Mathematical Sciences: Sampling Plans, Asymptotic Results, Resampling Algorithms, & Applications to Random Processes on the 3-dimensional Sphere

致：“Alan J.Izenman”Aizenman@nsf.gov来自：aizenman@nsf.gov(Alan J.Izenman)主题：Brown Abstract cc：bcc：x-attachments：9404130 Brown关于球面上各向同性随机场的研究主要集中在随机场的表示上，尽管已经有一些研究证明了球面上连续索引的随机场的中心极限定理。然而，从实践的角度来看，不可能在整个球体内连续采样，需要找到有限的全局采样计划，以便调查与全局数据相关的统计关系，特别是CLT和重采样算法。布朗1993年的工作为这一领域的未来研究奠定了基础，他已经将他的研究应用于全球陆地面积和海岸线数据。他的工作解决了在非参数设置下的上述统计问题，当球体半径增长时，使用弱的一般条件，而不受样本大小的限制。最近，在参数设置中，已经对球体上的整个随机过程进行了建模。这项研究将考虑非参数和参数两种情况，以及球体半径保持不变，而采样方案在球体上变得更稠密，并且球体半径随着样本大小增长而不受限制的情况下的渐近结果。具体地说，本研究将研究以下几个方面：有限全局抽样方案，不同抽样方案和一般统计学的渐近结果，不同抽样方案数据的重抽样机制，以及在这一新研究中的应用。通常，我们只有机会在全球有限数量的点上采样数据，这构成了一个全球采样计划。一旦制定了抽样计划，就在这些点位收集数据，并计算出感兴趣的数量。为了对感兴趣的数量做出决定，需要知道或估计其一些特征。由于通过抽样过程收集数据的成本很高，还需要开发一些重复使用(重新抽样)原始数据的方法。这项研究将研究以下领域：全球抽样计划、各种感兴趣的量的特征、全球数据的重抽样机制以及在这一新研究中的应用。

To: "Alan J. Izenman" aizenman@nsf.gov From: aizenman@nsf.gov (Alan J. Izenman) Subject: Brown Abstract Cc: Bcc: X-Attachments: 9404130 Brown Research on an isotropic random field on the sphere has mainly focused on the representation of the random field, although, there has been some research on proving a central limit theorem (CLT) for a continuously indexed random field on the sphere. From a practical point of view, however, it is impossible to sample continuously throughout the sphere and a finite global sampling plan needs to be found in order to investigate statistical relations associated with global data, in particular a CLT and resampling algorithms. Work by Brown 1993 lays the groundwork for future research in this area and he has applied his research to global land-area and coastline data. His work addresses the above statistical issues in a nonparametric setting using weak general conditions as the radius of the sphere grows without bound with the sample size. More recently, in the parametric setting there has been work done on modeling the entire random process on the sphere. This research will consider both nonparametric and parametric cases and asymptotic results for situations where the radius of the sphere remains fixed while the sampling plan gets more dense on the sphere and where the radius of the sphere grows without bound along with the sample size. Specifically, this research will study the following areas: finite global sampling plans, asymptotic results for different sampling plans and general statistics, resampling mechanisms of the data for various sampling plans, and applications to this new research. Typically, we only have an opportunity to sample data at a finite number of points on the globe, comprising a global sampling plan. Once a sampling plan is established, data is gathered at the points and quantities of interest are calculated. In order to make decisions about the quantity of interest, some of its characteristics need to be known or estimated. Since gathering data via the sampling process is expensive, some methods for reusing (resampling) the original data need to be developed as well. This research will study the following areas: global sampling plans, characteristics for various quantities of interest, resampling mechanisms for global data, and applications to this new research.