Elliptical Distributions Research Articles

Exploiting the diversity of modeling methods to probe systematic biases in strong lensing analyses

Challenges inherent to high-resolution and high signal-to-noise data as well as model degeneracies can cause systematic biases in analyses of strong lens systems. In the past decade, the number of lens modeling methods has significantly increased, from purely analytical methods, to pixelated and non-parametric ones, or ones based on deep learning. We embraced this diversity by selecting different software packages and use them to blindly model independently simulated Hubble Space Telescope (HST) imaging data. To overcome the difficulties arising from using different codes and conventions, we used the COde-independent Organized LEns STandard (COOLEST) to store, compare, and release all models in a self-consistent and human-readable manner. From an ensemble of six modeling methods, we studied the recovery of the lens potential parameters and properties of the reconstructed source. In particular, we simulated and inferred parameters of an elliptical power-law mass distribution embedded in a shear field for the lens, while each modeling method reconstructs the source differently. We find that, overall, both lens and source properties are recovered reasonably well, but systematic biases arise in all methods. Interestingly, we do not observe that a single method is significantly more accurate than others, and the amount of bias largely depends on the specific lens or source property of interest. By combining posterior distributions from individual methods using equal weights, the maximal systematic biases on lens model parameters inferred from individual models are reduced by a factor of 5.4 on average. We investigated a selection of modeling effects that partly explain the observed biases, such as the cuspy nature of the background source and the accuracy of the point spread function. This work introduces, for the first time, a generic framework to compare and ease the combination of models obtained from different codes and methods, which will be key to retain accuracy in future strong lensing analyses.

Maximum likelihood estimation of elliptical tail

This study is focused on the efficient estimation of the elliptical tail. Initially, we derive the density function of the spectral measure of an elliptical distribution concerning a dominating measure on the unit sphere, which consequently leads to the density function of the elliptical tail. Subsequently, we propose a maximum likelihood estimation based on the derived density function class. The resulting maximum likelihood estimator (MLE) is proven to be consistent and asymptotically normal. Moreover, it is demonstrated that the MLE is asymptotically efficient, with the added advantage that its asymptotic covariance matrix can be feasibly estimated at a low computational cost. A simulation study and real data analysis are conducted to illustrate the efficacy of the proposed method.

Investigating formability of AZ31B magnesium alloy sheet with different texture and thickness in combination of the Erichsen experiment and CPFEM model

In this study, the formability of the AZ31B magnesium alloy sheets was investigated in combination of experiment and CPFEM modelling. The deformation mechanisms during the Erichsen forming were revealed and the effect of the related parameters were discussed. It was shown that the simulation results based on the normalized Cockcroft-Latham criterion are consistent with the experimental results. The maximum stress and strain concentrate at the center of dome, and exhibit an elliptical distribution shape. The difference of asymmetry situation between the sheets results from the different textures. The basal <a> slip is the dominant mode, and the prismatic <a> slip is also active to accommodate plastic strain in sheet plane. The relative activity of the pyramidal <c+a> slip is low, but very effective to accommodate the plastic strain in thickness direction. The sheet with higher relative activity of the prismatic <a> and pyramidal <c+a> slips exhibit higher formability. The sheet thickness influences the formability through changing the stress-strain response, as well as the orientation-relationship between stress state and grains which can affect the deformation mechanisms. The formability depends on the plastic strain accommodation ability that is comprehensively related to the averaged plasticity, orientation-relationship between stress state and grains, r-value and n-value. CPFEM modeling is convenient pathway to predict the formability.

Early-type galaxy speciation: elliptical (E) and ellicular (ES) galaxies in the Mbh–M*,sph diagram, and a merger-driven explanation for the origin of ES galaxies, antitruncated stellar discs in lenticular (S0) galaxies, and the Sérsicification of E galaxy light profiles

ABSTRACT In a recent series of papers, supermassive black holes were used to discern pathways in galaxy evolution. By considering the black holes’ coupling with their host galaxy’s bulge/spheroid, the progression of mass within each component has shed light on the chronological sequence of galaxy speciation. Offsets between the galaxy-morphology-dependent $M_{\rm bh}$–$M_{\rm \star ,sph}$ scaling relations trace a pattern of ‘punctuated equilibrium’ arising from merger-driven transitions between galaxy types, such as from spirals to dust-rich lenticulars and further to ‘ellicular’ and elliptical galaxies. This study delves deeper into the distinction between the ellicular galaxies – characterised by their intermediate-scale discs – and elliptical galaxies. Along the way, it is shown how some antitruncated large-scale discs in lenticular galaxies can arise from the coexistence of a steep intermediate-scale disc and a relatively shallow large-scale disc. This observation undermines application of the popular exponential-disc plus Sérsic-bulge model for lenticular galaxies and suggests some past bulge mass measurements have been overestimated. Furthermore, it is discussed how merger-driven disc-heating and blending likely leads to the spheroidalisation of discs and the conglomeration of multiple discs leads to the (high-n) Sérsicification of light profiles. The ellicular and elliptical galaxy distribution in the $M_{\rm bh}$–$M_{\rm \star ,sph}$ diagram is explored relative to major-merger-built lenticular galaxies and brightest cluster galaxies. The super-quadratic $M_{\rm bh}$–$M_{\rm \star }$ relations, presented herein, for merger-built systems should aid studies of massive black hole collisions and the gravitational wave background. Finally, connections to dwarf compact elliptical and ultracompact dwarf galaxies, with their 100–1000 times higher $M_{\rm bh}/M_{\rm \star ,sph}$ ratios, are presented.

A mixture of ellipsoidal densities for 3D data modelling

We introduce a novel probability density function for modeling the distribution of points around an ellipsoidal surface. This density is part of the family of elliptical distributions. We establish the theoretical convergence properties of the parameter estimators and validate them using simulated data. Furthermore, we propose a mixture model utilizing this density, and we estimate its parameters using the Expectation-Maximization (EM) algorithm. To assess its performance, we compare the algorithm to a state-of-the-art ellipse fitting method and conduct experiments on 3D real data obtained from depth cameras.

Bivariate Tail Conditional Co-Expectation for elliptical distributions

In this paper, we consider a random vector X=(X1,X2) following a multivariate Elliptical distribution and we provide an explicit formula for E(X|X≤X˜), i.e., the expected value of the bivariate random variable X conditioned to the event X≤X˜, with X˜∈R2. Such a conditional expectation has an intuitive interpretation in the context of risk measures. Specifically, E(X|X≤X˜) can be interpreted as the Tail Conditional Co-Expectation of X (TCoES). Our main result analytically proves that for a large number of Elliptical distributions, the TCoES can be written as a function of the probability density function of the Skew Elliptical distributions introduced in the literature by the pioneering work of Azzalini (1985). Some numerical experiments based on empirical data show the usefulness of the obtained results for real-world applications.

Blood Plasma Film Multifractal Scanning in COVID-19 Consequences Diagnostics.

A 3D phase scanning method was applied to study blood plasma facies, generating layered polarization maps of the object field. The most sensitive parameters to changes in birefringence distribution were identified. Multifractal analysis using wavelet transforms and fractal dimension spectra provided specific insights into the scale self-similarity of the polarization maps. The multifractal spectra of ellipticity distributions were algorithmically derived, revealing that the third- and fourth-order statistical moments were most sensitive to changes in the supramolecular networks of the facies. These findings were successfully applied to differentiate post-COVID-19 effects with high accuracy.

Structural Deformations in Cucurbit[n]urils: Analysis, Host-Guest Dependence, and Automated Ellipticity Measurements Using ElliptiCB[n

Cucurbit[n]urils (CB[n]s) are cyclic macrocycles with rich host-guest chemistry. In many cases, guest binding in CB[n]s results in host structural deformations. Unfortunately, measuring such deformations remains a major challenge, with only a handful of manual estimations reported in the literature. To address this challenge, we have developed the public program ElliptiCB[n], which is available on GitHub, that provides a robust and automated method for measuring the elliptical deformations in CB[n] hosts. We outline the development and validation of this approach, apply ElliptiCB[n] to measure the ellipticity of the 1113 available CB[n] structures from the Cambridge Structural Database (CSD), and directly investigate the structural deformations of CB[5], CB[6], CB[7], CB[8], and CB[10] hosts. We also report the general landscape of accessible CB[n] elliptical deformations and compare ellipticity distributions across CB[n] hosts and host-guest complexes. We found that in almost all cases guest binding significantly impacts the distribution of host ellipticity distributions and that these distributions are dissimilar across host-guest complexes of differently sized CB[n]s. We anticipate that this work will provide a useful approach for understanding of the flexibility of CB[n] hosts and will also enable future measurement and standardization of ellipticity measurements of CB[n]s.

Probing gluon GTMDs of the proton in deep inelastic diffractive dijet production at HERA

We calculate several differential distributions for diffractive dijets production in ep→e′jet jetp in the perturbative QCD dipole approach using off diagonal unintegrated gluon distributions (generalized transverse momentum dependent distributions, GTMDs). Different models from the literature are used. We concentrate on the contribution from exclusive qq¯ dijets. Results of our calculations are compared to H1 and ZEUS data, including specific experimental cuts in our calculations. In general, except for one GTMD, our results are below the HERA data. The considered mechanism is expected to gives a sizeable contribution to the ZEUS data, while it is negligible in the kinematics of the H1 measurement. This is in contrast to recent results from the literature where the normalization was adjusted to some selected distributions of the H1 collaboration and no agreement with other observables was checked. The ZEUS data provide stricter limitations on the GTMDs than the H1 data. We conclude, based on comparison to different observables, that the calculated cross sections are only a small fraction of the measured ones which contain probably also processes with pomeron remnant. Alternatively one could try to explain the experimental data by inclusion of qq¯g component of the photon wave function. We present also azimuthal correlations between the sum and the difference of dijet transverse momenta. The cuts on transverse momenta of jets generate corresponding azimuthal correlations that can be misidentified as due to elliptic gluon distributions. Published by the American Physical Society 2024

Rank-based correlation matrix estimation for high dimensional microbiome data

In recent years, the progress made in high-throughout sequencing techniques supports the analysis of microbial communities greatly. The fact that real-world microbiome data lie in a high-dimensional simplex makes conventional statistical analysis unreliable owing to the sum-to-one constraint. Motivated by the urgent requirements of directly inferring the dependence pattern and dependence intensity simultaneously among microbial taxa, we address the challenge of correlation matrix estimation for this kind of data. Assuming that the unobserved log-basis random vectors follow the elliptical distribution, a robust method built on the Kendall's tau statistic is developed to estimate the basis generalized correlation matrix. Regardless of the existence of the covariance, the basis generalized correlation matrix is always a reliable measure of dependence structure. Theoretically, we establish the convergence rate of the proposed estimator under the spectral norm. Support recovery is discussed as well. Empirically, simulation studies demonstrate the outstanding numerical performance of our method. We also apply our method to analyse a microbiome dataset from human gut. We provide all the codes at https://github.com/lwfwhunanhero/CRCE.

Inter-individual variability in elliptical and diagonal error distributions potentially relevant to optimal motor planning in football instep kicking

The distribution of motor errors can influence optimal motor planning (where to aim). In football instep kicking, it was shown that ball landing locations exhibit the right-up-left-down elliptical distribution in right-footed kickers and vice versa. However, this was reported as a result of mixed multiple kickers; the individual-level error distribution has been unclear. Here we show substantial inter-individual variability in error shape and error direction in the 30 kicks aimed at a target (1.7 m high, 11.0 m in front) by 27 male football players. All players exhibit right-up-left-down distributions with ellipticity (minor/major radius ratio of the 95% confidence ellipse) ranging from 0.25 to 0.77 and major axis angle ranging from 13 to 67° from the horizontal axis. The mean absolute error and the area of the 95% confidence ellipse are not significantly correlated with major axis angle (ρ ≤ 0.312) and ellipticity (|r| ≤ 0.343). By simulating shots aimed at the top-right and top-left edges of a goal with these observed ranges and normalised ellipse area, we reveal a wide range of probability of shots on goal (top-right: 2.7-fold difference, top-left: 1.5-fold difference) due to inter-individual variability in error shape and direction independent of error size. Further simulation shows that, depending on the shape-direction combination, the aiming points with the same 80% probability of shots on goal change by up to 0.3 m vertically, even for the same minimal error size. We highlight the importance for football players to consider not only accuracy/precision, but also error shape and direction to optimise motor planning.

A general semi-parametric elliptical distribution model for semi-supervised learning

ABSTRACT This research proposes a novel semi-parametric elliptical distribution model for application in semi-supervised learning tasks. We use labelled and unlabelled data to develop a pseudo maximum likelihood method for estimation and classification. The proposed estimator outperforms the estimator based solely on labelled data and achieves the semi-parametric efficiency bound with a suitable size of unlabelled data. We efficiently maximise the objective function by utilising low-dimensional groupwise pseudo-likelihood functions in a block coordinate descent manner while ensuring numerical stability and convergence through appropriate bandwidth selectors and initial parameter estimates. Additionally, the study comprehensively investigates the impact of labelled and unlabelled data on the pseudo maximum likelihood estimator and classifier. Simulation studies and empirical data applications illustrate the superiority of our methodology.

Dynamic Evolution Law of Production Stress Field in Fractured Tight Sandstone Horizontal Wells Considering Stress Sensitivity of Multiple Media

Inter-well frac-hit has become an important challenge in the development of unconventional oil and gas resources such as fractured tight sandstone. Due to the presence of hydraulic fracturing fractures, secondary induced fractures, natural fractures, and other seepage media in real formations, the acquisition of stress fields requires the coupling effect of seepage and stress. In this process, there is also stress sensitivity, which leads to unclear dynamic evolution laws of stress fields and increases the difficulty of the staged multi-cluster fracturing of horizontal wells. The use of a multi-stage stress-sensitive horizontal well production stress field prediction model is an effective means of analyzing the influence of natural fracture parameters, main fracture parameters, and multi-stage stress sensitivity coefficients on the stress field. This article considers multi-stage stress sensitivity and, based on fractured sandstone reservoir parameters, establishes a numerical model for the dynamic evolution of the production stress field in horizontal wells with matrix self-supporting fracture-supported fracture–seepage–stress coupling. The influence of various factors on the production stress field is analyzed. The results show that under constant pressure production, for low-permeability reservoirs, multi-stage stress sensitivity has a relatively low impact on reservoir stress, and the amplitude of principal stress change in the entire fracture length direction is only within the range of 0.27%, with no significant change in stress distribution; The parameters of the main fracture have a significant impact on the stress field, with a variation amplitude of within 2.85%. The ability of stress to diffuse from the fracture tip to the surrounding areas is stronger, and the stress concentration area spreads from an elliptical distribution to a semi-circular distribution. The random natural fracture parameters have a significant impact on pore pressure. As the density and angle of the fractures increase, the pore pressure changes within the range of 3.32%, and the diffusion area of pore pressure significantly increases, making it easy to communicate with the reservoir on both sides of the fractures.

Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions

In this article, we propose the multivariate range Value-at-Risk (MRVaR) and the multivariate range covariance (MRCov) as two risk measures and explore their desirable properties in risk management. To facilitate analytical analyses, we derive explicit expressions of the MRVaR and the MRCov in the context of the multivariate (log-)elliptical distribution family. Frequently-used cases in industry, such as normal, student-t, logistic, Laplace, and Pearson type VII distributions, are presented with numerical examples. As an application, we propose a range-based mean-variance framework for optimal portfolio selection.

Adjusted location‐invariant U‐tests for the covariance matrix with elliptically high‐dimensional data

AbstractThis paper analyzes several covariance matrix U‐tests, which are constructed by modifying the classical John‐Nagao and Ledoit‐Wolf tests, under the elliptically distributed data structure. We study the limiting distributions of these location‐invariant test statistics as the data dimension may go to infinity in an arbitrary way as the sample size does. We find that they tend to have unsatisfactory size performances for general elliptical population. This is mainly because such population often possesses high‐order correlations among their coordinates. Taking such kind of dependency into consideration, we propose necessary corrections for these tests to cope with elliptically high‐dimensional data. For computational efficiency, alternative forms of the new test statistics are also provided. We derive the universal asymptotic null distributions of the proposed test statistics under elliptical distributions and beyond. The powers of the proposed tests are further investigated. The accuracy of the tests is demonstrated by simulations and an empirical study.

Failure analysis of wellhead casing and optimization of slip hanger in ultra deep wells

When replacing the sealing element and lifting the oil production spool in a certain oilfield in Xinjiang, it was found that the casing at the clamping point of the wellhead hanger slip was broken. Based on the Ф232.5 mm W-type slip hanger structure used in the X1 well on site, a three-dimensional finite element model of the casing’s grip by the slip hanger was established. Firstly, based on the third and fourth strength theories, the calculation formula for the ultimate load of the casing under the slip effect was derived. Furthermore, tensile experiments, chemical composition measurements, and microstructure analysis were conducted on the 155 V casing samples used, and it was found that the performance of the 155 V casing material meets the standard requirements, the data suggests that the slip may contribute to casing fracture, warranting further investigation. Therefore, based on the mechanical finite element analysis model, it is calculated that the maximum stress on the outer wall of the casing reaches 1146 MPa, which is higher than the yield strength of the 155 V casing. This will lead to the damage of the casing used on site. Therefore, the structure of the slip is optimized, and the analysis shows that the top of the slip tooth is optimized to have a long axis of 400 mm, the one-fourth elliptical distribution with a short axis of 1 mm and the optimization of the top angle of the slip teeth to 60°–70° will significantly decrease the maximum stress on the outer wall of the casing. However, the optimization scheme of changing the inclination angle of the slip teeth and increasing the clearance of the slip back cone angle cannot effectively improve the stress distribution and maximum value on the outer wall of the casing.

Tests for Large-Dimensional Shape Matrices via Tyler’s M Estimators

Tyler’s M estimator, as a robust alternative to the sample covariance matrix, has been widely applied in robust statistics. However, classical theory on Tyler’s M estimator is mainly developed in the low-dimensional regime for elliptical populations. It remains largely unknown when the parameter of dimension p grows proportionally to the sample size n for general populations. By using the eigenvalues of Tyler’s M estimator, this article develops tests for the identity and equality of shape matrices in a large-dimensional framework where the dimension-to-sample size ratio p/n has a limit in (0, 1). The proposed tests can be applied to a broad class of multivariate distributions including the family of elliptical distributions (see model (2.1) for details). To analyze both the null and alternative distributions of the proposed tests, we provide a unified theory on the spectrum of a large-dimensional Tyler’s M estimator when the underlying population is general. Simulation results demonstrate good performance and robustness of our tests. An empirical analysis of the Fama-French 49 industrial portfolios is carried out to demonstrate the shape of the portfolios varying. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Properties of a Lorentz correlated beam in uniaxial crystal

AbstractThe equation of a Lorentz‐correlated Schell‐model beam (LCSMB) in a uniaxial crystal is derived, and the relation of spreading properties and coherence with and are analyzed during propagation. The intensity of an LCSMB in the uniaxial crystal will evolve from Lorentz distribution into diamond distribution, and the LCSMB with smaller will evolve from Gaussian distribution into Lorentz distribution faster. The distribution of coherence of an LCSMB will lose the Lorentz distribution and become an elliptical distribution as increases.

Developing a novel Mg[sbnd]2Zn[sbnd]3Li[sbnd]1Gd alloy sheet with high room-temperature formability by introducing an elliptical texture distribution

In recent years, modification of texture distribution has been considered a valid approach to improve the room-temperature (RT) formability of magnesium (Mg) alloys. In this study, a novel Mg2Zn3Li1Gd alloy sheet with weak elliptical-texture was fabricated by cold rolling and subsequent annealing, and it showed an excellent Erichsen (IE) value near 7.1 mm. Both quasi-in-situ electron backscatter diffraction (EBSD) and transmission electron microscopy (TEM) analysis indicate that considerable basal and pyramidal dislocations can be activated in the cold rolling process. During annealing, these dislocations can induce nucleation and then cause preferential misorientation relationships around 〈uvt0〉 concerning the nuclei and parent grains, which can facilitate the formation of elliptical texture. Furthermore, the particle-stimulated nucleation (PSN) mechanism and the co-segregation of Zn and Gd at grain boundaries (GB) further weak texture intensity. Finally, the mechanical properties of the Mg2Zn3Li1Gd alloy sheet are significantly improved.

Low Surface Brightness Galaxies from BASS+MzLS with Machine Learning

From ∼5000 deg2 of the combination of the Beijing–Arizona Sky Survey and Mayall z-band Legacy Survey which is also the northern sky region of the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Surveys, we selected a sample of 31,825 candidates of low surface brightness galaxies (LSBGs) with the mean effective surface brightness 24.2<μ¯eff,g<28.8 mag arcsec−2 and the half-light radius 2.″5 < r eff < 20″ based on the released photometric catalog and the machine learning model. The distribution of the LSBGs is bimodal in the g − r color, indicating the two distinct populations of the blue (g − r < 0.60) and red (g − r > 0.60) LSBGs. The blue LSBGs appear spiral, disk or irregular while the red LSBGs are spheroidal or elliptical and spatially clustered. This trend shows that the color has a strong correlation with galaxy morphology for LSBGs. In the spatial distribution, the blue LSBGs are more uniformly distributed while the red ones are highly clustered, indicating that red LSBGs preferentially populate a denser environment than the blue LSBGs. Besides, both populations have a consistent distribution of ellipticity (median ϵ ∼ 0.3), half-light radius (median r eff ∼ 4″) and Sérsic index (median n = 1), implying the dominance of the full sample by the round and disk galaxies. This sample has definitely extended the studies of LSBGs to a regime of lower surface brightness, fainter magnitude and broader other properties than the previously Sloan Digital Sky Survey-based samples.