Elliptical Distributions Research Articles

Some results on multivariate measures of elliptical and skew-elliptical distributions: higher-order moments, skewness and kurtosis

<abstract><p>The kurtosis and skewness of distributions are important measures that can describe the shape of a distribution, and there have been many results for symmetric distributions, but there are still many difficulties and challenges in the characterization of skew distributions. Based on the results of Mardia's and Song's kurtosis measures of elliptical distributions obtained by Zografos <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, we generalize the results and study some measures for elliptical and skew-elliptical distributions. We also derive the expressions of moments of skew-elliptical distributions in terms of the ones of skew-normals and take skew-$ t $, skew-Pearson type Ⅶ and skew-Pearson type Ⅱ distributions as examples.</p></abstract>

Analysis of Coupling Efficiency in Optic Gyroscope Transceiver Module With Elliptical Core Fiber

Improving the coupling efficiency of the optical transceiver module in fiber optic gyroscopes (FOGs) benefits both the working efficiency of the module and the signal-to-noise ratio of the FOG system. In this work, a single-mode fiber (SMF) with an elliptical mode field distribution is proposed as the coupling fiber for the FOG transceiver module. The coupling efficiency of the transceiver module using fibers with different mode field distributions at the endface is analyzed. Two structure schemes of the transceiver module are compared: one focuses the beam directly onto the endface of the fiber, and the other collimates the beam and then converge it. The optimal coupling efficiency can reach above 91.5% when the horizontal and vertical mode field radii of the fiber endface are approximately 4.05 μm and 3.37 μm respectively. The influence of the axial and angular alignment errors of the pigtail on the coupling efficiency is also investigated. The maximum tolerance of the axial alignment reaches approximately ±40.10 μm and ±30.08 μm for the two structure schemes. The maximum tolerance of the angular alignment reaches approximately ±4.4° and ±5.2° in the horizontal direction, and approximately ±3.5° and ±3.6° in the vertical direction, respectively. These tolerances are favorable for practical applications.

R-NL: Covariance Matrix Estimation for Elliptical Distributions Based on Nonlinear Shrinkage

We combine Tyler's robust estimator of the dispersion matrix with nonlinear shrinkage. This approach delivers a simple and fast estimator of the dispersion matrix in elliptical models that is robust against both heavy tails and high dimensions. We prove convergence of the iterative part of our algorithm and demonstrate the favorable performance of the estimator in a wide range of simulation scenarios. Finally, an empirical application demonstrates its state-of-the-art performance on real data.

Tests for high-dimensional single-index models

In this paper, we aim to test the overall significance of regression coefficients in high-dimensional single-index models. We first reformulate the hypothesis testing problem under elliptical distributions for predictors. Applying distribution-based transformation, we introduce a high-dimensional score-type test statistic. Notably, no moment condition for the error term is required. Our introduced procedures are thus robust with respect to outliers in response. Moreover our procedure is free of variance estimation of the error term. We establish the test statistic’s asymptotic normality under null hypothesis. Power analysis is also investigated. To further improve computational efficiency and enhance empirical powers, we also introduce a two-stage test procedure under ultrahigh-dimensional settings based on random data splitting. To eliminate the additional randomness induced by data splitting, we further develop a powerful ensemble algorithm based on multiple data splitting. We show that the ensemble algorithm can control the type I error rate at a given significance level. Extension to partial significance testing problem is also investigated. Lastly, numerical studies and real data analysis are conducted to compare with existing approaches and to illustrate the robustness and validity of our proposed test procedures.

A Robust and Flexible EM Algorithm for Mixtures of Elliptical Distributions with Missing Data

This paper tackles the problem of missing data imputation for noisy and non-Gaussian data. A classical imputation method, the Expectation Maximization (EM) algorithm for Gaussian mixture models, has shown interesting properties when compared to other popular approaches such as those based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -nearest neighbors or on multiple imputations by chained equations. However, Gaussian mixture models are known to be non-robust to heterogeneous data, which can lead to poor estimation performance when the data is contaminated by outliers or have non-Gaussian distributions. To overcome this issue, a new EM algorithm is investigated for mixtures of elliptical distributions with the property of handling potential missing data. This paper shows that this problem reduces to the estimation of a mixture of angular Gaussian distributions under generic assumptions ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , each sample is drawn from a mixture of elliptical distributions, which is possibly different for one sample to another). In that case, the complete-data likelihood associated with mixtures of elliptical distributions is well adapted to the EM framework with missing data thanks to its conditional distribution, which is shown to be a multivariate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -distribution. Experimental results on synthetic data demonstrate that the proposed algorithm is robust to outliers and can be used with non-Gaussian data. Furthermore, experiments conducted on real-world datasets show that this algorithm is very competitive when compared to other classical imputation methods.

EARL: An Elliptical Distribution Aided Adaptive Rotation Label Assignment for Oriented Object Detection in Remote Sensing Images

EARL: An Elliptical Distribution Aided Adaptive Rotation Label Assignment for Oriented Object Detection in Remote Sensing Images

The density of the sample correlations under elliptical symmetry with or without the truncated variance-ratio

The probability density function (pdf) of the sample correlation coefficient, when the associated sample variance-ratio is variously truncated, is given for the bivariate elliptical distribution (elliptical symmetry) with some moments. It is derived that the joint pdf of the sample variance-ratios and correlation matrix with possible truncation under multivariate elliptical symmetry is unchanged irrespective of distinct elliptical distributions. A general condition for transformed sample variances and covariances to have an unchanged pdf under elliptical symmetry is given. Examples satisfying this condition are shown for the intraclass correlation, coefficient alpha and principal component analysis.

Multisensor Estimation Fusion on Statistical Manifold.

In the paper, we characterize local estimates from multiple distributed sensors as posterior probability densities, which are assumed to belong to a common parametric family. Adopting the information-geometric viewpoint, we consider such family as a Riemannian manifold endowed with the Fisher metric, and then formulate the fused density as an informative barycenter through minimizing the sum of its geodesic distances to all local posterior densities. Under the assumption of multivariate elliptical distribution (MED), two fusion methods are developed by using the minimal Manhattan distance instead of the geodesic distance on the manifold of MEDs, which both have the same mean estimation fusion, but different covariance estimation fusions. One obtains the fused covariance estimate by a robust fixed point iterative algorithm with theoretical convergence, and the other provides an explicit expression for the fused covariance estimate. At different heavy-tailed levels, the fusion results of two local estimates for a static target display that the two methods achieve a better approximate of the informative barycenter than some existing fusion methods. An application to distributed estimation fusion for dynamic systems with heavy-tailed process and observation noises is provided to demonstrate the performance of the two proposed fusion algorithms.

Semiparametric estimation of the high-dimensional elliptical distribution

This paper investigates semiparametric estimation of the multivariate elliptical distribution in case dimensionality increases with the sample size. We prove the almost sure convergence and derive the convergence rates of the estimator, depending on the sample size, dimensionality, and the bandwidth of the kernel. As an important by-product, we show almost sure convergence with the corresponding convergence rates for the sample covariance matrix under the Frobenius norm. An extensive simulation study has supported the theory.

Analytical Solutions of Steady a Seepage Field for Deep-Buried Tunnel with Grouting Ring Considering Anisotropic Flow

Difficulties related to seepage are frequently encountered in tunnel design and construction, especially in deep-buried tunnels. Nowadays, analytical solutions of steady seepage fields for deep-buried tunnel usually assume that the surrounding rock mass is homogeneous. In this study, analytical solutions of a steady seepage field for a deep-buried tunnel with grouting ring considering anisotropic flow are proposed. The proposed analytical solutions are verified by numerical simulations and parameter analysis are carried out. Results show that the seepage field of surrounding rocks around the deep buried circular tunnel is no longer uniformly distributed and presents elliptical distribution. The change of permeability coefficient of the lining structure has a great influence on the hydraulic head when the difference between permeability coefficient of lining structure and surrounding rock is not very large. The results show that the size of the grouting ring has more significant influence on the grouting effect.

Tidal Effect on Grouting in Karst Fracture with Flowing Water: Experimental Investigation and Its Application

To address the problem on grouting control of water inflow disaster in coastal area, a caisson-type grouting experimental system of a single fracture in flowing water under tidal action was developed independently. Through the tide simulation system, the tidal action of different intensity and frequency was effectively described. Tide simulation system, fracture system, grouting system, water supply system and information monitoring system were combined to realize the visualization of the whole grouting process under tidal action. Furthermore, the grouting experiments of a plate fracture under different tidal action were developed and the blocking mechanism under tidal action was revealed. The conclusions were as follow: 1) Under constant water head, the slurry had three typical diffusion patterns, which were circular distribution, double U-shaped distribution and elliptical distribution respectively. 2) Under tidal action, the pressure showed a periodic fluctuation growth, and the growth rate slowed down with the increase of fracture aperture. 3) Strong tidal action would promote the diffusion of and increased the outflow amount of slurry, which was not conducive to grouting. Based on the water inflow treatment project of limestone mine in Guangxi, Through the field grouting tests, the 2# water inflow point was blocked. The research results have a certain reference value for the technology optimization of grouting engineering in coastal areas.

Aerodynamic Shape Optimisation Using Parametric CAD and Discrete Adjoint

This paper presents an optimisation framework based on an open-source CAD system and CFD solver. In this work, the high-fidelity flow solutions and surface sensitivities are obtained using the primal and discrete adjoint formulations of SU2. This paper shows the direct use of CAD models for optimisation by developing a CAD system application programming interface and creating a link between the CAD-MESH-CFD analysis. A methodology to obtain geometric sensitivities is introduced, enabling the calculation of accurate gradients with respect to CAD variables and the deformation of the analysis mesh during the optimisation process. This methodology guarantees that the new surface mesh lies exactly on the CAD geometry. The optimisation framework is applied to a rectangular wing and a three section high-lift aerofoil configuration derived from the NASA CRM-HL configuration. Both geometries are created using FreeCAD. The performance objectives are to decrease the drag while constraining the lift to be above a desired value. The twist distribution of the wing was parameterised within the CAD system, allowing the minimisation of the induced drag by obtaining a nearly elliptical lift distribution. For the high-lift configuration, the position and rotation of the flap and slat were parameterised with respect to the original section; the final optimised positions yield a drag reduction of approximately 16.5%. These results show that the CAD parameterisation can be reliably used to obtain efficient optimums while operating directly on the CAD geometries.

Constructing portfolios using stable distributions: The case of S&P 500 sectors exchange-traded funds

Portfolio construction is an important practical problem in finance. In the traditional approach, introduced by Markowitz, one assumes normally distributed returns and constructs a portfolio with a minimum risk (measured by the standard deviation of portfolio returns) for a specified (and minimally acceptable) return.In practice, returns are not normally distributed and have heavy tails. As a result, the normality assumption severely underestimates risk. It has been long suggested that a more appropriate way is to model returns by using alpha-stable distributions.In this paper, we use elliptical stable distributions for optimal stable portfolio construction. We illustrate this by considering portfolios from S&P 500 sector exchange-traded funds (ETFs).Our main results indicate that stable portfolios are in general comparable with standard Markowitz portfolios. But we discovered a few properties of stable distributions that are promising for optimal portfolio selection problem. Stable risk estimation for next year is much closer to the actual portfolio risk, stable portfolios are more balanced, so they better handle maximum restriction on component weights, and stable portfolios are more resilient to extreme market conditions.Both stable and normal optimal portfolios outperform S&P 500 index and equally-weighted ETF portfolio in most years.We propose an investment strategy that uses both stable and normal distributions for building optimal portfolios. The return of this strategy exceeds the return of the strategies which use only stable or only normal distributions.

Centralized and Distributed Robust State Estimation Over Sensor Networks Using Elliptical Distribution

We consider the robust state estimation over sensor networks with non-Gaussian noise, which is often encountered in many applications. Motivated by the fact that the elliptical distribution is the natural extension of the Gaussian distribution and includes a variety of non-Gaussian distributions with heavy-tailed characteristics, we here adopt the elliptical distribution to model the heavy-tailed process and measurement noise. The general state evolution model is used to replace the process equation and the elliptical distribution is denoted as a Gaussian mixture form. Based on that, the posterior estimation of the system state together with the parameters of the process and measurement noises can be inferred by the variational Bayes method. Moreover, the corresponding distributed estimation algorithm is then provided, which enables distributed implementation. The target tracking over sensor networks is used to show the estimation accuracy of the proposed algorithms.

Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions

Let X be an M×N random matrix consisting of independent M-variate elliptically distributed column vectors x1,…,xN with general population covariance matrix Σ. In the literature, the quantity XX∗ is referred to as the sample covariance matrix after scaling, where X∗ is the transpose of X. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of XX∗ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distributions that x1,…,xN follow as M,N→∞ with M∕N→ϕ0>0 if the weak fourth moment of the radius of x1 exists. In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled largest eigenvalue is the celebrated Tracy-Widom law.

The Theoretical Value for the Tip Radius of Cracks and Notches

In this paper, using Creager and Paris's blunt elliptical hole stress distribution area equation, it is applied to crack and circular hole shaped defects using the theoretical radius value, which equalizes the maximum stress at the defect tip in terms of value to fracture toughness. By providing value equality, critical fracture stresses of all defect dimensions and tensile strength of the material were determined with a single mechanical test data. Compared with the predictions of other methodologies, it was determined that the obtained data gave results closer to the experimental values.

Robust tests for scatter separability beyond Gaussianity

Separability (a Kronecker product) of a scatter matrix is one of favorable structures when multivariate heavy-tailed data are collected in a matrix form, due to its parsimonious representation. However, little attempt has been made to test separability beyond Gaussianity. In this paper, we present nonparametric separability tests that can be applied to a larger class of multivariate distributions not only including elliptical distributions but also generalized elliptical distributions and transelliptical distributions. The proposed test statistic exploits robustness of Tyler's M (or Kendall's tau) estimator and a likelihood function of a scaled variable. Since its distribution is hard to specify, we approximate the p-value using a permutation procedure, whose unbiasedness is obtained from the permutation invariance of multivariate paired data. Our simulation study demonstrates the efficacy of our method against other alternatives, and we apply it to rhesus monkey data and corpus callosum data.

The Lensed Lyman-Alpha MUSE Arcs Sample (LLAMAS)

Aims. We present the Lensed Lyman-Alpha MUSE Arcs Sample (LLAMAS) selected from MUSE and HST observations of 17 lensing clusters. The sample consists of 603 continuum-faint (−23 &lt; MUV &lt; −14) lensed Lyman-α emitters (producing 959 images) with secure spectroscopic redshifts between 2.9 and 6.7. Combining the power of cluster magnification with 3D spectroscopic observations, we were able to reveal the resolved morphological properties of 268 Lyman-α emitters. Methods. We used a forward-modeling approach to model both Lyman-α and rest-frame UV continuum emission profiles in the source plane and measure spatial extent, ellipticity, and spatial offsets between UV and Lyman-α emission. Results. We find a significant correlation between UV continuum and Lyman-α spatial extent. Our characterization of the Lyman-α halos indicates that the halo size is linked to the physical properties of the host galaxy (SFR, Lyman-α equivalent width, Lyman-α line FWHM). We find that 48% of Lyman-α halos are best fit by an elliptical emission distribution with a median axis ratio of q = 0.48. We observe that 60% of galaxies detected both in UV and Lyman-α emission show a significant spatial offset (ΔLyα − UV). We measure a median offset of ΔLyα − UV = 0.58 ± 0.14 kpc for the entire sample. By comparing the spatial offset values with the size of the UV component, we show that 40% of the offsets could be due to star-forming sub-structures in the UV component, while the larger offsets (60%) are more likely due to greater-distance processes such as scattering effects inside the circumgalactic medium or emission from faint satellites or merging galaxies. Comparisons with a zoom-in radiative hydrodynamics simulation of a typical Lyman-α emitting galaxy show a very good agreement with LLAMAS galaxies and indicate that bright star-formation clumps and satellite galaxies could produce a similar spatial offset distribution.

Beam- and Band-Width Broadening of Intelligent Reflecting Surfaces Using Elliptical Phase Distribution

Reflectarray surfaces (RASs) can be useful as passive repeaters or intelligent reflecting surfaces (IRSs) in communication networks by providing nonspecular reflection of an incident wave. To provide adequate signal at a given location, the radar cross section (RCS), and therefore physical size, of the RAS needs to be maximized. This inevitably reduces the beamwidth and bandwidth of the RCS pattern, severely limiting their practical implementation. This article considers a shaped reflector profiled as a truncated ellipse that can give a widened “equi-ripple” beam that addresses these problems. It is shown that by careful selection of the phase profile, the beamwidth can be expanded by a factor of 3.5 (or even 7.5 if lower roll-off in the RCS pattern is allowed) with a corresponding increase in the useable bandwidth. The tradeoff is a reduction in the peak RCS. By increasing the physical size of the reflector, this loss of RCS can be compensated for, still resulting in at least 1.8 times the beamwidth of a flat reflector. The shaped reflector can be realized as an RAS using a closed-form expression. A typical design at 28 GHz is considered. In principle a 22 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times22$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.4\lambda $ </tex-math></inline-formula> spaced array offers illumination from 40° to 50° over 26–32 GHz, considerably better than the unshaped beam.

Experimental investigation on three-dimensional structures of wind wave surfaces

This study experimentally investigated the three-dimensional features of wind wave surfaces in a laboratory. Free surfaces of wind waves under five types of wind velocities at four wind fetches were reconstructed by a stereo camera system. The dispersion law of wind waves was obtained from the spatio–temporal spectrum and showed a Doppler shift owing to the drift current. The wavenumber spectrum in the crosswind and wind direction follow the same power law of k−3 in the equilibrium range. Dynamic mode decomposition (DMD) was used to investigate the three-dimensional structures of the wave surface. Two different modes—the parallel mode and the net mode—were defined to characterize the structure of the wave surface. The parallel mode represents the average wave behavior over time, whereas the net mode represents the three-dimensional characteristics of the wind waves. The ratio of the wave number in the crosswind to the wind direction decreases with increasing wind fetch and converges to approximately 0.5. The wind wave spectrum in the wavenumber space had an elliptical distribution. Using the dispersion law, the wind wave spectrum in the (ω,k) domain can be predicted by a quartic surface.