Problems Related to Gaussian Processes

与高斯过程相关的问题

• 批准号: 0504737
• 负责人: Jan Hannig
• 金额: $ 9.6万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504737&HistoricalAwards=false
• 关键词: Problems; Related; Gaussian; Processes

Abstract:The investigator studies research problems in theoretical probability. In particular he investigates the problem called small deviations. The PI is also working on extreme value theory for Gaussian random fields motivated by an application to nonparametric statistics. The problem of small deviations is related to the behavior of the probability that a stochastic process stays in a small neighborhood of the origin. As the neighborhood shrinks, this probability clearly tends to zero --- the question is at what rate? The PI calculates this rate for various processes, including the storage process that is used to model the amount of water available in a dam. Motivation of the other problem comes from nonparametric statistics. One of the main problem in modern nonparametric smoothing is concerned with finding the best smooth curve approximating the data. Chaudhuri & Marron (1999) proposed a tool for data exploration called SiZer. The visual display of SiZer, can be viewed as a summary of a large number (hundreds) of hypothesis test results. For reasonable statistical inference, care needs to be taken about the multiple comparison issue. The current implementations of SiZer is not addressing the issue of multiple testing adequately. To correct this one needs to study the distribution of the maximum of a particular discrete nonstationary Gaussian random field. The PI works on two major problems, small deviations for stochastic processes and extreme value theory for a particular random field. The area of small deviations is relatively new with major developments starting in the 1990's. The solution of the small deviation problems helps us understand the nature of certain rare events when the variability of a random process is much less then expected. Applications of this theory are currently pursued by many researchers. The investigator develops a new application of small deviation techniques in storage theory. The proposed extreme value problem is directly motivated by a statistical application. It aims at improving the performance of the data analysis tool SiZer. SiZer is currently widely used by many applied scientists. Successful applications of SiZer include internet traffic modeling, climatology and environment, and biology and genetics. Part of this proposal aims at significantly improving the SiZer tool to make it more reliable for applications.

翻译后摘要：调查研究的理论概率的研究问题。特别是他调查的问题称为小偏差。PI还致力于高斯随机场的极值理论，其动机是应用于非参数统计。小偏差问题与随机过程停留在原点的小邻域中的概率的行为有关。当邻域缩小时，这个概率明显趋向于零-问题是以多大的速率？PI计算各种过程的该速率，包括用于模拟大坝中可用水量的存储过程。另一个问题的动机来自于非参数统计。现代非参数光顺的主要问题之一是如何找到最佳的光滑曲线来逼近数据。Chaudhuri Marron（1999）提出了一种名为SiZer的数据探索工具。SiZer的可视化显示可以被视为大量（数百个）假设检验结果的汇总。为了进行合理的统计推断，需要注意多重比较问题。SiZer的当前实现没有充分解决多重测试的问题。为了纠正这一点，需要研究一个特定的离散非平稳高斯随机场的最大值的分布。PI主要研究两个问题，随机过程的小偏差和特定随机场的极值理论。小偏差领域相对较新，主要发展始于20世纪90年代。小偏差问题的解决方案，帮助我们了解某些罕见事件的性质时，随机过程的可变性是远远小于预期。这一理论的应用目前被许多研究人员所追求。研究人员开发了小偏差技术在存储理论中的新应用。所提出的极值问题的直接动机是统计应用。它旨在提高数据分析工具SiZer的性能。SiZer目前被许多应用科学家广泛使用。SiZer的成功应用包括互联网流量建模，气候学和环境，生物学和遗传学。该提案的一部分旨在显著改进SiZer工具，使其在应用中更加可靠。