Multivariate Nonparametric Methodology Studies

随着新的数据收集技术的出现，对统计方法的需求不断增长。按照统计标准，许多由此产生的数据集都是不同寻常的：海量和高度非线性的污染。有时，数据来自一种仅部分为人所知的机制。评估、测试、功能测试、模式发现、特征提取、可视化和比较等任务要求统计学家重新审视每个问题。在一维和二维中广泛使用的非参数方法也适用于这些更高的维度。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、稳健估计和降维。统计学家对科学方法的看法是，建立模型、收集数据、估计、批评和改进的过程是一个不断改进的过程。然而，许多执业统计学家因无法修复拟合不佳的模型而受阻。在这项研究中特别感兴趣的是提供关键诊断信息作为模型估计任务的一部分的方法。本研究的一个重点是一种特殊的基于最小距离数据的参数估计算法，该算法的稳健性得到了研究。该算法可应用于混合模型和样条线拟合。可以拟合不完整的密度模型，这是一种独特的能力，将在回归、图像处理、聚类、离群值检测和密度估计的背景下得到充分探索。函数数据分析的非参数方法将被设计并使用新的数据源进行测试。其他应用包括自适应小波阈值、混合回归问题的求解以及仅适用于数据子集的模型的应用。数据分析和统计建模的研究提供了智力挑战，在自然科学、社会科学和工程的几乎每个领域都有深入的应用。非参数统计领域为科学的成功做出了重大贡献，其算法是隐藏的，但即使在手机的内部工作中也是关键的。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时的关键统计需求：新的降维算法、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。这项拟议研究的结果直接影响到这四个关键机会中的三个。该项目代表着对多变量估计中的许多重要数据分析问题的全面和长期的攻击。研究生培训是该项目的重要组成部分。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。

The demands on statistical methodology have grown relentlessly as new technologies for data collection appear. Many of the resulting datasets are unusual by statistical standards: massive and highly nonlinear with contamination. Sometimes the data come from a mechanism which is only partially known. The tasks of estimation, testing, functional testing, pattern discovery, feature extraction, visualization, and comparison require the statistician look at each problem anew. Nonparametric methodology, which has been widely used in one and two dimensions, is also appropriate in these higher dimensions. Particular emphasis will be given to multivariate regression and density estimation problems, and closely related applications such as clustering, mixture estimation, pattern recognition, robust estimation, and dimension reduction. The statistician's view of the scientific method is as a continuously improving process of model building, data collection, estimation, criticism, and refinement. However, many practicing statisticians are stymied by an inability to repair poorly fitting models. Of particular interest in this research are methods which provide critical diagnostic information as part of the model estimation task. A focus of this research is a particular minimum-distance data-based parametric estimation algorithm, which has been investigated for its robustness properties. The algorithm can be applied to mixture models and spline fitting. An incomplete density model may be fitted, a unique capability that will be explored fully in the context of regression, image processing, clustering, outlier detection, and density estimation. Nonparametric methodology for functional data analysis will be devised and tested with novel data sources.Other applications include adaptive wavelet thresholding, solution of the mixture of regression problems, and application to models which apply to only a subset of the data.Research in data analysis and statistical modeling provides intellectual challenges with deep applications in almost every field of natural and social sciences and engineering. The field of nonparametric statistics has made a significant contribution to the success of science with algorithms that are hidden but critical even in the inner workings of cell phones. At a recent National Research Council workshop, numerous scientists identified critical statistical needs in their work with massive data sets: new dimension reduction algorithms, specialized visualization tools for exploring massive data, better clustering algorithms, and techniques for handling nonstationary data. Results from this proposed 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.