Copulae on products of compact Riemannian manifolds

Abstract One standard way of considering a probability distribution on the unit n -cube, [0 , 1]n , due to Sklar (1959), is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [0 , 1]n with uniform marginals. The definition of copula was extended by Jones et al. (2014) to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data.PostprintPeer reviewe

Modifications of the Rayleigh and Bingham Tests for Uniformity of Directions

AbstractThe general machinery of Cordeiro and Ferrari (1991, Biometrika78, 573–582) and Chandra and Mukerjee (1991, J. Multivariate Anal.36, 103–112) provides modifications of score test statistics which bring the null distributions close to their large-sample asymptotic distributions. These modifications are calculated here for the Rayleigh and Bingham tests of uniformity on spheres of arbitrary dimension, Stiefel manifolds, rotation groups, Grassmann manifolds and complex projective spaces

A general setting for symmetric distributions and their relationship to general distributions

A standard method of obtaining non-symmetrical distributions is that of modulating symmetrical distributions by multiplying the densities by a perturbation factor. This has been considered mainly for central symmetry of a Euclidean space in the origin. This paper enlarges the concept of modulation to the general setting of symmetry under the action of a compact topological group on the sample space. The main structural result relates the density of an arbitrary distribution to the density of the corresponding symmetrised distribution. Some general methods for constructing modulating functions are considered. The effect that transformations of the sample space have on symmetry of distributions is investigated. The results are illustrated by general examples, many of them in the setting of directional statistics.PostprintPeer reviewe

Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

The standard method of transforming a continuous distribution on the line to the uniform distribution on $[0,1]$ is the probability integral transform. Analogous transforms exist on compact Riemannian manifolds, \scr X, in that for each distribution with continuous positive density on \scr X, there is a continuous mapping of \scr X to itself that transforms the distribution into the uniform distribution. In general, this mapping is far from unique. This paper introduces the construction of an almost-canonical version of such a probability integral transform. The construction is extended to shape spaces, Cartan–Hadamard manifolds, and simplices. The probability integral transform is used to derive tests of goodness of fit from tests of uniformity. Illustrative examples of these tests of goodness of fit are given involving (i) Fisher distributions on $S^2$, (ii) isotropic Mardia–Dryden distributions on the shape space $Σ^5_2$. Their behaviour is investigated by simulation

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

A preliminary appeared as INRIA RR-5093, January 2004.International audienceIn medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and X² law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions

A van Trees inequality for estimators on manifolds

Van Trees' Bayesian version of the Cramér-Rao inequality is generalised here to the context of smooth loss functions on manifolds and estimation of parameters of interest. This extends the multivariate van Trees inequality of Gill and Levit (1995) [R.D. Gill, B.Y. Levit, Applications of the van Trees inequality: a Bayesian Cramér-Rao bound, Bernoulli 1 (1995) 59-79]. In addition, the intrinsic Cramér-Rao inequality of Hendriks (1991) [H. Hendriks, A Cramér-Rao type lower bound for estimators with values in a manifold, J. Multivariate Anal. 38 (1991) 245-261] is extended to cover estimators which may be biased. The quantities used in the new inequalities are described in differential-geometric terms. Some examples are given.Bayes risk Bias Cramer-Rao inequality Fisher information Hessian Proper dispersion model Tensor