Multivariate Depth and Quantile Functions: Foundations and Applications

高维和/或复杂性现在是统计数据分析应用中的标准，并且通常现在的数据是多变量的。计算资源的进步使实现相当复杂的方法成为可能。这支持开发强大的方法，系统地考虑到多变量数据集固有的特殊几何特征。尤其重要的是非参数多变量方法的设置。当然，这带来了概念上的挑战。特别是，多元深度和分位数函数现在提供了一种主要的方法，该方法在最近几年得到了很好的确立，但也在积极地进一步发展。在这个项目中，PI解决了这一方法的基础和应用方面的重大开放问题和方向。后者激励了前者，而前者为后者提供了工具。该项目通过为多变量深度和分位数函数开发有用的扩展基础和基础来推进核心统计科学。这些结果有更广泛的应用，并广泛地增强了统计科学在应用中的作用，使新类型的问题能够被更有意义和更有力地处理。该项目的中心主题是：1.产生统计程序的等方差和不变性的变换；2.基于空间深度的修剪，以产生稳健性，而不会造成不必要的计算负担；3.开发和利用深度函数方法和水平集方法之间的新的协同作用，以处理等高线。主题一提供了修改统计程序的工具，以便它们获得所需的某些相同或不变性质，否则这些性质可能不会成立。主题II研究了两个相关但不同的问题的最新解决方案：(I)空间分位数和突出度函数的粗暴，以及(Ii)同时计算容易、稳健和仿射等变的散布估计器。主题III研究了深度函数方法和水平集方法之间有希望但到目前为止尚未探索的协同作用。除了这些重大突破外，该项目还涉及多元L统计量的制定、深度-外延-分位数-等级范式的系统探索、综合数据深度的研究以及函数数据分析中深度方法的研究。作为一个整体，该项目旨在通过统计科学对现代数据处理方法产生变革影响。统计数据分析和建模现在适应了紧迫的新应用领域，涉及多变量数据，将许多变量放在一起。现在，科学、工程、政府和工业的所有领域都经常涉及多变量数据，通常结构复杂，变量数量多。该项目的三个关键技术重点解决了在处理多变量数据时出现的重要和及时的问题。例如，在处理多变量数据中的异常值时，我们需要对哪些点是异常值的分类不能简单地在坐标系发生变化时改变，例如公制到英制。此外，例如，应该有效和准确地确定描绘数据集的中间50%或75%或90%的等高线，而不会受到不是数据中心的极端边远数据点的不适当干扰。或者，例如，当在数据挖掘中发现显著的几何特征或图案时，有必要确定这些发现是真正的本质上有意义的特征，还是它们仅仅是已经被采用并被忽略的特定坐标系的伪像。该项目的另一个关键工作是开发一个新的框架，将最近独立开发的两种不同但相关的多变量分析方法(水平集和深度函数)结合在一起，并使它们能够以协调的方式一起应用。这加强了对这些方法在其特定应用领域中的理解和作用。该项目还通过让研究生和本科生参与，促进统计科学人力资源的教育和发展。项目参与者之间的交叉刺激，可能包括访问研究人员，通过定期会议和团队方法实现。代表人数不足的群体以及初级教员和专业人员的参与是由国际和平协会鼓励和促进的。该项目的结果和结论由国际宣传局通过重要的会议报告、期刊出版物、网站张贴和研究生课程介绍等方式进行传播。

High dimension and/or complexity is now standard in applications of statistical data analysis, and typically data now is multivariate. Advances in computational resources make it feasible to implement quite sophisticated methods. This supports the development of powerful approaches that systematically take into account the special geometric features intrinsic to multivariate data sets. Especially important is the setting of nonparametric multivariate methods. This, of course, presents conceptual challenges. In particular, multivariate depth and quantile functions now provide a major approach that has become well-established in recent years but also is in active further development. In this project, the PI addresses significant open issues and directions in both the foundations and the applications of this approach. The latter inspire the former, and the former yields tools for the latter. This project advances core statistical science by developing useful extended foundations and underpinnings for multivariate depth and quantile functions. The results have even wider application and broadly enhance the role of statistical science in applications, permitting new kinds of problems to be treated more meaningfully and more powerfully. Central themes of the project are: I. Transformations to Produce Equivariance and Invariance of Statistical Procedures, II. Spatial Depth-Based Trimming to Produce Robustness without Undue Computational Burden, and III. Development and Exploitation of a New Synergy Between Depth Function Methods and Level Set Methods for Treating Contours. Topic I provides tools for the modification of statistical procedures so that they acquire desired certain equivariance or invariance properties that may not hold otherwise. Topic II investigates recent solutions to two related but different problems: (i) robustification of the spatial quantile and outlyingness functions, and (ii) simultaneously computationally easy, robust, and affine equivariant scatter estimators. Topic III investigates a promising but hitherto unexplored synergy between depth function methods on one hand and level set methods on the other. Besides these major thrusts, the project also addresses formulation of multivariate L-statistics, systematic exploration of a depth-outlyingness-quantile-rank paradigm, studies on integrated data depth, and studies on depth methods in functional data analysis. As a whole, the project is intended to have transformative impacts on modern approaches to data handling through statistical science.Statistical data analysis and modeling now accommodates pressing new arenas of application involving data that is multivariate, using many variables taken together. All areas of science, engineering, government, and industry now routinely involve multivariate data, typically complex in structure and high in number of variables. The three key technical thrusts of this project address important and timely concerns arising in dealing with multivariate data. For example, in dealing with outliers in multivariate data, we need the classification of which points are outliers not to change simply when there is a simple change of coordinate system, such as metric to British. Also, for example, the contours that delineate the middle 50% or 75% or 90% of a data set should be determined efficiently and accurately without undue interference from extreme outlying data points not central to the data. Or, for example, when striking geometric features or patterns are discovered as in data mining, it is necessary to determine whether such findings are genuine features inherently meaningful or whether they are simply artifacts of the particular coordinate system that has been adopted and to be ignored. Another key effort of this project is to develop a new framework that brings together two different but related methodologies in multivariate analysis that have been recently developed independently (level sets and depth functions) and enables them to be applied together in a coordinated manner. This strengthens the understanding and the roles of these methodologies in their particular domains of application. The project also contributes to education and development of human resources in statistical science by involving graduate students and undergraduate students. Cross-stimulation among project participants, which may include visiting researchers, is achieved through regular meetings and a team approach. The participation of underrepresented groups and junior faculty and professionals is encouraged and fostered by the PI. The results and findings of the project are disseminated by the PI through high-profile conference presentations, journal publications, website postings, and introduction into the graduate curriculum.