Spatial Statistics

Abstract

When the distribution of X∈ℝ^p depends only on its distance to some θ₀∈ℝ^p, we discuss results from Hössjer and Croux and Neeman and Chang on rank score statistics. Similar results from Neeman and Chang are also given when X and θ₀ are constrained to lie on the sphere in ℝ^p. Results from Ko and Chang on M estimation for spatial models in Euclidean space and the sphere are also discussed. Finally we discuss a regression type model: the image registration problem. We have landmarks u_i on one image and corresponding landmarks V_i on a second image. It is desired to bring the two images into closest coincidence through a translation, rotation and scale change. The techniques and principles of this paper are summarized through extensive discussion of an example in three-dimensional image registration and a comparison of the L₁ and L₂ registrations. Two principles are important when working with spatial statistics: (1) Assumptions, such as that the distribution of X depends only on its distance to θ₀, introduce symmetries to spatial models which, if properly used, greatly simplify statistical calculations. These symmetries can be expressed in a more general setting by using the notion of statistical group models. (2) When working with a non-Euclidean parameter space Θ such as the sphere, techniques of elementary differential geometry can be used to minimize the distortions caused by using a coordinate system to reexpress Θ in Euclidean parameters.