Multivariate Nonparametric Methodology Studies

项目编号：DMS-0204723PI: David scott标题：多变量非参数方法学研究研究人员将研究新的非参数方法学，专注于中范围和高范围维度，以更好地理解数据建模，维度的灾难，以及与多变量回归和密度估计中的大量数据集相关的问题，以及与聚类，混合，模式识别和降维密切相关的问题。本文将研究一种新的基于数据的参数估计算法，该算法基于积分平方误差，具有灵活性和鲁棒性。将该准则应用于局部多项式的拟合，提出了一种新的鲁棒非参数回归算法，可用于亚原子探测器实验中数百条重叠轨迹的自动检测。该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更好地工作，将可行性扩展到几个额外的维度。特别有趣的是，该算法可用于拟合全混合模型的子集。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多个数据调查的结合工作扩展到有用的数据建模和因子及其协变量的条件估计量图。目前，变量的同时映射很难解释，因为只有在离散的空间区域（例如人口普查区）的数据可用性和两个感兴趣的变量的交叉制表。通过构建一个变量作为第二个变量变量的平滑映射，可以获得对空间关系更忠实和准确的理解。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。本研究侧重于中高维度，并提供了对维度诅咒和与大量数据集相关的问题对数据建模的影响的更深层次的理解。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。本提案探讨了新的观点，特别是与局部自适应和空间估计有关的观点，以及一些最近将非参数准则扩展到参数问题的观点。新的参数化方法具有新的非参数化公式和应用的潜力。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。研究结果将具有长期的理论价值，并将为现实世界的问题提供近期的解决方案。

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.