Spatial statistics: Marks, maps, and shapes

Abstract

ABSTRACTSpatial statistics is concerned with phenomena unfolding in space and possibly also evolving in time, expressing a system of interactions whereby an observation made at a (spatiotemporal) location is informative about observations made at other locations. In general, the interactions are best described probabilistically, rather than deterministically. Spatial scales range from the microscopic (for example, when describing interactions between molecules of a liquid) to planetary (for example, when studying the Earth’s ozone layer) or even larger; temporal scales are similarly varied.Marks indicate objects whose spatial locations are influenced by the presence and nature of other objects nearby: trees of the same or different species in a grove, molecules in a liquid, or galaxies throughout the universe. The statistical models are (marked) spatial point processes.Maps describe the variability of the values of a property across a geographical region. The Ising model of ferromagnetism describes collective properties of atoms arranged in a regular lattice. When mapping the prevalence or the incidence of a disease at the level of counties or parishes, the observations are associated with subsets of a region whose spatial relations are meaningful. Many maps are drawn based on observations made at a finite set of locations distributed either regularly or irregularly throughout a 2D or 3D spatial domain. For example, the mass fraction of uranium in soils and surface sediments across Colorado. Gaussian random functions are a model of choice for such quantities, possibly after re-expression.Shapes arise owing to modulated interactions between surface elements anchored to points in space — “generators” in the nomenclature of Ulf Grenander’s pattern theory. Probability distributions on spaces of generators and on spaces of interactions between them can then be used to describe variations on patterns and to fit shape models.