Elliptically Contoured Distributions Research Articles

An Outlier-Robust Kalman Filter With Adaptive Selection of Elliptically Contoured Distributions

In this paper, elliptically contoured (EC) distributions are used to model outlier-contaminated measurement noises. Exploiting a heuristic approach to introduce an unknown parameter, we present an analytical update form of the joint posterior probability density function of the state vector and auxiliary random variable, from which a novel robust EC distributions-based Kalman filtering framework is first derived. To illustrate the effectiveness of the proposed framework, the convergence, robustness, optimality and computational complexity analyses of the proposed method are then given. In addition, to cope with complex noise environments, the interaction multiple model is employed to achieve the adaptive selection of EC distributions such that well-behaved estimation performance can be obtained for different noise cases. Simulation results demonstrate the validity and superiority of the proposed algorithm.

Riemannian information gradient methods for the parameter estimation of ECD

Elliptically-contoured distributions (ECD) play a significant role, in computer vision, image processing, radar, and biomedical signal processing. Maximum likelihood estimation (MLE) of ECD leads to a system of non-linear equations, most-often addressed using fixed-point (FP) methods. Unfortunately, the computation time required for these methods is unacceptably long, for large-scale or high-dimensional datasets. To overcome this difficulty, the present work introduces a Riemannian optimisation method, the information stochastic gradient (ISG). The ISG is an online (recursive) method, which achieves the same performance as MLE, for large-scale datasets, while requiring modest memory and time resources. To develop the ISG method, the Riemannian information gradient is derived taking into account the product manifold associated to the underlying parameter space of the ECD. From this information gradient definition, we define also, the information deterministic gradient (IDG), an offline (batch) method, which is an alternative, for moderate-sized datasets. The present work formulates these two methods, and demonstrates their performance through numerical simulations. Two applications, to image re-colorization, and to texture classification, are also worked out.

Recursive estimation of the scatter matrix of ECD: the Riemannian Information Gradient method

ECD (elliptically-contoured distribution) models have been found remarkably successful in representing natural signals. At present, the estimation of these models is at the heart of numerous signal processing applications. Unfortunately, state-of-the-art methods for estimating the parameters of an ECD, especially its scatter matrix, may turn out to have excessive computational complexity. To remedy this problem, the present work introduces the Riemannian information gradient method, for recursive (i.e. online) estimation of the scatter matrix. It is shown that this method holds a significant advantage in terms of computational complexity, while still achieving the same performance as state-of-the-art methods.

Multivariate statistics and its applications

Multivariate statistics and its applications have received more and more attention since 1950s.Multivariate statistical research in China was initiated by Pao-Lu Hsu duringthe end of 1930s and the beginning of 1940s in the so-called “Southwest United University”.Modern big data analysis makes classical multivariate statistical theory unable to accurately andeffectively solve practical problems. The theory of generalized multivariate statistics has beendeveloping very fast since 1970s. This paper aims to introduce the summary contributions that Chinesescholars have made in the development of generalized multivariate statistics and its applications inseveral aspects: (1) multivariate statistics and generalized multivariate statistics; (2) general symmetricmultivariate distributions; and (3) growth curve modeling and miscellaneous directions.Generalized multivariate statistics is an extension to the traditional statistics under the normalassumption. It aims to generalize the traditional statistical methodologies like parametric estimation,hypothesis testing, and modeling to a much wider family of multivariate distributionsthat are called elliptically contoured distributions (ECDs). General symmetric multivariate distributionsform an even wider class of multivariate probability distributions that includes the ECDs as its special case.Growth curve modeling (GCM) includes statistical methods that allow for consideration of inter-individualvariability in intra-individual patterns of chance over time. Outlier detection and identification ofinfluential observations are important topics in the area of GCM.

High-dimensional rank-based inference

ABSTRACTExisting high-dimensional inferential methods for comparing multiple groups test hypotheses are formulated in terms of mean vectors or location parameters. These methods are applicable mainly for metric data. Furthermore, the mean-based methods assume that moments exist and the nonparametric (location-based) ones assume elliptical-contoured distributions for the populations. In this paper, a fully nonparametric (rank-based) method is proposed. The method is applicable for metric as well as non-metric data and, hence, is applicable for ordered categorical as well as skewed and heavy tailed data. To develop the theory, we prove a novel result for studying asymptotic behaviour of quadratic forms in ranks. Simulation study shows that the developed rank-based method performs comparably well with mean-based methods when the assumptions of those methods are satisfied. However, it has significantly superior power for heavy tailed distributions with the possibility of outliers. The rank method is applied to an EEG data with the objective of examining associations between alcohol use and change in brain function.

The Class of (p,q)-spherical Distributions with an Extension of the Sector and Circle Number Functions

For evaluating the probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based on the Gauss–Laplace law. The latter will be considered here as an element of the newly-introduced family of ( p , q ) -spherical distributions. Based on a suitably-defined non-Euclidean arc-length measure on ( p , q ) -circles, we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly to with elliptically-contoured distributions and more general homogeneous star-shaped ones. This is demonstrated by the generalization of the Box–Muller simulation method. In passing, we prove an extension of the sector and circle number functions.

Flexible Generalized Mixture Model Cluster Analysis with Elliptically-Contoured Distributions

Flexible Generalized Mixture Model Cluster Analysis with Elliptically-Contoured Distributions

Effective sample size of spatial process models

This paper focuses on the reduction of sample sizes due to the effect of autocorrelation for the most common models used in spatial statistics. This work is an extension of a simple illustration highlighted in several books for an autoregressive-type correlation structure. The paper briefly reviews existing proposals to quantify the effective sample size and proposes a new definition that is a function of the correlation structure, sample size, and dimension of the space where the coordinates are defined. It describes the properties of and explicit expression for the effective sample size for processes with patterned correlation matrices, including elliptical contoured distributions. The estimation of the effective sample size is achieved using restricted maximum likelihood. Additionally, the paper describes the monotonicity of the effective sample size when two random points are uniformly distributed on the unit sphere and includes several Monte Carlo simulations to explore monotonic features of the effective sample size and to compare its behavior with respect to other proposals. Finally, this paper analyzes two real datasets, and the discussion includes topics that should be addressed in further research.

Improved preliminary test and Stein-rule Liu estimators for the ill-conditioned elliptical linear regression model

Recently, Liu (1993) estimator draws an important attention to estimate the regression parameters for an ill-conditioned linear regression model when the vector of errors is distributed according to the law belonging to the class of elliptically contoured distributions (ECDs). This paper proposed some improved Liu type estimators, namely, the unrestricted Liu estimator (ULE), restricted Liu estimator (RLE), preliminary test Liu estimator (PTLE), shrinkage Liu estimator (SLE) and positive rule Liu estimator (PRLE) for estimating the regression parameters @b. The performance of the proposed estimators is compared based on the quadratic bias and risk functions under both null and alternative hypotheses, which specify certain restrictions on the regression parameters. The conditions of superiority of the proposed estimators for parameter d and non-centrality parameter @D are given.

Invariance properties of the likelihood ratio for covariance matrix estimation in some complex elliptically contoured distributions

The likelihood ratio (LR) for testing if the covariance matrix of the observation matrix X is R has some invariance properties that can be exploited for covariance matrix estimation purposes. More precisely, it was shown in Abramovich et al. (2004, 2007, 2007) that, in the Gaussian case, LR(R0|X), where R0 stands for the true covariance matrix of the observations X, has a distribution which does not depend on R0 but only on known parameters. This paved the way to the expected likelihood (EL) approach, which aims at assessing and possibly enhancing the quality of any covariance matrix estimate (CME) by comparing its LR to that of R0. Such invariance properties of LR(R0|X) were recently proven for a class of elliptically contoured distributions (ECD) in Abramovich and Besson (2013) and Besson and Abramovich (2013) where regularized CME were also presented. The aim of this paper is to derive the distribution of LR(R0|X) for other classes of ECD not covered yet, so as to make the EL approach feasible for a larger class of distributions.

Shrinkage Ridge Estimators in Linear Regression

The problem of estimation of the regression coefficients in a multiple regression model (MRM) is considered under multicollinearity situation. Further it is suspected that the regression coefficients may be restricted to a subspace. In this approach, we present the estimators of the regression coefficients combining the idea of preliminary test estimator and Stein-rule estimator with the ridge regression methodology for elliptically contoured distributions (ECD). Accordingly, their exact risk expressions in addition to biases are derived. The estimators under study are compared and the regions of optimality of the estimators are exactly determined along with some numerical analysis. In this regard, the ridge parameter is determined in different disciplines.

On the Fisher Information Matrix for Multivariate Elliptically Contoured Distributions

The Slepian-Bangs formula provides a very convenient way to compute the Fisher information matrix (FIM) for Gaussian distributed data. The aim of this letter is to extend it to a larger family of distributions, namely elliptically contoured (EC) distributions. More precisely, we derive a closed-form expression of the FIM in this case. This new expression involves the usual term of the Gaussian FIM plus some corrective factors that depend only on the expectations of some functions of the so-called modular variate. Hence, for most distributions in the EC family, derivation of the FIM from its Gaussian counterpart involves slight additional derivations. We show that the new formula reduces to the Slepian-Bangs formula in the Gaussian case and we provide an illustrative example with Student distributions on how it can be used.

Shrinkage Estimation Under Multivariate Elliptic Models

The estimation of the location vector of a p-variate elliptically contoured distribution (ECD) is considered using independent random samples from two multivariate elliptically contoured populations when it is apriori suspected that the location vectors of the two populations are equal. For the setting where the covariance structure of the populations is the same, we define the maximum likelihood, Stein-type shrinkage and positive-rule shrinkage estimators. The exact expressions for the bias and quadratic risk functions of the estimators are derived. The comparison of the quadratic risk functions reveals the dominance of the Stein-type estimators if p ≥ 3. A graphical illustration of the risk functions under a “typical” member of the elliptically contoured family of distributions is provided to confirm the analytical results.

Estimation in nonlinear mixed-effects models using heavy-tailed distributions

Nonlinear mixed-effects models are very useful to analyze repeated measures data and are used in a variety of applications. Normal distributions for random effects and residual errors are usually assumed, but such assumptions make inferences vulnerable to the presence of outliers. In this work, we introduce an extension of a normal nonlinear mixed-effects model considering a subclass of elliptical contoured distributions for both random effects and residual errors. This elliptical subclass, the scale mixtures of normal (SMN) distributions, includes heavy-tailed multivariate distributions, such as Student-t, the contaminated normal and slash, among others, and represents an interesting alternative to outliers accommodation maintaining the elegance and simplicity of the maximum likelihood theory. We propose an exact estimation procedure to obtain the maximum likelihood estimates of the fixed-effects and variance components, using a stochastic approximation of the EM algorithm. We compare the performance of the normal and the SMN models with two real data sets.

Multivariate Statistical Models for Image Denoising in the Wavelet Domain

We model wavelet coefficients of natural images in a neighborhood using the multivariate Elliptically Contoured Distribution Family (ECDF) and discuss its application to the image denoising problem. A desirable property of the ECDF is that a multivariate Elliptically Contoured Distribution (ECD) can be deduced directly from its lower dimension marginal distribution. Using the property, we extend a bivariate model that has been used to successfully model the 2-D joint probability distribution of a two dimension random vector--a wavelet coefficient and its parent--to multivariate cases. Though our method only provides a simple and rough characterization of the full probability distribution of wavelet coefficients in a neighborhood, we find that the resulting denoising algorithm based on the extended multivariate models is computably tractable and produces state-of-the-art restoration results. In addition, we discuss the equivalence relation between our denoising algorithm and several other state-of-the-art denoising algorithms. Our work provides a unified mathematic interpretation of a type of statistical denoising algorithms. We also analyze the limitations and advantages of algorithms of this type.

Random Regression Models Based On The Elliptically Contoured Distribution Assumptions With Applications To Longitudinal Data

Bartolucci et al.(2003) extended the distribution assumption from the normal (Lyles et al., 2000) to the elliptical contoured distribution (ECD) for random regression models used in analysis of longitudinal data accounting for both undetectable values and informative drop-outs. In this paper, the random regression models are constructed on the multivariate skew ECD. A real data set is used to illustrate that the skew ECDs can fit some unimodal continuous data better than the Gaussian distributions or more general continuous symmetric distributions when the symmetric distribution assumption is violated. Also, a simulation study is done for illustrating the model fitness from a variety of skew ECDs. The software we used is SAS/STAT, V. 9.13.

Improved estimation of a covariance matrix in an elliptically contoured matrix distribution

In this paper, the problem of estimating the covariance matrix of the elliptically contoured distribution (ECD) is considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean is proposed. It is shown that this new estimator dominates the unbiased estimator under the squared error loss function. Two special classes of ECD, namely, the multivariate-elliptical t distribution and the ε-contaminated normal distribution are considered. A simulation study is carried out and indicates that this new shrinkage estimator provides a substantial improvement in risk under most situations.

ESTIMATION OF VARIANCE AND COVARIANCE COMPONENTS IN ELLIPTICALLY CONTOURED DISTRIBUTIONS

In this paper, we consider the problems of estimating the univariate and multivariate components of variance in elliptically contoured distribution (ECD) model in a decision-theoretic setup. Empirical Bayes or generalized Bayes estimators and several other positive or nonnegative (definite) estimators improving upon usual ANOVA (unbiased) estimators of the variance components are obtained. The robustness of these dominance results is also investigated, and it is shown that all dominance results under normal models remain true within a specific class of distributions.

Robust improvement in estimation of a covariance matrix in an elliptically contoured distribution

This paper derives an extended version of the Haff or, more appropriately, Stein-Haff identity for an elliptically contoured distribution (ECD) . This identity is then used to show that the minimax estimators of the covariance matrix obtained under normal models remain robust under the ECD model.

Some Inference Problems for Matrix Variate Elliptically Contoured Distributions

In this paper, maximum likelihood estimators for a broad class of matrix variate elliptically contoured distributions have been derived. Optimality properties of estimators in the multivariate regression models are studied when the error term has a matrix variate elliptically contoured distribution. Least-square estimators are also obtained for the multivariate random effect regression model when the underlying distribution is elliptically contoured.