David E. Tyler

Complex Elliptically Symmetric Distributions: Survey, New Results and Applications

Complex elliptically symmetric (CES) distributions have been widely used in various engineering applications for which non-Gaussian models are needed. In this overview, circular CES distributions are surveyed, some new results are derived and their applications e.g., in radar and array signal processing are discussed and illustrated with theoretical examples, simulations and analysis of real radar data. The maximum likelihood (ML) estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness, and for convergence of the iterative fixed point algorithm are established. Specific ML-estimators for several CES distributions that are widely used in the signal processing literature are discussed in depth, including the complex t -distribution, K-distribution, the generalized Gaussian distribution and the closely related angular central Gaussian distribution. A generalization of ML-estimators, the M-estimators of the scatter matrix, are also discussed and asymptotic analysis is provided. Applications of CES distributions and the adaptive signal processors based on ML- and M-estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for Direction-of-Arrival (DOA) estimation and beamforming. Furthermore, experimental validation of the usefulness of CES distributions for modelling real radar data is given.

Robust Computer Vision

This special issue is dedicated to examining the use of techniques from robust statistics in solving computer vision problems. It represents a milestone of recent progress within a subarea of our field that is nearly as old as the field itself, but has seen rapid growth over the past decade. Our Introduction considers the meaning of robustness in computer vision, summarizes the papers, and outlines the relationship between techniques in computer vision and statistics as a means of highlighting future directions. It complements the available reviews on this topic [12, 13].

Maximum likelihood estimation for the wrapped Cauchy distribution

The wrapped Cauchy distribution is an alternative to the Fisher-von Mises distribution for modeling symmetric data on the circle, and its maximum likelihood estimate (m.l.e.) represents a robust alternative to the mean direction for estimating the location for circular data. Surprisingly, there appear to be no previous results on the m.l.e. for the wrapped Cauchy distribution. It is shown that for sample sizes greater than two, the m.l.e. exists, is unique, and can be found by solving the likelihood equations. Also, a simple algorithm is presented which converges to the m.l.e.

A curious likelihood identity for the multivariate t-distribution

It is shown that maximum likelihood estimates of the location vector and scatter matrix for a multivariate t-distribution in p dimensions with v≥1 degrees of freedom. can be identified with the maximum likelihood estimates for a scatter-only estimation problem from a (p+1)-dimensional multivariate the t-distribution with v−1>0 degrees of freedom. The t-distribution is only distribution for which this dual formulation is possible. Since the existence and uniqueness properties of maximum likelihood estimates are straightforward to prove for general scatter-only problems. we are able to immediately deduce existence and uniqueness results for the trickier location-scatter problem in the special case of the t-distribution. Each of these two formulations gives rise to an EM algorithm to maximize the likelihood. though the two algorithms are slightly different. The limiting Cauchy case v=1 requires some special treatment.

ROBUST FUNCTIONAL PRINCIPAL COMPONENTS: A PROJECTION-PURSUIT APPROACH

In many situations, data are recorded over a period of time and may be regarded as realizations of a stochastic process. In this paper, robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting. Our approach combines robust projection-pursuit with different smoothing methods. Consistency of the estimators are shown under mild assumptions. The performance of the classical and robust procedures are compared in a simulation study under different contamination schemes. 1. Introduction. Analogous to classical principal components analysis (PCA), the projection-pursuit approach to robust PCA is based on finding projections of the data which have maximal dispersion. Instead of using the variance as a measure of dispersion, a robust scale estimator sn is used for the maximization problem. This approach was introduced by Li and Chen (1985), who proposed estimators based on maximizing (or minimizing) a robust scale. In this way, given a sample xi ∈ R d ,1 ≤ i ≤ n, the first robust principal component vector is defined as

Robust regression for data with multiple structures

In many vision problems (e.g., stereo, motion) multiple structures can occur in the data, in which case several instances of the same model need to be recovered from a single data set. However, once the measurement noise becomes significantly large relative to the separation between the structures, the robust statistical methods commonly used in the vision community tend to fail. In this paper, we show that all these techniques are special cases of the general class of M-estimators with auxiliary scale, and explain their failure in the presence of noisy multiple structures. To be able to cope with data containing multiple structures the techniques innate to vision (Hough and RANSAC) should be combined with the robust methods customary in statistics. The implications of our analysis are illustrated by introducing a simple procedure for 2D multistructured data problematic for all known current techniques.

Compound-Gaussian Clutter Modeling With an Inverse Gaussian Texture Distribution

The compound-Gaussian (CG) distributions have been successfully used for modelling the non-Gaussian clutter measured by high-resolution radars. Within the CG class, the complex K -distribution and the complex t-distribution have been used for modelling sea clutter which is often heavy-tailed or spiky in nature. In this paper, a heavy-tailed CG model with an inverse Gaussian texture distribution is proposed and its distributional properties such as closed-form expressions for its probability density function (p.d.f.) as well as its amplitude p.d.f., amplitude cumulative distribution function and its kurtosis parameter are derived. Experimental validation of its usefulness for modelling measured real-world radar lake-clutter is provided where it is shown to yield better fits than its widely used competitors.

On the Optimality of the Simultaneous Redundancy Transformations.

The objective of this paper is to introduce and motivate additional properties and interpretations for the redundancy variables. It is shown that these variables can be derived by application of certain invariance arguments and without reference to the index of redundancy. In addition, an optimality property for the variables is presented which is important whenever one restricts attention in a study to a subset of the redundancy variables. This optimality property pertains to the subset rather than to the individual variables.

Regularized M-estimators of scatter matrix

In this paper, a general class of regularized M-estimators of scatter matrix are proposed that are suitable also for low or insufficient sample support (small n and large p) problems. The considered class constitutes a natural generalization of M-estimators of scatter matrix (Maronna, 1976) and are defined as a solution to a penalized M-estimation cost function. Using the concept of geodesic convexity, we prove the existence and uniqueness of the regularized M-estimators of scatter and the existence and uniqueness of the solution to the corresponding M-estimating equations under general conditions. Unlike the non-regularized M-estimators of scatter, the regularized estimators are shown to exist for any data configuration. An iterative algorithm with proven convergence to the solution of the regularized M-estimating equation is also given. Since the conditions for uniqueness do not include the regularized versions of Tylers M-estimator, necessary and sufficient conditions for their uniqueness are established separately. For the regularized Tylers M-estimators, we also derive a simple, closed form, and data-dependent solution for choosing the regularization parameter based on shape matrix matching in the mean-squared sense. Finally, some simulations studies illustrate the improved accuracy of the proposed regularized M-estimators of scatter compared to their non-regularized counterparts in low sample support problems. An example of radar detection using normalized matched filter (NMF) illustrate that an adaptive NMF detector based on regularized M-estimators are able to maintain accurately the preset CFAR level.

Some Issues in the Robust Estimation of Multivariate Location and Scatter

This paper gives a selective overview and discusses some contemporary issues in the robust estimation of multivariate location and scatter. Special emphasis is placed on the seeming trade off between local and global robustness properties of affine equivariant estimators, and in particular of M-estimators, MVE-estimators and S-estimators. Furthermore, a reappraisal of the breakdown point of redescending M-estimators is made.