Problems Related to Gaussian Processes

摘要：研究者从理论概率的角度研究研究问题。特别是，他研究了被称为小偏差的问题。PI还致力于高斯随机场的极值理论，其动机是将其应用于非参数统计。小偏差问题与随机过程停留在原点附近的概率行为有关。随着社区的缩小，这个概率显然趋于零-问题是以多大的速度？PI计算各种过程的这一比率，包括用于对大坝中可用水量进行建模的存储过程。另一个问题的动机来自于非参数统计。现代非参数光顺中的一个主要问题是如何找到最接近数据的光滑曲线。Chaudhuri&Amp；Marron(1999)提出了一个称为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。对于合理的统计推断，需要注意多重比较问题。Sizer的当前实现没有充分解决多个测试的问题。要纠正这一点，需要研究特定离散非平稳高斯随机场的极大值的分布。PI处理两个主要问题：随机过程的小偏差和特定随机场的极值理论。小偏差是一个相对较新的领域，主要的发展始于1990年代的S。小偏差问题的解决有助于我们理解某些罕见事件的性质，当随机过程的变异性比预期的要小得多的时候。这一理论的应用目前正受到许多研究者的追寻。研究人员开发了小偏差技术在存储理论中的新应用。所提出的极值问题是由一个统计应用直接引起的。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、气候学和环境以及生物学和遗传学。该提案的一部分旨在显著改进Sizer工具，使其更适用于应用程序。

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.