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27
Directional Statistics and Shape Analysis
, 1995
"... There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various c ..."
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Cited by 794 (33 self)
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There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various concepts are connected. In particular, certain distributions of directional statistics have emerged in shape analysis, such a distribution is Complex Bingham Distribution. This paper first gives some background to shape analysis and then it goes on to directional distributions and their applications to shape analysis. Note that the idea of using tangent space for analysis is common to both manifold as well. 1 Introduction Consider shapes of configurations of points in Euclidean space. There are various contexts in which k labelled points (or "landmarks") x 1 ; :::; x k in IR m are given and interest is in the shape of (x 1 ; :::; x k ). Example 1 The microscopic fossil Globorotalia truncat...
Semiparametrically efficient rankbased inference for shape I: Optimal rankbased tests for sphericity
 Ann. Statist
, 2006
"... A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under a ..."
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Cited by 48 (32 self)
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A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under any radial density g, without any moment assumptions, and semiparametrically efficient at some prespecified density f. When based on normal scores, they are uniformly more efficient than the traditional normaltheory estimator based on empirical covariance matrices (the asymptotic normality of which, moreover, requires finite moments of order four), irrespective of the actual underlying elliptical density. They rely on an original rankbased version of Le Cam’s onestep methodology which avoids the unpleasant nonparametric estimation of crossinformation quantities that is generally required in the context of Restimation. Although they are not strictly affineequivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finitesample performances. 1. Introduction. 1.1. Rankbased inference for elliptical families. An elliptical density over Rk is determined by a location center θ ∈ Rk, a scale parameter σ ∈ R + 0, a realvalued positive definite symmetric k × k matrix V = (Vij) with V11 = 1,
Efficient Markov Chain Monte Carlo methods for decoding population spike trains
 TO APPEAR, NEURAL COMPUTATION
, 2010
"... Stimulus reconstruction or decoding methods provide an important tool for understanding how sensory and motor information is represented in neural activity. We discuss Bayesian decoding methods based on an encoding generalized linear model (GLM) that accurately describes how stimuli are transformed ..."
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Cited by 33 (14 self)
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Stimulus reconstruction or decoding methods provide an important tool for understanding how sensory and motor information is represented in neural activity. We discuss Bayesian decoding methods based on an encoding generalized linear model (GLM) that accurately describes how stimuli are transformed into the spike trains of a group of neurons. The form of the GLM likelihood ensures that the posterior distribution over the stimuli that caused an observed set of spike trains is logconcave so long as the prior is. This allows the maximum a posteriori (MAP) stimulus estimate to be obtained using efficient optimization algorithms. Unfortunately, the MAP estimate can have a relatively large average error when the posterior is highly nonGaussian. Here we compare several Markov chain Monte Carlo (MCMC) algorithms that allow for the calculation of general Bayesian estimators involving posterior expectations (conditional on model parameters). An efficient version of the hybrid Monte Carlo (HMC) algorithm was significantly superior to other MCMC methods for Gaussian priors. When the prior distribution has sharp edges and corners, on the other hand, the “hitandrun” algorithm performed better than other MCMC methods. Using these
Model Acquisition Using Stochastic Projective Geometry
, 1993
"... This thesis presents a methodology for scene reconstruction that is based on the principles of projective geometry, while dealing with uncertainty at a fundamental level. Uncertainty in geometric features is represented and manipulated using probability density functions on projective space, allowin ..."
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Cited by 14 (1 self)
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This thesis presents a methodology for scene reconstruction that is based on the principles of projective geometry, while dealing with uncertainty at a fundamental level. Uncertainty in geometric features is represented and manipulated using probability density functions on projective space, allowing geometric constructions to be carried out via statistical inference. The main contribution of this thesis is the development of stochastic projective geometry, a formalism for performing uncertain geometric reasoning during the scene reconstruction process. The homogeneous coordinates of points and lines in the projective plane are represented by antipodal pairs of points on the unit sphere, and geometric uncertainty in their location is represented...
Theory of Cross Sectionally Contoured Distributions and Its Applications
 Economics, University of Tokyo
, 1996
"... We discuss generalization of elliptically contoured distributions to densities whose contours are arbitrary cross sections in the framework of group invariance. This generalization leads to much richer family of distributions compared to the elliptically contoured distributions. The basic property o ..."
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Cited by 3 (3 self)
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We discuss generalization of elliptically contoured distributions to densities whose contours are arbitrary cross sections in the framework of group invariance. This generalization leads to much richer family of distributions compared to the elliptically contoured distributions. The basic property of the elliptically contoured distribution is the independence of the "length" and the "direction" of the random vector. We show that in our generalized framework this independence still holds if we define the length appropriately. Our examples include "starshaped distributions" and their generalization to random matrices. Key words: Elliptically contoured distribution, starshaped distribution, group action, invariance, relatively invariant measure. 1 Introduction Consider a continuous elliptically contoured distribution in R p . Its density f(x) is written as f(x) = h(x 0 6 01 x); (1) where x is considered as a p dimensional column vector and 6 is a positive definite matrix. Let ...
Modelling of Single Mode Distributions of Colour Data Using Directional Statistics
, 1997
"... Three different statistical models of colour data for use in segmentation or tracking algorithms are proposed. Results of a performance comparison of a tracking algorithm, applied to two separate applications, using each of the three different types of underlying model of the data are presented. Fro ..."
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Cited by 3 (2 self)
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Three different statistical models of colour data for use in segmentation or tracking algorithms are proposed. Results of a performance comparison of a tracking algorithm, applied to two separate applications, using each of the three different types of underlying model of the data are presented. From these a comparison of the performance of the statistical colour models themselves is obtained. 1. Introduction The work presented here is part of a longerterm investigation into techniques for tracking country roads and lanes in sequences of images to assist the navigation of cross country autonomous land vehicles. There are several sources of information that might be useful in constructing a system to perform this task. Greylevel pixel data can be used to provide some degree of discrimination between different regions in an image and many segmentation and tracking techniques are based on the examination of greylevel data, e.g., [1,2]. However, the extra dimensionality in colour data...
Probabilistic frames: An overview
"... Abstract Finite frames can be viewed as mass points distributed in Ndimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of probabilistic frames, and we characterize one of the ..."
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Cited by 2 (1 self)
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Abstract Finite frames can be viewed as mass points distributed in Ndimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of probabilistic frames, and we characterize one of their subclasses in terms of minimizers of some appropriate potential function. In addition, we survey a range of areas where probabilistic frames, albeit, under different names, appear. These areas include directional statistics, the geometry of convex bodies, and the theory of tdesigns. 1
Starshaped distributions and their generalizations
, 2006
"... Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to “starshaped distributions ” with concentric starshaped contours and show that many results in ..."
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Cited by 1 (1 self)
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Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to “starshaped distributions ” with concentric starshaped contours and show that many results in the former case continue to hold in group invariance so that the results can be applied to other cases as well, especially those involving random matrices. Key words: elliptically contoured distribution, equivariance, global cross section, group action, Haar measure, invariance, isotropy subgroup, normalizer, orbital decomposition, starshaped set. 1
The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions. ArXiv eprints 1309.1915
, 2013
"... The asymptotic efficiency of the spatial sign covariance matrix (SSCM) relative to affine equivariant estimates of scatter is studied in detail. In particular, the SSCM is shown to be asymptoticaly inadmissible, i.e. the asymptotic variancecovariance matrix of the consistency corrected SSCM is unif ..."
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Cited by 1 (0 self)
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The asymptotic efficiency of the spatial sign covariance matrix (SSCM) relative to affine equivariant estimates of scatter is studied in detail. In particular, the SSCM is shown to be asymptoticaly inadmissible, i.e. the asymptotic variancecovariance matrix of the consistency corrected SSCM is uniformly smaller than that of its affine equivariant counterpart, namely Tyler’s scatter matrix. Although the SSCM has often been recommended when one is interested in principal components analysis, the degree of the inefficiency of the SSCM is shown to be most severe in situations where principal components are of most interest. A finite sample simulation shows the inefficiency of the SSCM also holds for small sample sizes, and that the asymptotic relative efficiency is a good approximation to the finite sample efficiency for relatively modest sample sizes. 1. Introduction Multivariate procedures are implemented to help understand the relationships between several quantitative variables of interest. The most commonly used methods rely heavily on the sample variancecovariance matrix. It is now well known that the sample variancecovariance matrix is highly nonrobust, being extremely sensitive to outliers and being very inefficient at longer tailed distributions. Consequently, there have been many proposed robust alternatives to the sample variancecovariance matrix,
by
, 2009
"... A generalization of Tyler's Mestimators to the case of incomplete data Discussion papers in statistics and econometrics, No. 3/07 Provided in Cooperation with: ..."
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A generalization of Tyler's Mestimators to the case of incomplete data Discussion papers in statistics and econometrics, No. 3/07 Provided in Cooperation with: