Mathematical Sciences: Sampling Designs for Spatially Distributed Data

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Many important sampling problems occur within a spatial context. Examples range from estimating the total concentration of DDT at a dumpsite to counting the number of polar bears in a region of Alaska to determining the number of distinct species in an ecosystem. This project involves the problem of designing sampling schemes for spatial data where information about the degree of clustering of the elements being sampled is utilized. Initial research will consider univariate point processes with extensions to the multivariate case. The basic premise is that some extant sampling schemes can be modified to take advantage of available knowledge such as that assumed about the population units themselves or additional information obtained during sampling. The intended research is divided into several areas. The first is based on a univariate model. Two approaches will be considered. One is a "non-parametric" sequential sampling effort where no assumptions are made about the population under study. Here, information collected during sampling will be used to provide an initial characterization of the spatial aspects of the population. The other approach is parametric in that it will be assumed that the population is generated via a general Poisson cluster process (or the more specific Neyman-Scott process). Sampling will then be used to estimate the parameters as a means of estimating the population size. The latter is useful when it is of interest to describe a more general situation than exhibited by the current population (a single realization from the cluster process). Later, the results will be extended to the multivariate case. Multivariate data models of interest include a spatial marked point process such as would arise from a superposition of spatial Poisson processes.