Problems Related to Gaussian Processes

摘要：研究者研究理论概率论中的研究问题。他特别研究了一个叫做小偏差的问题。PI也在研究高斯随机场的极值理论，这是由一个非参数统计的应用所激发的。小偏差问题与随机过程停留在原点的小邻域内的概率行为有关。随着邻域的缩小，这个概率明显趋向于零——问题是以多大的速度？PI计算了各种过程的速率，包括用于模拟大坝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。现代非参数平滑的主要问题之一是如何找到最接近数据的光滑曲线。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，可以看作是对大量（数百）个假设检验结果的总结。为了进行合理的统计推断，需要注意多重比较问题。当前的SiZer实现并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1990年代。当随机过程的可变性远远小于预期时，小偏差问题的解决有助于我们理解某些罕见事件的本质。这一理论的应用目前被许多研究人员所追求。研究者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。目前，SiZer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，使其对应用程序更可靠。

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.