Mathematical Sciences: Multivariate Nonparametric Methodology Studies

非参数方法学在一维和二维中得到了广泛的应用，但在高维中应用较少。本研究计划著重于中程维度，以试图加深对维度诅咒的含义的理解。将特别强调多元回归和密度估计问题，以及密切相关的应用。轶事证据表明，在非参数方法的明显成功和理论预测的糟糕表现之间存在差距。我们将研究新的观点，特别是与自适应估计有关的观点。高质量的估计通常需要使用负核，但最近的研究表明，在黑森不确定的区域，通常在高维中占主导地位的尾部，等效增益是可能的。其他最近的工作表明，交叉验证算法在一个维度上被认为是边际实用价值，在多元情况下显着改善。我们发现许多带宽选择算法分为两类，并提出对这两类算法进行表征和研究。中维数据的处理在数据可视化中引起了许多问题，我们提出要对这些问题进行研究。多变量可视化需要诸如专家之类的辅助和指导。我们计划将我们的密度估计可视化能力扩展到回归曲面以及视觉聚类和识别等应用。我们提出将基于模态树和仿真的单变量模态估计和测试思想扩展到多个维度。将简要地考虑多处理器和并行体系结构的算法开发。非参数方法学似乎在专家手中工作得很好，本研究的目的不仅是帮助专家，而且是为了促进更广泛的受众使用该方法学。他最近完成了一本关于多元密度和回归估计以及相关应用的书，特别关注直方图及其逻辑扩展（Scott, 1992）。具有实际影响的理论难题仍然大量存在。然而，广泛的应用反映了非参数方法的普遍接受。科学可视化领域的发展为这些探索性程序提供了肥沃的土壤。该项目试图利用大型数据库的现有投资，通过开发灵活的技术，试图提取隐藏在高维数据中的最大数量的信息和结构。

Nonparametric methodology is widely used in one and two dimensions, but not in high dimensions. This research proposal focuses on the mid-range dimensions in an attempt to foster a deeper understanding of the implications of the curse of dimensionality. Particular emphasis will be given to multivariate regression and density estimation problems, and closely related applications. Anecdotal evidence suggests a gap exists between the apparent successes of nonparametric methodology and the poor performance predicted by theory. We will examine new points of view, especially related to adaptive estimation. Higher quality estimation has often required use of negative kernels, but recent research and shown that equivalent gains are possible in regions where the Hessian is indefinite, often in the tails which dominate in higher dimensions. Other recent work suggests that cross-validation algorithms which are considered of marginal practical value in one dimension, improve dramatically in the multivariate case. We have found the many bandwidth selection algorithms cluster into two cases, and propose to characterize and investigate these classes. Dealing with medium dimensional data gives rise to many problems in data visualization which we propose to investigate. Multivariate visualization requires aids and guides such as cognostics. We plan to extend our density estimation visualization capabilities to regression surfaces as well as applications such as visual clustering and discrimination. We propose to extend univariate ideas of mode estimation and testing based on the mode tree and simulation to several dimensions. Algorithmic development for multiprocessor and parallel architectures will be briefly considered. Nonparametric methodology seems to work well in the hands of experts, and this research is designed to not only aid the expert but to facilitate the use of the methodology by a wider audience. The proposer has recently completed a book on the topic of multivariate density and regression estimation, and related applications, particularly focusing on histograms and their logical extensions (Scott, 1992). Difficult theoretical problems with practical consequences still abound. However, widespread application reflects the general acceptance of nonparametric methodology. The growth in the field of scientific visualization is fertile ground for these exploratory procedures. This project attempts to capitalize on existing investments in large data bases, by developing flexible techniques that attempt to extract the maximum amount of information and structure hidden in the high dimensional data.