Multivariate Nonparametric Methodology Studies

提案ID：DMS-0204723PI：David Scott标题：多变量非参数方法论研究研究人员将研究新的非参数方法论，重点放在中高维，以更好地理解数据建模、维度诅咒、多元回归和密度估计中与海量数据集相关的问题，以及在聚类、混合、模式识别和降维方面密切相关的问题。研究了一种新的基于积分平方误差的基于数据的参数估计算法，该算法具有较强的灵活性和鲁棒性。通过将该准则应用于局部多项式的拟合，提出了一种新的稳健的非参数回归算法，并将其应用于亚原子探测器实验中数百个重叠航迹的自动检测。这个项目将研究密度估计的半参数模型，它可以比普通的非参数算法工作得更好，将可行性扩展了几个额外的维度。特别值得注意的是，该算法可用于拟合完全混合模型的子集。应用包括回归、图像处理、聚类、离群点检测、密度估计和可视化。该项目将把空间建模和综合多种数据调查的工作扩展到有用的数据建模和因素及其协变量的条件估计值地图。目前，由于只在离散的空间区域(如人口普查区域)获得数据，而且两个感兴趣的变量交叉制表，因此很难解释变量的同时映射。通过构建一个变量随着第二个变量的变化而变化的平滑映射，可以更真实和准确地理解空间关系。非参数方法在一维和二维中得到了广泛的应用，但在更高的维度中应用较少。这项研究侧重于中高端维度，并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提供了更深入的理解。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，如聚类、混合估计、模式识别和降维。这一建议考察了新的观点，特别是与局部自适应和空间估计有关的新观点，以及最近非参数标准对参数问题的一些扩展。这种新的参数方法具有开发新的非参数公式和应用的潜力。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。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。

Proposal ID: DMS-0204723PI: David ScottTitle: Multivariate nonparametric methodology studiesThe investigators will study new nonparametric methodology focusing on the mid-range and high-range dimensions to better understand data modeling, the curse of dimensionality, and problems associated with massive data sets in multivariate regression and density estimation as well as closely related problems in clustering, mixtures, pattern recognition, and dimension reduction. A new data-based parametric estimation algorithm, based upon integrated squared error, will be investigated for its flexibility and robustness. By applying the criterion to the fitting of local polynomials, a new robust nonparametric regression algorithm can be proposed, which will be applied to automatic detection of hundreds of overlapping tracks in subatomic detector experiments. This project will examine semiparametric models for density estimation that can work better than ordinary nonparametric algorithms, extending feasibility by several extra dimensions. Of special interest, this algorithm can be used to fit subsets of a full mixture model. Applications include regression, image processing, clustering, outlier detection, density estimation, and visualization. The project will extend work on spatial modeling and the combination of multiple data surveys into useful data modeling and maps of conditional estimators of factors and their covariates. Currently, simultaneous mapping of variables is difficult to interpret, due to the availability of data only in discrete spatial areas (e.g. census tracts) and cross-tabulation of the two variables of interest. By constructing a smooth map of one variable as a second variable varies, a more faithful and accurate understanding of the spatial relationship may be obtained.Nonparametric methodology is widely used in one and two dimensions, but less so in higher dimensions. This research focuses on the mid-range and high-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 will be given to multivariate regression and density estimation problems, and closely related applications such as clustering, mixture estimation, pattern recognition, and dimension reduction. This proposal examines new points of view, especially related to locally adaptive and spatial estimation, as well as some recent extensions of nonparametric criteria to parametric problems. The new parametric approach has potential for new nonparametric formulations and applications. 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. The results will be of long-term theoretical interest and will provide near-term solutions to real-world problems.