Species diversity in microbiome is a cutting-edge concept in metagenomic research. In this study, we propose a multifractal analysis for metagenomic research. From the chaos game representation (CGR) visualization of simulated and real metagenomes, we find that there exists self-similarity in the visualization of metagenomes. Then we compute the multifractal dimensions for simulated and real metagenomes. For simulated metagenomes, we also compute their diversity indices, such as species richness indices, Shannon’s diversity indices and Simpson’s diversity indices respectively for varying value of . Fom the Pearson correlation coefficients between their multifractal dimensions and traditional species diversity indices, we find that the correlation coefficients between the multifractal dimensions and species richness indices and Shannon diversity indices reach their maximums at respectively. The correlation coefficients between the multifractal dimensions and Simpson’s diversity indices reach their maximums at nearly. So the traditional diversity indices can be unified by the frame of multifractal analysis. These results coincided with the similar results in macrobial ecology. Finally, we apply our methods to real metagenomes of 100 infants’ gut microbiomes when they are newborn, 4 months and 12 months. Our results show that multifractal dimensions of infants’ gut microbiomes can discriminate the age difference.

This research investigates the complex transformation of the Baltic Clean and Dirty Tankers markets from 1998 to 2023. Our study quantifies the market multifractality in two distinct periods, from 1998 to 2010 and 2010 to 2023. We compare multifractal measures of the original, surrogated, and shuffled data, which indicate high multifractality and a complex market structure influenced by temporal organization and inherent volatility. Shuffled data analysis emphasizes the critical role of temporal sequences within market behaviour. Additionally, the study delves into external factors such as regulatory changes, economic disturbances, technological innovation, and environmental concerns contributing to market complexity. A distinct shift in multifractal values between the two periods suggests an evolution from crisis-induced complexities to a market influenced by diversified and modern dynamics. Our findings offer a concise understanding of the multifractal nature of the tanker markets and how global crises can impact the markets, emphasizing the importance of considering systemic risks in freight market analysis.

This study presents an application of fractal mathematics to financial markets through the utilization of the Multifractal Model of Asset Returns (MMAR). The model is employed to analyse a 30-year dataset encompassing three distinct financial markets: the USD/NOK currency pair, the OMXS30 stock index, and the 12-month LIBOR rate. The MMAR model successfully captures essential stylized facts about financial markets, including high kurtosis, non-independent price movements, and clustered volatile days. Simulations generated by the model demonstrate realistic behaviour and multifractal characteristics. However, the model lacks a statistical method for assessing its accuracy and may underestimate kurtosis in high-kurtosis markets. Notably, the model is capable of producing simulations across a wide range of kurtosis values and can be instrumental in stress-testing portfolios. Despite its potential, the model is subject to limitations, such as the need for human judgment, ambiguous predictions, and interpretability issues. Furthermore, the model may become unreliable following a high-kurtosis event.

This paper examined the multifractal properties of six acknowledged agro-meteorological parameters, such as reference evapotranspiration (ET0), wind speed (U), incoming solar radiation (SR), air temperature (T), air pressure (P), and relative air humidity (RH) of five stations in California, USA. The investigation of multifractality of datasets from stations with differing terrain conditions: Dagget, Bakersfield, Santa Maria, Los Angeles and San Diego using the Multifractal Detrended Fluctuation Analysis showed the existence of a long term persistence and multifractality irrespective of the location. The scaling exponents of SR and ET0 time series are found to be higher for stations with higher altitudes. Subsequently, this study proposed using the novel multifractal cross correlation (MFCCA) method to examine the multiscale-multifractal correlations properties between ET0 and other investigated variables. MFCCA could successfully capture the scale dependent association of different variables and the dynamics in the nature of their associations from seasonal to multi-annual time scale. The multifractal exponents of pressure and relative air humidity are consistently lower than the exponents of ET0, irrespective of station location. This study found that joint scaling exponent was nearly the average of scaling exponents of individual series in different pairs of variables. Additionally, the α-values of joint multifractal spectrum were lower than the α values of both of the individual spectra, validating two universal properties in the mutifractal cross correlation studies for agro-meteorological time series. The temporal evolution of cross-correlation showed similar pattern for all pair-wise associations involving ET0, except for the RH-ET0 link.

Revealing the spatial distribution mechanism of the coupling effect between soil water and salt is essential for managing saline-alkali farmland. This study utilized the multifractal method to analyze soil water and salt under varying water content levels in arid saline-alkali farmland. Soil samples were collected on the second (S1), fifth (S2), eighth (S3), eleventh (S4), and fourteenth (S5) days after a rainfall event. The findings showed that a significant decrease in soil water content, contrasting with an increase in soil salt content throughout the soil layers post-rainfall. As water content decreased, the spatial variability of soil water initially increased from S1-3 and then decreased, while the spatial variability of soil salt decreased. The spatial distribution between soil water and salt exhibited a high correlation at S3-4 (with relative water content of soil ranging from 0.52 to 0.75) due to their coupling effect. However, soil salt was unevenly leached by rainfall at high water content levels (S1-2) and precipitated at low water content levels (S5), resulting in low spatial variability correlations between soil water and salt. This study elucidated the coupling process of soil water and salt, identifying their spatiotemporal distribution mechanism in dryland agricultural areas.

Increasing groundwater levels (GWL) may become one of the most serious issues for Odessa City, Ukraine. This study investigated the spatial distribution characteristics and multifractal scaling behaviour of groundwater level/depth fluctuation for a Quaternary aquifer in Odessa City using a geostatistical approach and a Multifractal Detrended Fluctuation Analysis (MF-DFA). These two methods were applied to monthly GWL fluctuation time series from 1970 to 2020 to monitor 72 hydrogeological wells situated in different parts of Odessa City. The spatial distribution of GWL revealed an overall trend of decline and recovery from 1970 to 2020 in the study area, except for most of the southern region, where a persistent recovery of groundwater depth was observed. The MF-DFA results suggest that the dynamics of GWL fluctuations have multifractal characteristics in the Odessa City area. In addition, both long-range correlation and fat-tail probability distribution contribute to multifractality. However, long-range correlations among fluctuations made a major contribution to the observed multifractality of the GWL fluctuations time series. The generalized Hurst exponent shows a wide range of change (0.20 < h(q) < 2.85), indicating the sensitivity of GWL fluctuations to changes in small scale factors and large-scale factors. Regarding the long range correlations of GWL depth, the Hurst exponents (q = 2) demonstrated the positive persistence of groundwater depth recovery in the southern region and the persistence of groundwater depth variation in the other regions of the study area. The dynamic changes in GWL depth in the Odessa City area may be affected by both natural structural and anthropogenic factors.

Based on the analysis of the spatial distribution of uranium grade in 348 boreholes of a uranium deposit in Xinjiang, the enrichment and spatial variation of uranium ore in two stopes of the deposit are discussed by using multifractal theory. The distribution characteristics of uranium ore of the two stopes are studied by multifractal parameters: the scaling exponent of mass , the scaling exponent of each sub-set and its corresponding fractal dimension , the fractal dimension D0 and information dimension D1. The differences of uranium distribution in two stopes can be well quantified by multifractal spectrum and multifractal parameters( , , ).10m×10m is defined as a fence unit, and the window sizes are respectively set, the singularity exponent of the two stopes are calculated by this element concentration-area(C-A) method. The results show that the multifractal theory and model can organically combine spatial structure information, scale change information and anisotropy information to obtain low-grade and weak mineral resources information, and can effectively distinguish complex and superimposed anomalies. This will provide a basis for local concentration and spatial variation rules of uranium distribution and the design of the parameters of leaching uranium mining well site.

Urban morphology exhibits fractal characteristics, which can be described by multifractal scaling. Multifractal parameters under positive moment orders primarily capture information about cen-tral areas with relatively stable growth, while those under negative moment orders mainly reflect information about marginal areas with more active growth. However, effectively utilizing mul-tifractal spectrums to uncover the spatio-temporal variations of urban growth remains a challenge. To addresses this issue, this paper proposes a multifractal measurement by combining theoretical principles and empirical analysis. To capture the difference between growth stability in central areas and growth activity in marginal areas, an index based on generalized correlation dimension Dq is defined. This index takes the growth rate of Dq at extreme negative moment order as the numerator, and that at extreme positive moment order as the denominator. During the stable stage of urban growth, the index demonstrates a consistent pattern over time. While during the active stage, the index may exhibit abnormal fluctuations or even jumps. This indicates that the index can reveal spatio-temporal information about urban evolution that cannot be directly observed through multifractal spectrums alone. By integrating this index with multifractal spectrums, we can more comprehensively characterize the evolutionary characteristics of urban spatial structure.

The aim of this paper is to investigate whether multifractal analysis can be applied to study choroidal blood vessels and help ophthalmologists for the early diagnosis of retinitis pigmentosa (RP). In a case study, we use Spectral Domain Optical Coherence Tomography (SDOCT), which is a non-invasive and highly sensitive imaging technique of the retina and choroid. The image of a choroidal branching pattern can be regarded as a multifractal. Therefore, we calculate generalized Renyi point-centered dimensions, which is considered a measure of inhomogeneity of data, to prove that it increases in the patients with RP with respect to the control group.

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, 5,376 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. The 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.

In many soil processes, including solute and gas dynamics, the architecture of intra-aggregate pores is a crucial component. Soil management practices and wetting-drying (W-D) cycles—the latter having a significant impact on pore aggregation—are two key factors that shape pore structure. This study examines the effects of W-D cycles on the architecture of intra-aggregate pores under three different soil management systems: no-tillage (NT), minimum tillage (MT), and conventional tillage (CT). The soil samples were subjected to 0 and 12 W-D cycles, and the resulting pore structures were scanned using X-ray micro-computed tomography, generating reconstructed 3D volumetric data. The analysis was conducted in terms of multifractal spectra, normalized Shannon entropy, lacunarity, porosity, anisotropy, connectivity, and tortuosity. The morphological and geometric properties of the soil pores indicated that W-D cycles did not cause significant changes across the management systems studied. Furthermore, multifractal analysis revealed that the porous systems exhibited fractal behavior rather than multifractality. The results demonstrate that, within the resolution limits of the microtomography analysis, pore architecture remained resilient to changes, despite some observable trends in specific parameters.