Multivariate Nonparametric Methodology Studies

9971797本研究项目致力于发展中期维度中的非参数密度和回归方法。以一种新的观点考察了密切相关的应用，如聚类、混合估计和降维，将局部自适应和空间估计以及非参数准则的最新扩展联系到参数问题。研究了基于积分平方误差的基于数据的参数估计算法的灵活性和稳健性。关于局部自适应曲线估计的带宽选择问题的对偶解的研究仍在继续，令人惊讶的发现是，其中一个解是渐近的大常数。我们将研究寻找有趣的子空间的三种算法。一种是测量模式的数量；第二种是寻找最大偏差子空间；第三种是新的最小正态准则。可视化工作继续进行，收购了ImmersaDesk，这将允许改进算法的实施，如密度大巡视。一种比较新的方法被称为变量聚类分析，它有助于半参数密度估计、数据分析和可解释的降维。在聚类方面，提出了一种基于EM的复杂混合模型的简化算法，以及一种新的组元个数估计和检验算法。该项目继续进行空间建模方面的创新工作，并与农业部的研究人员合作，将许多数据调查结合成有用的数据建模和因素及其协变量的条件估计。非参数方法在一维和二维中得到广泛使用，但在较高维度中使用较少。这项研究侧重于中级维度，并提供了对维度灾难对数据建模的影响以及与海量数据集相关的问题的更深层次的理解。特别强调了多元回归和密度估计问题，以及与之密切相关的应用，如聚类、混合估计和降维。在处理中维数据和不断增长的海量数据集时，可视化尤其重要。特别感兴趣的是在视觉集群和视觉辨别应用中发现和显示数据。莱斯大学收购了一台ImmersaDesk，这将允许在虚拟现实环境中实施最近开发的算法。在国家研究委员会最近的一次研讨会上，许多科学家确定了他们在处理海量数据集工作中的关键统计需求：主成分的替代方案、用于探索海量数据的专门可视化工具、更好的集群算法以及处理非平稳数据的技术。这项研究的结果直接影响到这四个关键机会中的三个。该程序代表了对多变量估计中的一系列重要数据分析问题的全面和长期的攻击。研究生培训是该项目的重要组成部分。这一结果将具有长期的理论意义，并将为现实世界的问题提供短期解决方案。

9971797This research project focuses on the development of nonparametric density and regression methodology in mid-range dimensions. Closely related applications such as clustering, mixture estimation, and dimension reduction are examined with a new point of view, relating locally adaptive and spatial estimation and recent extensions of nonparametric criteria to parametric problems. The new data-based parametric estimation algorithm, based upon integrated squared error, is investigated for its flexibility and robustness. An investigation of the dual solutions to the bandwidth choice problem for locally adaptive curve estimates continues, with the surprising finding that one solution is asymptotically a large constant. Three algorithms for finding interesting subspaces will be investigated. One measures the number of modes; a second finds maximal bias subspaces; and a third is a new least-normal criterion. Visualization work continues with the acquisition of an ImmersaDesk, which will allow improved implementations of algorithms such as the density grand tour. A somewhat new methodology is called Variable Clustering Analysis, which assists in semiparametric density estimation, data analysis, and interpretable dimension reduction. Also in the area of clustering, an algorithm for simplifying complex mixture models fitted by EM is developed, as is a new estimation and testing algorithms for the number of components. The project continues innovative work on spatial modeling and combining many data surveys into useful data modeling and conditional estimation of factors and their covariates in collaboration with researchers in the Department of Agriculture.Nonparametric methodology is widely used in one and two dimensions, but less so in higher dimensions. This research focuses on the mid-range dimensions and provides a deeper understanding of the implications to data modeling of the curse of dimensionality and problems associated with massive data sets. Particular emphasis is given to multivariate regression and density estimation problems, and closely related applications such as clustering, mixture estimation, and dimension reduction. Visualization is especially important when dealing with medium-dimensional data and the growing body of massive data sets. Of special interest is discovering and displaying data in visual clustering and visual discrimination applications. Rice University has acquired an ImmersaDesk, which will allow the implementation of recently developed algorithms in a virtual reality environment. At a recent National Research Council workshop, numerous scientists identified critical statistical needs in their work with massive data sets: alternatives to principal components, specialized visualization tools for exploring massive data, better clustering algorithms, and techniques for handling nonstationary data. Results from this research directly impact three of these four critical opportunities. This program represents a comprehensive and long-term attack on a host of important data analytic problems in multivariate estimation. Graduate training is significant component of this project. The results will be of long-term theoretical interest and will provide short-term solutions to real-world problems.