Multivariate Nonparametric Methodology Studies

9971797本研究项目的重点是非参数密度和回归方法在中程维度的发展。密切相关的应用，如聚类，混合估计和降维与一个新的观点进行了检查，涉及局部自适应和空间估计和非参数标准的参数问题的最新扩展。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鲁棒性。继续研究了局部自适应曲线估计的带宽选择问题的对偶解，惊人地发现其中一个解是渐近的大常数。本文将研究三种寻找感兴趣子空间的算法。一种测量模式的数量；第二步求最大偏置子空间；第三个是新的最低正常标准。随着ImmersaDesk的收购，可视化工作将继续进行，这将允许改进算法的实现，如密度大巡展。一种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。在聚类领域，提出了一种简化EM拟合的复杂混合模型的算法，以及一种新的分量数估计和测试算法。该项目继续在空间建模方面进行创新工作，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。非参数方法学在一维和二维中应用广泛，但在高维中应用较少。本研究侧重于中等维度，并提供了对维度诅咒和与大规模数据集相关的问题对数据建模的影响的更深入理解。特别强调多元回归和密度估计问题，以及密切相关的应用，如聚类、混合估计和降维。在处理中等维度数据和不断增长的海量数据集时，可视化尤为重要。特别感兴趣的是在视觉聚类和视觉识别应用中发现和显示数据。莱斯大学已经收购了一个ImmersaDesk，它将允许在虚拟现实环境中实现最近开发的算法。在最近的一次国家研究委员会研讨会上，许多科学家确定了他们在处理大量数据集时的关键统计需求：主成分的替代方案、用于探索大量数据的专门可视化工具、更好的聚类算法和处理非平稳数据的技术。这项研究的结果直接影响了这四个关键机会中的三个。这个程序代表了对多元估计中重要数据分析问题的主机的全面和长期攻击。研究生培训是这个项目的重要组成部分。研究结果将具有长期的理论价值，并将为现实世界的问题提供短期的解决方案。

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.