Joint Multifractal Analysis Research Articles

A Joint Multifractal Approach to Solar Wind Turbulence

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

Relações de escala de artrópodes do solo e a estrutura da vegetação de fitofisionomias de Cerrado

ABSTRACT The objective of this study was to assess the scale relationships of soil arthropods and the vegetation structure of Cerrado phytophysiognomies. The experimental plots were set in areas with dense Cerrado (T1), typical Cerrado (T2), and sparse Cerrado (T3). The edaphic fauna was collected at 128 points through pitfall traps, and the vegetation was evaluated in subplots of 9 m2. The data were evaluated using descriptive statistics, geostatistics, multifractal analysis, and joint multifractal analysis. The soil arthropods and vegetation structure were adjusted to a geostatistical model, except for arborescent plants (T1) and arthropod abundance and arboreal plants (T2), which showed a pure nugget effect. The studied variables showed different degrees of multifractality. The graphs of joint multifractal dimension showed circular lines with high values of joint correlation for the pairs of arthropod richness versus the abundance of plant strata (r = -0.498), arthropod richness versus herbaceous plants (r = 0.323), and arthropod richness versus arboreal plants (r = 0.451) at T1. The soil fauna was influenced by the composition of the plant strata. The plots with dense Cerrado (T1) and sparse Cerrado (T3) showed the greatest spatial dependence between the samples. The multifractal analysis showed that the plot with sparse Cerrado (T3) had the greatest heterogeneity of measurement along the geometric support. In contrast, the greatest asymmetry of the singularity spectrum was described for the plot with dense Cerrado (T1). The use of geostatistical and multifractal analysis tools enabled us to characterize the scale relationships between the variables.

Multifractal and joint multifractal analysis of soil invertebrate fauna, altitude, and organic carbon

ABSTRACT The objectives of this study were to evaluate the degree of multifractality of the spatial distribution of altitude, organic carbon concentration, and invertebrate fauna diversity, and to characterize the degree of joint multifractal association among these variables. Soil sampling was performed every 20 m across a 2,540 m transect, with a total of 128 sampling points in a sugarcane area in Goiana municipality, Pernambuco State. For each sampling point, the altitude, organic carbon concentration, and macrofauna diversity (diversity indices and functional groups) were evaluated. Spatial distributions of altitude, organic carbon concentration, and macrofauna diversity were characterized by the generalized dimension spectrum (Dq) and singularity spectrums [f(α) versus α], which presented multifractal behavior with different degrees of heterogeneity in scales. Joint multifractal analysis was useful for revealing the relationships at multiple scales between the studied variables, as demonstrated by the non-detected associations using traditional statistical methods. To quantify the spatial variability of edaphic fauna based on the multiple scales and association sets in the joint dimension, the impact of agricultural production systems on biological diversity can be described. All of the studied variables displayed a multifractal behavior with greater or lower heterogeneity degree depending on the variable, with altitude and organic carbon being the most homogeneous attributes.

TESTING FOR EFFECTS OF CROSS-CORRELATIONS ON JOINT MULTIFRACTALITY

Joint multifractality has been studied in many areas of applied sciences, but few studies focused on the sources of joint multifractality, especially the cross-correlations between two records which play a crucial role in the presence of joint multifractality. To test the effect of cross-correlations on joint multifractality, we propose a de-cross-correlation method via the shifting technique based on the framework of multifractal detrended cross-correlation analysis (MF-DCCA), namely, multifractal detrended de-cross-correlation analysis (MF-DDA). It keeps the original detrending procedure of MF-DCCA. The proposed method is validated via some simulation and real data including multifractal random walk (MRW), daily water levels, and daily stock returns. Results of validation show that the MF-DDA can detect the effects of cross-correlations between simulation series and between real series, and are consistent with the simulation setting of MRW and the actual situation of real data. This proposed method can be extended to other similar joint multifractal analysis methods.

A MIXED MULTIFRACTAL ANALYSIS FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.

Coupled multifractal methods to reveal changes in nitrogen dioxide and tropospheric ozone concentrations during the COVID-19 lockdown

Due to COVID-19 pandemic, the lockdown effects on air pollution level are undeniable. Several studies around the world have detected an uneven behaviour in tropospheric ozone (O3) concentrations. In this work, Seville (Spain) is used as example of faced to traffic place in which the nitrogen dioxide (NO2) is drastically reduced (41%) while O3 has no significant changes. In order to evaluate the existence of differences in O3 behaviour that is not detected by statistical procedures, a multifractal approach was used to assess the coupled scale relationship between NO2 and O3 during the 2020 lockdown against a period reference (2017–2019). For this purpose, the two main coupled multifractal method were employed: multifractal detrended cross-correlation and joint multifractal analysis. While cross-correlation analysis did not detect differences between the cross-correlated fluctuations of NO2 and O3 in the periods analysed, the joint multifractal analysis, based on the partition function and the method of moments, found a loss of variability in O3 during the lockdown. This leads to a loss of multifractal characteristic of O3 time series. The drastically reduction of primary pollutants during the lockdown might be the responsible of the tendency to monofractality in O3 time series. These differences were found for a wide temporal extent ranging from 80 min to ~28 days.

Joint Multifractal Analysis and Source Testing of River Level Records Based on Multifractal Detrended Cross-Correlation Analysis

The joint multifractal analysis is usually conducted in two different variables for their cross-correlations but rarely used for two records of one variable collected at two different places. It is important for the detection of change in multifractality in space. Besides, the cross-correlations in two analyzed series make the analysis of sources of joint multifractality difficult. There are few studies on the source of joint multifractality. We focus on the two issues for two level records at pairs of adjacent sites along one river and carry out an extension of our previous work which is about the single multifractality of one record with the same data set. The data set is collected from 10 observation stations of a northern China river and contains about two million high-frequency river level records. Results of joint multifractal analysis via multifractal detrended cross-correlation analysis show that the change in joint multifractality at pairs of adjacent sites caused by weak cross-correlations can be detected by comparing the single generalized Hurst exponent with the joint scaling exponent function and reveal the effects of human activities on joint multifractality. This analysis provides an approach for detecting the change in multifractality. Following the idea of our previous work, two robust hypothesis tests via a set of pairs of surrogate series are proposed for the source testing of joint multifractality. The analysis of the effects of cross-correlations is carried out via a proposed simultaneously half-shifting technique which can both minimize the cross-correlations between original series and make full use of records. Results of source analysis show not only the effects of autocorrelations in series and probability distribution of river levels but also the effects of cross-correlations between series.

Multifractal and joint multifractal analysis of soil micronutrients extracted by two methods along a transect in a coarse textured soil

AbstractUnderstanding the spatial behaviour of soil nutrients is essential for fertilizer management. Traditionally, comparison of soil testing methods has been performed using simple correlation analysis. The aims of this work were: (a) to characterize patterns of spatial variability of micronutrient concentrations obtained by two different soil testing methods, using single multifractal spectra, and (b) to compare the scale‐dependent relationship between the two datasets by joint multifractal analysis. The soil study site was located in Pernambuco state, Brazil. A 384‐m transect was marked every 3 m on an Orthic Podzol, cropped to sugar cane, in Pernambuco state, Brazil. Both, Mehlich‐3 and DTPA‐extracted micronutrients were measured in each of the 132 collected soil samples. The spatial distributions of available Fe, Mn, Cu and Zn could be fitted with multifractal models, regardless of the soil testing method. The generalized dimension, Dq versus q, and singularity spectra, f(α) versus α, showed a multifractal nature. Except for Fe, the scaling heterogeneity of the spatial distributions was higher for the Mehlich‐3 than for the DTPA datasets. Moreover, Zn exhibited the highest multifractality in both cases. Joint multifractal analysis demonstrated strong positive correlations between the scaling indices of the studied micronutrients extracted by the two methods. There were, however, different degrees of association between the scaling indices of these micronutrients, which ranked as: Zn > Cu > Fe > Mn. Except for Mn, Pearson product moment correlations between concentrations of available micronutrients extracted by the two solutions at the single or point measurement scale were weaker than those of the respective scaling indices at multiple scales. Therefore, single multifractal analysis captured a great complexity in the spatial distribution of available Fe, Mn, Cu and Zn. Knowledge of the scaling behaviour of these available nutrients could be useful for transferring information about variability from one scale to another for soil fertility management. This study suggests that correlation analysis at the single or point measurement scale may not be sufficient to fully characterize the concentrations of microelements extracted by two different methods, whereas joint multifractal analysis showed potential to improve calibration of soil micronutrient testing.Highlights Mehlich‐3 and DTPA‐extracted micronutrients showed high statistical variability across a transect. All the spatial distributions of Fe, Mn, Cu and Zn displayed a well‐defined multifractal structure. Joint correlations of extracted micronutrients were stronger than single statistical correlations. Our approach fully characterized complex spatial distributions arising from nonlinear processes.

Temporal scaling study of particulate matter ([formula omitted]) and solar radiation influences on air temperature in the Caribbean basin using a 3D joint multifractal analysis

In literature, many studies have highlighted the health impact of African dust. To our knowledge, few of them have analyzed the impact of mineral dust on the climate of the Caribbean basin in a multifractal way. In this study, the joint multifractal analysis was performed for three variables coexisting in the same geometric support in order to outline the influence of Particulate Matter (PM10) and Solar Radiation (SR) on air Temperature (TEMP) over a wider range of temporal scales and exponents order. To conduct this analysis, six years of daily time series recorded in Guadeloupe archipelago were used. A 3D joint multifractal analysis was conducted by considering both the strange attraction formalism and the method of moments. The scale dependence among the studied variables was highlighted and the capability of joint multifractal analysis to completely characterize the complex temporal geometry among these non linear natural phenomena was demonstrated. Saharan dust seasonality strongly influenced TEMP values across analyzed temporal scales. We found that multifractality degree is more significant for high dust season. In addition, the single multifractal spectrum always showed the monofractal pattern of TEMP while the fractal dimension differs according to dust seasonality and explanatory variable values, i.e. PM10 and SR. All these results have clearly highlighted the strong influence of PM10 concentration and SR intensity on TEMP value.

Joint multifractal analysis of air temperature, relative humidity and reference evapotranspiration in the middle zone of the Guadalquivir river valley

Previous works have analysed the relationship existing between reference evapotranspiration (ET0) and other climatic variables under a one-at-a-time perturbation condition. However, due to the physical relationships between these climatic variables is advisable to study their joint influence on ET0. The box-counting joint multifractal algorithm describes the relations between variables using relevant information extracted from the data singularities. This work investigated the use of this algorithm to describe the simultaneous behaviour of ET0, calculated by means of Penman–Monteith (PM) equation, and the two main climatic variables, relative humidity (RH) and air temperature (T), influencing on it in the middle zone of the Guadalquivir river valley, Andalusia, southern Spain. The studied cases were grouped according to the fractal dimension values, obtained from the global multifractal analysis, which were related to their probability of occurrence. The most likely cases were linked to smooth behaviour and weak dependence between variables, both circumstances were detected in the local multifractal analysis. For these cases, the rest of Penman Monteith (PM) equation variables, neither the T nor the RH, seemed to influence on ET0 determination, especially when low T values were involved. By contrast, the least frequent cases were those with variables showing high fluctuations and strong relationship between them. In these situations, when T is low, the ET0 is affected by the rest of PM equation variables. This fact confirmed T as main driver of ET0 because the higher T values the lesser influence of other climate variables on ET0. This condition could not be extended to RH because the variability in ET0 singularities was not significantly influenced by low or high values of this variable. These results show that the joint multifractal analysis can be regarded as a suitable tool for describing the complex relationship between ET0, T and RH, providing additional information to that derived from descriptive statistics.Although, joint multifractal analysis shows some limitations when it is applied to large number of variables, the results reported are promising and suggest the convenience of exploring the relationships between ET0 and other climatic variables not considered here with this framework such as wind speed and net radiation.

A joint multifractal analysis of vector valued non Gibbs measures

Abstract The multifractal formalism for measures holds whenever the existence of corresponding Gibbs-like measures supported on the singularities sets holds. In the present work we tried to relax such a hypothesis and introduce a more general framework of joint multifractal analysis where the measures constructed on the singularities sets are not Gibbs but controlled by an extra-function allowing the multifractal formalism to hold. We fall on the classical case by a particular choice of such a function. An answer to a question raised in [2] on which gauge function φ shall we get a finite, infinite or zero value of H μ , φ q , t ( K ) for the singularities set K is provided.

Joint multifractal spectrum analysis for characterizing the nonlinear relationship among hydrological variables

Joint multifractal spectrum analysis (JMS) is an effective approach for investigation of relationship among hydrological variables and is regarded as one potential way to reveal nonlinear relationship in hydrology. In this study, the joint multifractal spectrum analysis was for the first time used to analyze soil moisture-soil temperature-precipitation (SM-ST-P) relationship. Pearson’s linear correlation coefficient considering lag effects and scale effects was also applied for the aims of comparison. Soil data were collected at four sites in Tibet and three depths for each site (10 cm, 30 cm and 60 cm) at 2-hourly frequency from November 2016 to November 2017. Correlation analysis showed that the coefficients varied at every measurement point but failed to provide general and comprehensive analysis of SM-ST-P relationship. On the contrary, multifractal spectra of SM-ST-P confirmed that the relationship was strong for all measurement points. Furthermore, JMS analysis provided holistic results including verification of multifractal of SM-ST-P relationship, relationship evaluation for each part of variables separately (high value, middle value, low value), intuitive diagrammatic comparison of SM-ST and SM-P relationships, etc. Seasonal JMS analysis was adopted to investigate the freezing-thawing process in the study area as well. The multifractal spectra of frozen soil site and no frozen soil site are totally different for dry cold period but similar for rainy warm period. The difference in multifractal of SM-ST-P relationship with and without frozen soil presented the latent application of JMS to quantitatively detect the process of frozen soil. This study implied the capability of JMS analysis in hydrology, especially for multivariable relationship like SM-ST-P relationship.

Multifractal and joint multifractal description of available nutrients concentrations extracted by two methods along short transects

ABSTRACTSpatial variability of available P, K, Ca and Mg, sampled along two short transects and determined by two different extraction methods, was characterized using multifractal and joint multifractal analysis. Sixty-six soil samples were collected every 0.8 m, both in a vineyard and in a polyculture field. Available nutrients were extracted with an ion exchange resin and the Mehlich 3 solution. Shape and width of singularity, f(α) versus α, and generalized dimension, Dq, spectra, showed that spatial distributions obtained by the two tests behaved as quasi-monofractals in the vineyard, but were fitted with multifractal models under polyculture. Joint multifractal analysis demonstrated positive correlations between the scaling indices of P, K, Ca and Mg, extracted by resin and Mehlich 3, which were stronger under polyculture. Pearson correlations between available nutrients determined by two tests at the single scale ranged from R2 = 0.54 to 0.95 and were weaker than those of the respective scaling indices across the range of scales studied, which varied from R2 = 0.63 to 0.97. The joint multifractal approach showed that single scale analysis may be insufficient to compare relationships between available nutrient tests. Moreover, multifractal analysis may be useful to upscaling/downscaling nutrient availability.

Joint Multifractal and Lacunarity Analysis of Image Profiles for Manufacturing Quality Control

The modern manufacturing industry faces increasing demands to customize products according to personal needs, thereby leading to the proliferation of complex designs. To cope with design complexity, manufacturing systems are increasingly equipped with advanced sensing and imaging capabilities. However, traditional statistical process control methods are not concerned with the stream of in-process imaging data. Also, very little has been done to investigate nonlinearity, irregularity, and inhomogeneity in the image stream collected from manufacturing processes. This paper presents the joint multifractal and lacunarity analysis to characterize irregular and inhomogeneous patterns of image profiles, as well as detect the hidden dynamics in the manufacturing process. Experimental studies show that the proposed method not only effectively characterizes surface finishes for quality control of ultraprecision machining but also provides an effective model to link process parameters with fractal characteristics of in-process images acquired from additive manufacturing. This, in turn, will allow a swift response to processes changes and consequently reduce the number of defective products. The proposed multifractal method shows strong potentials to be applied for process monitoring and control in a variety of domains such as ultraprecision machining and additive manufacturing.

Joint multifractal analysis for three variables: Characterizing the effect of topography and soil texture on soil water storage

Multifractal analysis describes heterogeneity in the distribution of a variable by characterizing and summarizing the variability across scales. Joint multifractal analysis has been widely employed to characterize scale relationships between two variables co-existing along a single geometric support. In this study, joint multifractal analysis was used for three variables coexisting in the same geometric support to describe the influence of topography (relative elevation) and soil texture (sand content) on water storage within a soil profile. Soil water storage, as well as sand content and relative elevation, were measured down to 1.4 m depth along a 576 m long transect in the hummocky landscape of central Saskatchewan, Canada. Joint multifractal analysis was conducted to consider both the strange attractor formalism and the method of moments. The variability in soil water storage, sand content and relative elevation was scale dependent and showed multifractal behavior. The spatial variability in relative elevation was strongly reflected on water storage across the analyzed spatial scales but the joint multifractal spectrum for sand content and water storage suggested a lower degree of correlation. The change in multifractality was also observed when there was high variability in relative elevation and texture. This clearly demonstrated the capability of joint multifractal analysis to completely characterize the scaling behavior among three variables.

Multivariate multifractal analysis

Abstract We show how a joint multifractal analysis of a collection of signals unravels correlations between the locations of their pointwise singularities. The multivariate multifractal formalism, reformulated in the general setting supplied by multiresolution quantities, provides a framework which allows to estimate joint multifractal spectra. General results on joint multifractal spectra are derived, and illustrated by the theoretical derivation and practical estimation of the joint multifractal spectra of simple mathematical models, including correlated binomial cascades.

Multifractal and joint multifractal analysis of general soil properties and altitude along a transect

Multifractal characterisation of soil spatial variability has the potential for providing a better understanding of the distribution patterns of data values, and may contribute to improved resource management. We examined the scaling heterogeneity and multiple scale relationships of soil general properties and topography using multifractal and joint multifractal techniques. Soil samples were collected down to 0.20 m depth and altitude was recorded at equal intervals of 3 m along a 396 m transect in an Orthic Podzol at Pernambuco, Brazil. Soil properties studied were: textural fractions, pH, organic carbon (OC), exchangeable cations, exchangeable acidity (H + Al), sum of bases (SB), cation exchange capacity (CEC) and percent base saturation (V). The spatial distribution of altitude and soil general properties, characterised through generalised dimension, Dq, and singularity spectra, f(α)–α, showed a well-defined multifractal structure. Notwithstanding, these variables displayed several degrees of scaling heterogeneity, which was lowest for pH, sand and clay contents and highest for exchangeable cations and silt content. Joint multifractals showed that correlations between pairs of variables may or may not be stronger at the observation scale than across a range of spatial scales. Hence, soil OC and pH showed higher relationships to CEC, (H + Al), SB and V at the observation than at multiple spatial scales, while local topography effects on pH and CEC were greater at multiple scales. Multifractal and joint multifractal analysis provided new insights to characterise the spatial patterns and the relationships between soil properties at multiple scales, and to evaluate the effect of topography on soil heterogeneity.

Joint multifractal analysis based on wavelet leaders

Mutually interacting components form complex systems and the outputs of these components are usually long-range cross-correlated. Using wavelet leaders, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call joint multifractal analysis based on wavelet leaders (MF-X-WL). We test the validity of the MF-X-WL method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. Both experiments indicate that MF-X-WL is capable to detect the cross correlations in synthetic data with acceptable estimating errors. We also apply the MF-X-WL method to the pairs of series from financial markets (returns and volatilities) and online worlds (online numbers of different genders and different societies) and find an intriguing joint multifractal behavior.

Evaluation of Infiltration Models and Variability of Soil Infiltration Properties at Multiple Scales

AbstractSoil infiltration is characterized by scale dependence, so the accuracy and reliability of indirect methods used to estimate soil infiltration properties from available soil physical data are often below expectations. This study aimed to determine the scale‐dependent relationship between soil infiltration and soil physical properties and to establish pedotransfer functions (PTFs) for estimating normalization factors (FC). Field studies were conducted using double‐ring infiltrometers in the first (sandy loam) and third (clay loam) terraces in Yangling District. Multiple scale variability and relationships were studied by multifractal and joint multifractal techniques. Results indicated that the Kostiakov–Lewis model best described the relationship between cumulative infiltration and time in the study areas, and that the normalization equation for predicting cumulative infiltration was accurate. Joint multifractal analysis showed that the variability of FC in the first terrace was related largely to sand content, soil bulk density, and silt content. FC in the third terrace depended on the contents of sand, silt, initial soil water, and clay at multiple scales. The PTFs that were established based on the results of joint multifractal analysis reliably estimated FC values. Copyright © 2017 John Wiley & Sons, Ltd.

Asymmetric joint multifractal analysis in Chinese stock markets

In this paper, the asymmetric joint multifractal analysis method based on statistical physics is proposed to explore the asymmetric correlation between daily returns and trading volumes in Chinese stock markets. The result shows asymmetric multifractal correlations exist between return and trading volume in Chinese stock markets. Moreover, when the stock indexes are upward, the fluctuations of returns are always weaker than when they are downward, whether the trading volumes are more or less.