Colin Goodall

The Kriged Kalman filter

In recent years there has been growing interest in spatial-temporal modelling, partly due to the potential of large scale data in pollution and global climate monitoring to answer important environmental questions. We consider the Kriged Kalman filter (KKF), a powerful modelling strategy which combines the two wellestablished approaches of (a) Kriging, in the field of spatial statistics, and (b) the Kalman filter, in general state space formulations of multivariate time series analysis. We give a brief introduction to the model and describe its various properties, and highlight that the model allows prediction in time as well as in space, simultaneously. Some special cases of the time series model are considered. We give some procedures to implement the model, also illustrated through a practical example. The paper concludes with a discussion.

Projective Shape Analysis

Abstract Emulating human vision, computer vision systems aim to recognize object shape from images. The main difficulty in recognizing objects from images is that the shape depends on the viewpoint. This difficulty can be resolved by using projective invariants to describe the shape. For four colinear points the cross-ratio is the simplest statistic that is invariant to projective transformations. Five coplanar sets of points can be described by two independent cross-ratios. Using the six-fold set of symmetries of the cross-ratio, corresponding to six permutations of the points, we introduce an inverse stereographic projection of the linear cross-ratio (c) to a stereographic cross-ratio (ξ). To exploit this symmetry, we study the distribution of cos 3ξ when the four points are randomly distributed under appropriate distributions and find the mapping of the cross-ratio so that the distribution of ξ is uniform. These mappings provide a link between projective invariants and directional statistics so that we...

A geometrical derivation of the shape density

The density for the shapes of random configurations of N independent Gaussiandistributed landmarks in the plane with unequal means was first derived by Mardia and Dryden (1989a). Kendall (1984), (1989) describes a hierarchy of spaces for landmarks, including Euclidean figure space containing the original configuration, preform space (with location removed), preshape space (with location and scale removed), and shape space. We derive the joint density of the landmark points in each of these intermediate spaces, culminating in confirmation of the MardiaDryden result in shape space. This three-step derivation is an appealing alternative to the single-step original derivation, and also provides strong geometrical motivation and insight into Kendalls hierarchy. Preform space and preshape space are respectively Euclidean space with dimension 2(N 1) and the sphere in that space, and thus the first two steps are reasonably familiar. The third step, from preshape space to shape space, is more interesting. The quotient by the rotation group partitions the preshape sphere into equivalence classes of preshapes with the same shape. We introduce a canonical system of preshape coordinates that include 2(N 2) polar coordinates for shape and one coordinate for rotation. Integration over the rotation coordinate gives the Mardia-Dryden result. However, the usual geometrical intuition fails because the set of preshapes keeping the rotation coordinate (however chosen) fixed is not an integrable manifold. We characterize the geometry of the quotient operation through the relationships between distances in preshape space and distances among the corresponding shapes. SIZE-AND-SHAPE; PRESHAPE; LANDMARKS; OFFSET NORMAL; REPEATED NORMAL INTEGRAL; RIEMANNIAN SUBMERSION

The noncentral Bartlett decompositions and shape densities

In shape analysis, it is usually assumed that the matrix X:N-K of the co-ordinates of landmarks in K is isotropic Gaussian. Let Y:(N-1)-K be the centered matrix of landmarks from X so that Y ~ N([mu], [sigma]2I). Let Y=TT be the Bartlett decomposition of Y into lower triangular, T, and orthogonal, [Gamma], components. The matrix T denotes the size-and-shape of X. For N-1>=K (the usual case in multivariate analysis is N-1 =2 the distribution of T is related to the noncentral Wishart distribution, an integral over the orthogonal group, [Gamma]=±1. To derive the distribution of T when [Gamma]=+1, so that [Gamma] is a rotation, we investigate extending the method of random orthogonal transformations, especially when rank [mu]=K>=2. The case K=2 is tractable, but the case K=3 is not. However, by a direct method we obtain the shape density when rank [mu]=K=3 and [Gamma]=1 as a computable double-series of trigonometric integrals. However, for K>3, the density is not tractable which is not surprising in view of the same problem for the standard non-central Wishart distribution.

Distributions of projective invariants and model-based machine vision

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

A simple objective method for determining a percent standard in mixed reimbursement systems

In a mixed system for hospital rate setting, reimbursement is set at the weighted average of provider-specific and standard unit costs. With one or more explanatory variables, forward and reverse regression is used to motivate the simple but objective choice of one minus the squared correlation coefficient as the proportion standard. Special treatment is given to nuisance variables that help explain cost but not reasonable cost. Efrons bootstrap provides confidence intervals for the proportion standard. This regression approach is contrasted with the conventional use of the coefficient of variation, and with economic models for the optimal proportion.