John T. Kent

Spatial classification using fuzzy membership models

In the usual statistical approach to spatial classification, it is assumed that each pixel belongs to precisely one of a small number of known groups. This framework is extended to include mixed-pixel data; then, only a proportion of each pixel belongs to each group. Two models based on multivariate Gaussian random fields are proposed to model this fuzzy membership process. The problems of predicting the group membership and estimating the parameters are discussed. Some simulations are presented to study the properties of this approach, and an example is given using Landsat remote-sensing data. >

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.

Sampling realistic protein conformations using local structural bias.

The prediction of protein structure from sequence remains a major unsolved problem in biology. The most successful protein structure prediction methods make use of a divide-and-conquer strategy to attack the problem: a conformational sampling method generates plausible candidate structures, which are subsequently accepted or rejected using an energy function. Conceptually, this often corresponds to separating local structural bias from the long-range interactions that stabilize the compact, native state. However, sampling protein conformations that are compatible with the local structural bias encoded in a given protein sequence is a long-standing open problem, especially in continuous space. We describe an elegant and mathematically rigorous method to do this, and show that it readily generates native-like protein conformations simply by enforcing compactness. Our results have far-reaching implications for protein structure prediction, determination, simulation, and design.

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.

Consistency of Procrustes Estimators

Lele has shown that the Procrustes estimator of form is inconsistent and raised the question about the consistency of the Procrustes estimator of shape. In this paper the consistency of estimators of form and shape is studied under various assumptions. In particular, it is shown that the Procrustes estimator of shape is consistent under the assumption of an isotropic error distribution and that consistency breaks down if the assumption of isotropy is relaxed. The relevance of these results for practical shape analysis is discussed. As a by-product, some new results are derived for the offset uniform distribution from directional data.

Eigenvalue expansions for diffusion hitting times

Consider a non-singular diffusion on an interval (ro, rt) and let r 0 < a < b < r~. Set zab to be the first time the diffusion hits b, starting at a, with moment generating function (m.g.f.) ~,b,b(2)=E{exp(2Zab)}. Since we shall be concerned with the behaviour of 4)(2) for positive 2, it is more convenient to work with m.g.f.s than with Laplace transforms. We shall show that all such diffusion hitting times are generalized convolutions of mixtures of exponential distributions (g.c.m.e.d.s). If r o is not a natural boundary, more can be said; then tab can be written as an infinite convolution of elementary mixtures of exponential distributions. The parameters are given by the eigenvalues of associated Sturm-Liouville expansions. Furthermore, normalizing z,b to have mass 1 and letting aJ, r o leads to an infinite convolution of exponential densities. Sections 2 and 3 summarize the necessary information needed about g.c.m.e.d.s and diffusion theory, respectively. The expansion for zab when r o is not natural is derived in Sects. 4 and 5. A related series expansion for the density of Z,b is discussed in Sect. 6, and Sect. 7 gives the general result which holds for all diffusion hitting times. An example based on the Bessel diffusion process is analyzed in Sect. 8. Some simple formulae for the first two moments are given in Sect. 9.

Fitting straight lines and planes with an application to radiometric dating

Conventional practice in geochronology is to fit a straight line or “isochron” to data consisting of two isotopic ratios by a method (e.g. that of 1 , 2 , or perhaps the more modern version of Titterington and Haliday, 1979 ) that takes into account that fact that both ratios are measured with error. In this paper we use matrix algebra to lay out a general method for fitting linear relations between any number of variables, all subject to errors with known variances and covariances and the well-known Newton-Raphson method to do the optimization. This leads to a good computational algorithm which may also be used e.g. to check whether coefficients in several linear relations are the same. In many fields of science one needs to fit linear relations so our method is of wide utility; its use is in no way restricted to radiometric dating.

Ridge Curves and Shape Analysis.

Ridge curves are important features in human vision (see Koenderink, 1990, p.295). In this paper we apply a simple algebraic criterion of Porteous (1994) to the problem of nding ridge curves in machine vision. This work provides a simpler approach than other existing methods (see for example, Thirion and Gourdon, 1992, and Hosaka, 1992). To identify ridge curves in practice on surface data it is necessary rst to smooth the data, which we do using thin-plate splines and related kriging procedures. After validating our methodology, we illustrate its use on a set of laser range data of the human head.

Data analysis for shapes and images

Abstract Modern computing power has made it increasingly feasible to collect data on 2D and 3D objects in the form of images. The statistical analysis of such data is typically based on the location of landmarks on the objects and on further information such as the tangent directions and curvatures at the landmarks. This paper gives a review of the field and makes three new contributions: (a) a new method for computing an average shape for landmark data which is resistant to outliers, (b) a method for visualizing between-groups and within-groups shape variation for two groups of landmark data, and (c) an assessment of the usefulness of incorporating tangent and curvature information when summarizing the shape of objects.

Simulation for the complex Bingham distribution

The complex Bingham distribution is relevant for the shape analysis of landmark data in two dimensions. In this paper it is shown that the problem of simulating from this distribution reduces to simulation from a truncated multivariate exponential distribution. Several simulation methods are described and their efficiencies are compared.