Alfred Kume

The Fréchet mean shape and the shape of the means

We identify the Fréchet mean shape with respect to the Riemannian metric of a class of probability measures on Booksteins shape space of labelled triangles and show, in contrast to the case of Kendalls shape space, that the Fréchet mean shape of the probability measure on Booksteins shape space induced from independent normal distributions on vertices, having the same covariance matrix σ2 I 2, is not necessarily the shape of the means.

Detection of shape changes in biological features.

The question of analysing shape changes over time, such as during growth and during the progress of disease, is an important fundamental issue for many applications. Recent mathematical developments in the understanding of the detailed structure of shape spaces have made possible the quantitative study of shape variation. In this paper, we combine the classical multidimensional scaling technique with knowledge of the geometry of shape spaces to examine the role played by the geodesics in shape spaces. We present some promising early results answering questions about shape changes over time.

Sampling from compositional and directional distributions

This paper describes a method for sampling from a non-standard distribution which is important in both population genetics and directional statistics. Current approaches rely on complicated procedures which do not work well, if at all, in high dimensions and usual parameter set-ups. We use a Gibbs sampler which seems necessary in practical situations of high dimensions.

Calculating the normalising constant of the Bingham distribution on the sphere using the holonomic gradient method

In this paper we implement the holonomic gradient method to exactly compute the normalising constant of Bingham distributions. This idea is originally applied for general Fisher–Bingham distributions in Nakayama et al. (Adv. Appl. Math. 47:639–658, 2011). In this paper we explicitly apply this algorithm to show the exact calculation of the normalising constant; derive explicitly the Pfaffian system for this parametric case; implement the general approach for the maximum likelihood solution search and finally adjust the method for degenerate cases, namely when the parameter values have multiplicities.

On the Fisher---Bingham distribution

This paper primarily is concerned with the sampling of the Fisher–Bingham distribution and we describe a slice sampling algorithm for doing this. A by-product of this task gave us an infinite mixture representation of the Fisher–Bingham distribution; the mixing distributions being based on the Dirichlet distribution. Finite numerical approximations are considered and a sampling algorithm based on a finite mixture approximation is compared with the slice sampling algorithm.

An EM algorithm for markovian arrival processes observed at discrete times

The present paper contains a specification of the EM algorithm in order to fit an empirical counting process, observed at discrete times, to a Markovian arrival process. The given data are the numbers of observed events in disjoint time intervals. The underlying phase process is not observable. An exact numerical procedure to compute the E and M steps is given.

Maximum-likelihood estimation for the offset normal shape distributions using EM

The offset-normal shape distribution is defined as the induced shape distribution of a Gaussian distributed random configuration in the plane. Such distributions were introduced by Dryden and Mardia (1991) and represent an important parameterized family of shape distributions for shape analysis. This article reports a method for performing maximum likelihood estimation of parameters involved. The method consists of an EM algorithm with simple update rules and is shown to be easily applicable in many practical examples. We also show the necessary adjustments needed for using this algorithm for shape regression, missing landmark data, and mixtures of offset-normal shape distributions.

ON FRECHET MEANS IN SIMPLEX SHAPE SPACES

By making use of the geometric properties of simplex shape spaces, this paper investigates the problems relating to the estimation of the Fréchet means of the random shapes of simplices in Euclidean spaces and also, for the random shapes induced by certain normally distributed simplices, the problems relating to the location of these Fréchet means. In particular, we obtain an algorithm for computing sample mean shapes in simplex shape spaces which converges reasonably fast.

Estimating Fréchet means in Bookstein's shape space

In [8], Le showed that procrustean mean shapes of samples are consistent estimates of Fréchet means for a class of probability measures in Kendalls shape spaces. In this paper, we investigate the analogous case in Booksteins shape space for labelled triangles and propose an estimator that is easy to compute and is a consistent estimate of the Fréchet mean, with respect to sinh(δ/√2), of any probability measure for which such a mean exists. Furthermore, for a certain class of probability measures, this estimate also tends almost surely to the Fréchet mean calculated with respect to the Riemannian distance δ.

On the Bingham distribution with large dimension

In this paper, we investigate the Bingham distribution when the dimension p is large. Our approach is to use a series expansion of the distribution from which truncation points can be determined yielding particular errors. A point of comparison with the approach of Dryden (2005) is highlighted.