Multivariate Nonparametric Methodology Studies

随着数据收集新技术的出现，对统计方法的要求不断增长。 从统计标准来看，由此产生的许多数据集都是不寻常的：大量且高度非线性且带有污染。 有时，数据来自仅部分已知的机制。估计、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。非参数方法已在一维和二维中广泛使用，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，例如聚类、混合估计、模式识别、鲁棒估计和降维。 统计学家对科学方法的看法是模型构建、数据收集、估计、批评和完善的不断改进的过程。然而，许多执业统计学家因无法修复拟合不佳的模型而陷入困境。这项研究特别感兴趣的是提供关键诊断信息作为模型估计任务一部分的方法。 这项研究的重点是一种特定的基于最小距离数据的参数估计算法，并对其鲁棒性进行了研究。该算法可应用于混合模型和样条拟合。可以拟合不完整的密度模型，这是一种独特的功能，将在回归、图像处理、聚类、异常值检测和密度估计的背景下得到充分探索。 将设计用于功能数据分析的非参数方法并使用新颖的数据源进行测试。其他应用包括自适应小波阈值、混合回归问题的解决方案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为自然科学和社会科学和工程的几乎每个领域的深入应用提供了智力挑战。非参数统计领域通过隐藏但至关重要的算法为科学的成功做出了重大贡献，即使在手机的内部运作中也是如此。 在最近的国家研究委员会研讨会上，许多科学家确定了在处理海量数据集时的关键统计需求：新的降维算法、用于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。该计划代表了对多元估计中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。 研究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.