, complex processes of dilution and enrichment made it difficult to be studied by conventional methods. One of the most important methods to separate anomalous areas are fractal and multifractal analysis; however, some techniques of the fractal analysis which are more popular and applicable, such as concentration-area, suffer from lack of a reliable interpolated map of stream sediments data. In this study, we introduce a simple workflow to separate anomalous areas in three steps: 1. a grid interpolation method is developed to produce continues geochemical surface based on geometry of Sample Catchment Basin (SCB), 2. Although U-statistics, a powerful spatial based method, has no permission to work with Stream sediment data, the grid interpolation prepares a convenience (by converting stream sediment data from vector nature into point or pixel-based data) to apply it for this type of geochemical data, 3. Finally, the fractal analysis is applied for anomaly separation. In other words, the transformation into pixel-based nature in the first step is followed by using U-statistics (or any spatial method) to compensate spatial relation among the pixels. Once an accurate interpolated map is computed in the first two steps, fractal analysis is capable to effectively separate different populations (i.e. background and anomaly) with the highest accuracy and validity. This workflow is implemented on 305 stream sediment samples from Saqez area located on western part of Iran to delineate Au mineralization. Results show that anomalous areas of Au are successfully separated in the final categorical map, which is highly matched with the location of 5 high potential zones of Au known in this area.

Hydrological applications such as storm-water management or flood design usually deal with and are driven by region-specific reference rainfall regulations or guidelines based on Intensity-Duration-Frequency (IDF) curves. IDF curves are usually obtained via frequency analysis of rainfall data using which the exceedance probability of rain intensity for different durations are determined. It is also rather common for reference rainfall to be expressed in terms of precipitation P, accumulated in a duration D (related to rainfall intensity ), with a return period T (inverse of exceedance probability). Meteorological modules of hydro-meteorological models used for the aforementioned applications therefore need to be capable of simulating such reference rainfall scenarios. The multifractal cascade framework, since it incorporates physically realistic properties of rainfall processes (non-homogeneity or intermittency, scale invariance and extremal statistics) seems to suit this purpose. Here we propose a discrete-in-scale universal multifractal (UM) cascade based approach. Daily, Hourly and six-minute rainfall time series datasets (with lengths ranging from 100 to 15 years) over three regions (Paris, Nantes, and Aix-en-Provence) in France that are characterized by different climates are analyzed to identify scaling regimes and estimate corresponding UM parameters (α, C1) required by the UM cascade model. Suitable renormalization constants that correspond to the P, D, T values of reference rainfall are used to simulate an ensemble of reference rainfall scenarios, and the simulations are finally compared with datasets. Although only purely temporal simulations are considered here, this approach could possibly be generalized to higher spatial dimensions as well.

We are interested in the behavior of relative multifractal box-dimensions, density results and multifractal dimensions under projections onto a lower- dimensional linear subspace. The results in this paper establish the connections with various multifractal and fractal dimensions and their projections, and generalize many known results about Hausdor and packing measures and dimensions of a subset of Rn.

We characterized the multifractality and power-law cross-correlation of mitochondrial genomes of various species through the recently developed method which combines the chaos game representation theory and 2D-multifractal detrended cross-correlation analysis. In the present paper, we analyzed 32 mitochondrial genomes of different species and the obtained results show that all the analyzed data exhibit multifractal nature and power-law cross-correlation behaviour. Further, we performed a cluster analysis from the calculated scaling exponents to identify the class affiliation and its outcome is represented as a dendrogram. We suggest that this integrative approach may help the researchers to understand the phylogeny of any kingdom with their varying genome lengths and also this approach may find applications in characterizing the protein sequences, mRNA sequences, next-generation sequencing, and drug development, etc.

Massive seamounts have been surveyed and documented in the last decades. However, the morphologies of seamounts are usually described in qualitative manners, yet few quantitative detections have been carried out. Here, based on the high-resolution multi-beam bathymetric data, we report a recently-surveyed guyot on the Caroline Ridge in the West Pacific, and the large-scale volcanic structures and small-scale erosive-depositional landforms in the guyot area have been identified. The multifractal features of the guyot are characterized for the first time by applying multifractal detrended fluctuation analysis on the surveyed bathymetric data. The results indicate that the multifractal spectrum parameters of the seafloor have strong spatial dependency on the fluctuations of local landforms. Both small- and large-scale components contribute to the degree of asymmetry of the multifractal spectrum (B), while the fluctuations of B are mostly attributed to the changes in small-scale roughness. The maximum singularity strength (α0) correlates well with the roughness of large-scale landforms and likely reflects the large-scale topographic irregularity. Comparing to traditional roughness parameters or monofractal exponents, multifractal spectra are able to depict not only the multiscale characteristics of submarine landforms, but also the spatial variations of scaling behaviors. Although more comparative works are required for various seamounts, we hope this study, as a case of quantifying geomorphological characters and multiscale behaviors of seamounts, can encourage further studies on seamounts concerning geomorphological processes, ocean bottom circulations, and seamount ecosystems.

This paper examines the nonlinear and dynamic cross-correlations between SHIBOR and Chinese stock market liquidity by employing MF-DCCA method. The cross-correlations display weak persistence and multifractal characteristics, explaining the variations in the relationship between them. The multifractality strength of the cross-correlations decreases after a recent liberalization reform. Moreover, interest rates have a significantly strong influence on stock market liquidity during tight monetary policy and emergencies, indicating the asymmetric and time-varying impact of interest rates on stock market liquidity. In addition, the effectiveness of interest rate transmission decreases in the period of the COVID-19 pandemic.

The shale nanopore size (diameter [less than] 100 nm) distribution heterogeneity (SNDH) is one of the important factors affecting gas production. However, quantitative analysis of the SNDH and the applicability of single and multifractal models need to be further studied. Here, based on low-temperature liquid nitrogen and carbon dioxide tests of organic-rich shale in Qinshui Basin, multifractal dimension variation of micropores ([less than] 2 nm) and mesopores (2–100 nm) are studied, and the multifractal factors that affect the distribution of nanopores are determined. Additionally, the differences between single fractal and multifractal results are compared. Based on this, dynamic variation of porosity and permeability under the constraints of the nanopore structure is discussed from the perspective of multifractal variation. The results of this study are as follows: (1) The pore size distribution of micropores and meso–macropores in shale samples exhibit typical multifractal behavior. The overall distribution heterogeneity of meso–macropores is mainly affected by the distribution of pores in the low-value area of pore volume (LAPV), while the overall distribution heterogeneity of micropores is affected by the distribution of the high-value area of the pore volume. The multifractal parameters and influencing factors of micropores and meso–macropores are clearly different. (2) The single fractal dimension D2 calculated using the Frenkel–Halsey–Hill model has a negative correlation with the multifractal parameters, implying that the distribution heterogeneity of the LAPV gradually decreases with the increase of the D2 value, indicating that the physical meaning of the two models is obviously different. (3) The pore distribution heterogeneity affects permeability variation and diffusion process of shale reservoir. With the increase of the multifractal dimension of meso–macropores, the damaging effect of stress on permeability is stronger. The more heterogeneous the micropore size distribution, the smaller is the ″modification effect″ of stress on the diffusion coefficient.

The present study reports measurements of radon in groundwater conducted in China between January and December 2013. During the measurement period, a great Ms = 6.6 earthquake occurred in Gansu Province, China. The paper analyses data derived from three nearby stations via monofractal and multifractal detrended fluctuations analysis and explores whether pre-earthquake fractal and long-memory trends exist in the time series. Several critical epochs with characteristic pre-seismic long-lasting fractal patterns are identified in segmented parts of the series. In the whole data series, the fluctuation function, scaling exponent and generalised Hurst exponent depend on the fractal scales, which indicates multifractality. The multifractal spectrum of the raw, shuffled and truncated series showed a significant increase in the degree of fluctuation that can be explained as a pre-earthquake indicator attributable to the Gansu earthquake due to its very large magnitude.

Atmosphere and ocean dynamics display many complex features and are characterized by a wide variety of processes and couplings across different timescales. Here we demonstrate the application of multivariate empirical mode decomposition (MEMD) to investigate the multivariate and multiscale properties of a reduced order model of the ocean–atmosphere coupled dynamics. MEMD provides a decomposition of the original multivariate time series into a series of oscillating patterns with time-dependent amplitude and phase by exploiting the local features of the data and without any a priori assumptions on the decomposition basis. Moreover, each oscillating pattern, usually named multivariate intrinsic mode function (MIMF), represents a local source of information that can be used to explore the behavior of fractal features at different scales by defining a sort of multiscale and multivariate generalized fractal dimensions. With these two complementary approaches, we show that the ocean–atmosphere dynamics presents a rich variety of features, with different multifractal properties for the ocean and the atmosphere at different timescales. For weak ocean–atmosphere coupling, the resulting dimensions of the two model components are very different, while for strong coupling for which coupled modes develop, the scaling properties are more similar especially at longer timescales. The latter result reflects the presence of a coherent coupled dynamics. Finally, we also compare our model results with those obtained from reanalysis data demonstrating that the latter exhibit a similar qualitative behavior in terms of multiscale dimensions and the existence of a scale dependency of the statistics of the phase-space density of points for different regions, which is related to the different drivers and processes occurring at different timescales in the coupled atmosphere–ocean system. Our approach can therefore be used to diagnose the strength of coupling in real applications.

This paper proposes a new strategy for portfolio selection in the Brazilian equity market with the use of multifractal detrended fluctuation analysis (MF-DFA) as a mechanism to select assets based on their efficiency levels. Empirical analysis uses daily prices to compose minimum variance (MVP) and maximum Sharpe ratio (MSR) long-only portfolios, and also includes their performances during the COVID-19 pandemic. MF-DFA indicated a multifractal nature for asset price returns, generally associated with long-term persistence. The strategy using the most efficient equities resulted in portfolios with lower levels of systematic risk (betas), indicating that the lack of efficiency is related to higher sensitivity to macroeconomic and conjuncture changes. The MVP portfolio produces higher performance than the alternatives in terms of risk and return. Finally, during the COVID-19 pandemic, besides its consistent negative impacts, MVP and MSR portfolios verified lower losses than the IBOVESPA.

The Three Gorges Reservoir (TGR) in China is the largest hydroelectric project in the world, but the threat of sediment affecting ecological sustainability of the reservoir is a topic of concern. Sediment particle-size distribution (PSD) is informative in understanding sediment transport dynamics and biochemical functions. It is, therefore, important to quantitatively characterize the distribution of sediment particles. In the current study, fractal theory is applied to determine the PSD of suspended sediment in the TGR. The results show that the volumetric fractal dimension ( D v ) exhibits a significant seasonal difference (p D ( q ) – q , and multifractal singularity spectrum, f [ α ( q ) ] – α ( q ) , were calculated for each suspended sediment sample. Thereafter, the parameters, D(0), D(1), D(2), α(0), Δα(q), and Δ f [ α ( q ) ] , were determined to characterize the PSD. As a result, the coarser suspended sediment during the wet season is characterised by a more complex PSD pattern, with a wider range of particle sizes, greater heterogeneity, and greater homogeneity of distribution over the measurement interval. However, the multifractal structure of the PSD of suspended sediment is more complex during the dry season than during the wet season, with higher local dispersion and variability. The findings of the current study highlight that multifractal analysis provides important insight for understanding the PSD of suspended sediment in the TGR.

It is of great significance to study the spatial distribution patterns and petrophysical complexity of volcanic vesicles which determine whether the reservoir spaces of the volcanic rocks can accumulate oil and gas and enrich high yields or not. In this paper, the digital images of three different textures of vesicular andesite samples, including spherical vesicular andesite, shear deformation vesicular andesite, and secondary filling vesicular andesite, are obtained by microscopic morphology X-CT imaging technology. The spatial micro-vesicle heterogeneity of vesicular andesite samples with different textures is quantitatively analyzed by fractal and multifractal methods such as box-counting dimension and the moment method. It is found that the shear stress weakens the spatial homogeneity since vesicles rupture are accelerated, elongated directionally, and connected with one another under the strain; the secondary filling breaks the vesicles, which significantly enhances the spatial heterogeneity. In addition, shear stress and secondary filling increase the complexity of vesicle microstructures characterized by different fractal and multifractal parameters. These conclusions will provide important theoretical and practical insights into understanding the degassing of volcanic rocks and prediction of high-quality volcanic reservoirs.

PurposeBy combining econometrics and multifractal methods, utilizing a financial framework, this paper will examine with objectivity the economic, financial and social impact of coronavirus disease 2019 (COVID-19) on society.Design/methodology/approachThrough Granger causality, the authors test the effect of the COVID-19 pandemic on excessive gaming and gambling activities, and through econometrics and multifractal methods, they combine the results to analyze a possible long-run relationship.FindingsThe COVID-19 confirmed cases Granger cause all examined stocks. Based on the co-integration technique, and the multifractal cross-correlation analysis, a long-run relationship exists between all examined stocks and COVID-19.Originality/valueThis is an empirical examination of a very important subject in the field of economics, namely, the consequences of the COVID-19-related events on the behavior of global citizens. It proposes a different and more objective approach (than the interviews and questionnaires) in the examination of this specific subject, through a financial framework, depicting the stock performance of the gaming and online gambling-related companies, and reflecting on the activity of these companies. It combines two different approaches from two different disciplines, namely econometrics and multifractal analysis, to test and describe the causal and the long-run relationship between the phenomena examined, combining the results to an overall and multidimensional view of this occurrence.

Several studies have been conducted on droughts, precipitation, and temperature, whereas none have addressed the underlying relationship between nonlinear dynamic properties and patterns of two main hydrological parameters, precipitation and temperature, and meteorological and hydrological droughts. Monthly datasets of Midlands in the UK between 1921 and 2019 were collected for analysis. Subsequent to apply a multifractal approach to attain the nonlinear features of the datasets, the relationship between two hydrological parameters and droughts was investigated through the cross-correlation technique. A similar process was performed to analyze the relationship between multifractal strength variations in time series of precipitation and temperature and droughts. The nonlinear dynamic results indicated that droughts (meteorological and hydrological) were substantially affected by precipitation than temperature. In other words, droughts were more sensitive to precipitation fluctuations than temperature fluctuations. Concerning temperature, meteorological, and hydrological droughts were dependent on the minimum and maximum temperatures ( $$T_{{{\text{min}}}}$$ and $$T_{{{\text{max}}}}$$ ), respectively. The correlation between precipitation and meteorological drought was more long-range persistence than precipitation and hydrological drought. Besides, the correlation between $$T_{{{\text{max}}}}$$ and droughts was more long-range persistence than $$T_{{{\text{min}}}}$$ and droughts. Analysis of nonlinear dynamic patterns proved that the multifractal strength of meteorological drought depended on the multifractal strength of precipitation and $$T_{{{\text{max}}}}$$ , whereas the multifractal strength of hydrological drought depended on the multifractal strength of the $$T_{{{\text{min}}}}$$ . The correlation between precipitation and drought indices exhibited more multifractal strength than temperature and drought indices. Finally, the pivotal role of maximum temperature on drought events was quite alerting due to global warming intensification.

This paper investigates the long-term dynamic cross-correlation evolution between US economic policy uncertainty index (USEPU) and Guangdong carbon emission trading price (GDCP) from the multifractal detrended cross-correlation analysis (MF-DCCA) perspective. With the calculation of correlation statistics and fluctuation function, the beginning procedures of MF-DCCA, we find that the cross-correlation between USEPU and GDCP is significant and presents power law property. Also, with the Hurst exponent, we find that the long-horizon correlations between series are persistent. Moreover, we perform Rényi exponent and spectrum singularity check. The empirical findings reveal that the all the correlations are of multifractality and the correlation of GDCP holds the highest degree.

In this paper, fractal theory and wavelet transform are combined to detect and classify self-extinguishing and fugitive scenarios of power quality disturbances (PQDs). After deciding whether the disturbance is simple or complex, the additional voltage is denoised through Discrete Wavelet Transform (DWT); the denoising process is adapted according to whether the distorted voltage contains oscillatory transients or not. At the detection stage, the grille fractal dimension of the DWT decomposition detail is computed. Then, a threshold is deduced to detect the start and end moments of the disturbance. The results reveal that the proposed detection scheme yields accurate location of PQDs even in the presence of high oscillatory transients. An algorithm based on geometric and statistical approaches is developed at the classification stage to recognize PQDs automatically. The geometric classification is based on Continuous Wavelet Transform (CWT), whereas the statistical classification is based on Multifractal Detrended Fluctuations Analysis (MFDFA) and an energy metric. The results prove that the combination of geometric and statistical classification can serve as an effective discrimination tool for PQDs. The major strength of the proposed approach is its ability to interpret the impact of each disturbance on the multifractal behavior of the nominal voltage, thus giving the possibility to draw the necessary generalizations for real-time applications.

During the last few decades, the urban hydrological cycle has been strongly modified by the built environment, resulting in fast runoff and increasing the risk of waterlogging. Nature-based solutions (NBSs), which apply green infrastructures, have been more and more widely considered as a sustainable approach for urban storm water management. However, the assessment of NBS performance still requires further modelling development because of hydrological modelling results strongly depend on the representation of the multiscale space variability of both the rainfall and the NBS distributions. Indeed, we initially argue this issue with the help of the multifractal intersection theorem. To illustrate the importance of this question, the spatial heterogeneous distributions of two series of NBS scenarios (porous pavement, rain garden, green roof, and combined) are quantified with the help of their fractal dimension. We point out the consequences of their estimates. Then, a fully distributed and physically based hydrological model (Multi-Hydro) was applied to consider the studied catchment and these NBS scenarios with a spatial resolution of 10 m. A total of two approaches for processing the rainfall data were considered for three rainfall events, namely gridded and catchment averaged. These simulations show that the impact of the spatial variability in rainfall on the uncertainty of peak flow of NBS scenarios ranges from about 8 % to 18 %, which is more significant than those of the total runoff volume. In addition, the spatial variability in the rainfall intensity at the largest rainfall peak responds almost linearly to the uncertainty of the peak flow of NBS scenarios. However, the hydrological responses of NBS scenarios are less affected by the spatial distribution of NBSs. Finally, the intersection of the spatial variability in rainfall and the spatial arrangement of NBSs produces a somewhat significant effect on the peak flow of green roof scenarios and the total runoff volume of combined scenarios.

Particle size distribution (PSD) is a basic soil property, closely related to main soil physical and chemical attributes. Soil PSD determined by laser diffraction (LD) may provide additional information to soil texture determined by traditional methods. In turn, LD may be implemented using either Mie or Fraunhofer calculation methods, based on different assumptions. The multifractal approach has been demonstrated to be useful for characterizing the inner structure of soil PSD. We analysed the PSDs of 18 soil horizons sampled on the most characteristic soil types from Argentina with a wide range of textural classes, from clayey to sandy, mainly developed over loess material. Our aims were to assess the multifractality between PSDs computed using either Mie scattering or Fraunhofer diffraction models, and to compare the scaling properties and multifractal behavior of PSDs from soils with contrasting textures. Fraunhofer model provided PSD curves, which were shifted toward coarser particles compared to those obtained with Mie model. Regardless calculation method, all the PSDs studied displayed a well-defined multifractal structure, as shown by singularity, f(α) versus α, and by Renyi spectra, Dqvs q. Moreover, all singularity spectra were strong asymmetric, right deviating curves, which is consistent with a greater heterogeneity of the low values of volume frequencies for all the PSDs analysed. Soil PSDs computed by Fraunhofer model showed higher asymmetry than those computed by Mie model for 16 out of 18 horizons studied; this suggests that scaling heterogeneity mainly depends on the support length, which is larger for the PSDs computed by Mie. In addition, other features of the PSDs stemming from soil processes such as weathering intensity of loess materials may also influence multifractality. Stronger correlations were found between multifractal parameters from PSDs calculated by the two models for the most positive q moments of singularity and Renyi spectra, i.e. α10, D10, than for those gathered from the central and negative q moments. The entropy dimension, D1, significantly (P

The results of comparative multifractal studies of the grain structure of different ceramic piezoelectric materials are presented. Hard and soft ferroelectric ceramic materials as well as materials...

This paper takes 11-year 5-minute high-frequency trading data of the Shanghai and Shenzhen 300 Index (CSI300) as a research sample. First, it proposes a method to define the normal state and the state of attention of the financial market based on multi-fractal characteristics, and randomly owes it Sampling (RU), synthetic minority oversampling (SMOTE) and traditional support vector machine (SVM) are combined to propose an improved SVM model—RU-SMOTE-SVM model to predict extreme risks in China’s financial market, and compare Traditional SVM, SMOTE-SVM, RU-SMOTE-NN and RU-SMOTE-DT are compared. The empirical results show that the price fluctuations of China’s emerging financial markets have significant multi-fractal characteristics; the normal and concerned states defined based on the multi-fractal feature parameters are not only accurate, but also have obvious statistical test significance and clear practical significance; and traditional SVM and Compared with BP neural network (NN), RU-SMOTE-SVM is not only significantly higher in prediction accuracy, but also in terms of prediction stability. That is, RU-SMOTE-SVM can effectively solve the problems of other early warning models to solve the symmetrical sample problem.

Measurements of the roughness parameters and fractal dimension of the surface of Cu/Co superlattices are carried out by scanning tunneling microscopy. The fractal analysis of the superlattice surface shows that the fractal-dimension curve can be divided into two sections with different slopes depending on the scanning step. On the fractal-dimension curve, two intervals can be distinguished that define groups of objects that are heterogeneous in morphology: the fractal dimension of the nanostructure grains and the fractal dimension of the surface microrelief. In this regard, the parameters of the root-mean-square roughness of the surface relief and the fractal dimension of the surface morphology can be used as quantitative parameters for analyzing the structural features of the surface of superlattices at the final stage of their synthesis.

Wavelet and multifractal analysis has shown that alterations in oscillatory activity accompanied by long-term correlations between successive values of these oscillations occur as a result of movement disorders in the structure of patterns of involuntary oscillatory hand movements that arise during motor task performance. These alterations cause a significant increase in the variation of the amplitude of involuntary oscillatory hand movements in an individual with Parkinson’s disease compared to healthy individuals. The mechanism of the appearance of correlated dynamics is associated with an increase in the contribution of strong fluctuations of successive values of involuntary oscillations. A decrease in the variation of the amplitude of these oscillations and the energy of their wavelet spectrum in association with antiparkinsonian drugs is accompanied by a decrease in long-term correlations; multifractal characteristics tend to be attributed to the range that is characteristic of healthy individuals.

The issues of joint estimation of the scaling, singular, multifractal, and anisotropic properties of image texture are considered. A technique for forming a fundamentally new class of topological textural-multifractal descriptors for joint estimation of various fractal properties of a texture is described. The efficiency of segmentation of real radar and optical images by the Fuzzy c-means clustering algorithms in the space of the proposed class of descriptors is estimated. Based on the results and their comparison with the accuracy of segmentation by the existing descriptor, the conclusion about the higher informativity of the new class of topological fractal descriptors is made.

We are concerned with some density estimations for relative multifractal measures. We tried, through this article, to improve many known density results. The results in this paper establish the con...

In view of the inaccuracy of the traditional correlation analysis method, this paper proposes a correlation analysis method between the multifractal characteristics of regional landforms and the development of geological disasters. Firstly, the multifractal characteristics of regional landforms are described by using the basic fractal characteristics of self-similarity or scale invariance. Then the corresponding relation table is established according to the width of the fractal spectrum and the number of landslides and hidden dangers, and the spatial relationship of geological disaster development is analyzed. Combined with the above-mentioned spatial relationship of geological disaster development and the multifractal characteristic data of regional landforms, the correlation coefficient between the two is calculated to complete the correlation analysis between the multifractal characteristics of regional geomorphology and the development of geological disasters. The experimental results show that compared with the traditional correlation analysis method, the correlation analysis results of the multifractal characteristics of regional geomorphology and the development of geological disasters are more accurate.

Recently, a multivariate multifractal analysis for pointwise regularities based on hierarchical multiresolution quantities was developed. General bounds between the Hausdorff dimension of the inter...

In this paper, we attain the multifractal Hewitt–Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt–Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples.

Saliva has emerged as an efficient screening sample for early stage detection of oral cancer (OC) owing to non-invasiveness coupled with high sensitivity and specificity. Although spectroscopic characterization of saliva in oral potentially malignant disorders OPMDs) and OC is extensively studied, its potential as imaging biomarker is sparsely explored. Further, the literature on crystalline pattern of saliva for other diseases or different physiological conditions is mostly qualitative. This paper proposed multifractal based methodology to quantitatively study alteration of the salivary fern pattern in different OPMDs and OC in relation to normal counterpart. The fern pattern of dried saliva is captured by stereo-zoom microscope in reflective mode and an image dataset is developed. We resort to two dimensional multi-fractal detrended fluctuation analysis (2d MFDFA) to elucidate the complexity and heterogeneity of these micro-structured patterns. Existence of multifractal nature embedded in salivary fern has been validated for the first time. Long range spatial correlation is found to be the origin of multifractality. Variation in multi-scale self-similarity of irregular pattern in different study groups is demonstrated by four features extracted from MFDFA. Statistical analysis shows discriminating nature of these features for combinations of pairwise interclass classification. This study sheds light on acceptability of microscopic images of arborized saliva in fast and cost effective screening of different oral lesions.

The purpose of this contribution is to highlight four topics of regional and worldwide mineral resource prediction: (1) use of the jackknife for bias elimination in regional mineral potential assessments; (2) estimating total amounts of metal from mineral potential maps; (3) fractal/multifractal modeling of mineral deposit density data in permissive areas; and (4) worldwide and large-areas metal size-frequency distribution modeling. The techniques described in this paper remain tentative because they have not been widely researched and applied in mineral potential studies. Although most of the content of this paper has previously been published, several perspectives for further research are suggested.

The production of magnetically hard Fe–Cr–Co alloy components by an additive method—specifically, selective laser melting (SLM) of metal powder produced by atomization of melt—is considered. The structure and magnetic properties of the resulting samples are investigated. Their magnetic properties are superior to those of cast samples.

The cycle-to-cycle variations (CCVs) in reciprocating internal combustion engines may cause negative influence on diving performance, fuel economy, and emissions. Especially, lean-burn technology or exhaust gas recirculation (EGR) was used to improve engine combustion efficiency and reduce NOx, and the combustion boundary was limited by increased CCVs. Therefore, it was important to identify the complex dynamics of CCVs and to take measures for inhibiting them. The CCVs based on indicated mean effective pressure (IMEP) time series were examined in a lean-burn natural gas engine with a non-uniform pre-mixture based on statistical and multifractal theories. Tests were conducted at an engine speed of 1000 rpm and low loads of 10% and 25%, and combustion data at an engine load of 10% were analysed in detail because the CCVs are less sensitive to changes of gas injection timing (GIT) at the higher engine load. The nonlinear dynamics of the CCVs was revealed at the different GIT from 1° to 120°CA ATDC. The statistical properties of IMEP time series were characterised by distributions of probability density functions (PDF), the multifractal complexities of the combustion fluctuations were quantitatively analysed by the singularity spectra in terms of the Holder exponent based on the theory of wavelet transform modulus maxima, and the primary source leading to the increased CCVs and complex dynamics in a natural gas engine with a non-uniform pre-mixture was identified using 3D-computational fluid dynamics simulations. Results show that as the GIT increased, the kurtosis of the IMEP time series systematically decreased from 592 to 1.8, the fast dynamics transitions from super-Gaussian to quasi-Gaussian distributions in combustion system were revealed, and the lower value of kurtosis implied the lower degree of intermittency. Except for the GIT of 60°CA ATDC, the value of the Holder exponent $$h_{0} > 0.5$$ , which implies that the CCVs for the other GITs behaved like a persistent walk or positive correlation. For GITs of 60° and 90°CA ATDC, the narrow broadness of the singularity spectrum implicated a monofractality, while the obvious multifractal properties could be identified for the other GITs. The transitions of the dynamic behaviours may be caused by the degree of mixture in-homogeneity combined with a new mechanism of “prior-cycle effects” proposed in this paper, and the effect mechanism was not only associated with the residual gas in the cylinder but also with the residual gas fuel in the intake port, rather than independent effect of residual gas in cylinder reported in previous researches. Our research results provided the deeper understanding on the dynamics of combustion system in multi-point injection natural gas engines and may be beneficial to achieve nonlinear prediction and to develop improved control strategies for inhibiting the CCVs.