Multivariate Nonparametric Methodology Studies

9971797本研究项目的重点是发展非参数密度和回归方法在中等范围的尺寸。 密切相关的应用程序，如聚类，混合估计，降维检查与一个新的观点，有关局部自适应和空间估计和最近扩展的非参数标准参数问题。 研究了基于积分平方误差的参数估计算法的灵活性和鲁棒性。 对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，令人惊讶的发现一个解是渐近的大常数。 三个算法寻找有趣的子空间将进行调查。 一个措施的模式的数量;第二个发现最大偏差子空间;和第三个是一个新的最小正常的标准。 随着ImmersaDesk的收购，可视化工作继续进行，这将允许改进算法的实现，如密度大巡游。 一种新的方法叫做变量聚类分析，它有助于半参数密度估计，数据分析和可解释的降维。 此外，在聚类方面，开发了一种简化EM拟合的复杂混合模型的算法，以及一种新的组分数估计和检验算法。 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到有用的数据建模和因子及其协变量的条件估计中。 这项研究的重点是中档尺寸，并提供了一个更深入的理解的影响，数据建模的维数灾难和问题与海量数据集。 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。 在处理中维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ImmersaDesk，这将允许在虚拟现实环境中实施最近开发的算法。 在最近的一次国家研究理事会研讨会上，许多科学家确定了他们在处理海量数据集时的关键统计需求：主成分的替代品，探索海量数据的专用可视化工具，更好的聚类算法，以及处理非平稳数据的技术。 这项研究的结果直接影响这四个关键机会中的三个。 该程序代表了对多变量估计中许多重要数据分析问题的全面和长期攻击。 研究生培训是该项目的重要组成部分。 这些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.