Mathematical Sciences: Sampling Designs for Spatially Distributed Data

许多重要的抽样问题都发生在空间环境中。例子从估计垃圾场的滴滴涕总浓度到计算阿拉斯加地区的北极熊数量，再到确定生态系统中不同物种的数量，不一而足。该项目涉及设计空间数据抽样方案的问题，其中利用了有关被抽样要素的聚集度的信息。最初的研究将考虑单变量点过程，并将其扩展到多变量情况。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，例如关于人口单位本身的假设知识或在抽样期间获得的额外信息。拟进行的研究分为几个领域。第一种是基于单变量模型的。将考虑两种方法。一种是“非参数”序贯抽样，不对研究中的总体做出任何假设。在这里，抽样期间收集的信息将用于提供人口空间方面的初步特征。另一种方法是参数的，因为它假设总体是通过一般的泊松集群过程(或更具体的Neyman-Scott过程)产生的。然后，将使用抽样来估计参数，作为估计人口数量的一种手段。如果有兴趣描述比当前人口所表现出的更一般的情况(来自分组进程的单一实现)，则后者是有用的。稍后，结果将推广到多元情形。感兴趣的多变量数据模型包括空间标记点过程，例如从空间泊松过程的叠加而产生的过程。

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.