Multivariate Nonparametric Methodology Studies

提案ID：DMS-0204723 PI：大卫斯科特标题：多变量非参数方法研究研究人员将研究新的非参数方法，重点是中范围和高范围的维度，以更好地理解数据建模，维数灾难，以及与多元回归和密度估计中的大量数据集相关的问题，以及聚类，混合，模式识别和降维中密切相关的问题。一种新的基于数据的参数估计算法，基于集成平方误差，将研究其灵活性和鲁棒性。将该准则应用于局部多项式的拟合，可以提出一种新的鲁棒非参数回归算法，该算法将应用于亚原子探测器实验中数百条重叠径迹的自动检测。这个项目将研究半参数模型的密度估计，可以比普通的非参数算法更好地工作，通过几个额外的维度扩展可行性。特别感兴趣的是，该算法可以用于拟合全混合模型的子集。应用包括回归，图像处理，聚类，离群值检测，密度估计和可视化。该项目将把空间建模和多种数据调查相结合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量图。 目前，变量的同时绘图难以解释，因为只有离散空间区域（如普查区）的数据可用，而且两个相关变量交叉制表。 通过构造一个变量随第二个变量变化的光滑映射，可以更真实、更准确地理解空间关系。非参数方法在一维和二维中应用广泛，但在高维中应用较少。这项研究的重点是中范围和高范围的维度，并提供了一个更深入的了解的影响，数据建模的维数灾难和与海量数据集相关的问题。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。该建议探讨了新的观点，特别是有关局部自适应和空间估计，以及一些最近的扩展参数问题的非参数标准。新的参数化方法有可能为新的非参数配方和应用。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。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.