Mathematical Sciences: Sampling Designs for Spatially Distributed Data

许多重要的采样问题都发生在空间背景下。 这方面的例子包括估计倾倒场滴滴涕的总浓度、计算阿拉斯加某地区北极熊的数量以及确定生态系统中不同物种的数量。 这个项目涉及的问题，设计抽样方案的空间数据的信息聚类的程度被抽样的元素被利用。 最初的研究将考虑单变量点过程的扩展到多变量的情况。 基本前提是，可以修改一些现有的抽样方案，以利用现有的知识，如假设的人口单位本身或在抽样过程中获得的额外信息。 拟进行的研究分为几个领域。 第一种是基于单变量模型。将考虑两种办法。 一种是“非参数”序贯抽样，即对所研究的人口不作任何假设。 在此，将利用取样期间收集的信息初步确定人口的空间特征。 另一种方法是参数化的，因为它将假设人口是通过一般的泊松聚类过程（或更具体的内曼-斯科特过程）生成的。 然后将使用抽样来估计参数，作为估计群体大小的方法。当需要描述一个比当前总体更一般的情况时，后者是有用的（聚类过程的单一实现）。 稍后，结果将被扩展到多变量的情况。 感兴趣的多变量数据模型包括空间标记点过程，例如由空间泊松过程的叠加引起的。

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.