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的标准，通常数据现在是多变量的。计算资源的进步使实现相当复杂的方法成为可能。这支持了系统地考虑多元数据集固有的特殊几何特征的强大方法的发展。特别重要的是非参数多元方法的设置。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提供了近年来已经建立起来的一种主要方法，并且还在积极地进一步发展。在这个项目中，PI解决了该方法的基础和应用中的重要开放问题和方向。后者激励前者，而前者又为后者提供工具。该项目通过开发有用的扩展基础和支撑多元深度和分位数函数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，使新类型的问题得到更有意义和更有力的处理。该项目的中心主题是：1 .产生统计程序的等变性和不变性的转换；基于空间深度的修剪产生鲁棒性，而没有不必要的计算负担；深度函数法与水平集法在轮廓处理中的新协同作用的发展与利用。主题1提供了用于修改统计程序的工具，以便它们获得所需的某些等变性或不变性属性，否则可能无法保持。主题II研究了两个相关但不同的问题的最新解决方案：(i)空间分位数和离群函数的鲁棒性，以及（II）同时计算简单，鲁棒和仿射等变散射估计。主题III研究了深度函数方法和水平集方法之间有前途但迄今尚未探索的协同作用。除了这些重点外，本项目还涉及多元l统计的制定、深度-离群度-分位数-秩范式的系统探索、综合数据深度研究以及功能数据分析中的深度方法研究。总体而言，该项目旨在通过统计科学对现代数据处理方法产生变革性影响。统计数据分析和建模现在适应了涉及多变量数据的应用程序的新领域，这些数据使用许多变量组合在一起。科学、工程、政府和工业的所有领域现在都经常涉及多变量数据，这些数据通常结构复杂，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离群点时，我们需要对哪些点是离群点的分类不能在简单的坐标系变化（如公制到英制）时简单地改变。此外，例如，描绘数据集中间50%或75%或90%的轮廓应该有效而准确地确定，而不会受到非数据中心的极端离群数据点的不当干扰。或者，例如，当在数据挖掘中发现引人注目的几何特征或模式时，有必要确定这些发现是具有内在意义的真实特征，还是仅仅是采用的特定坐标系统的工件而被忽略。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（可能包括访问研究人员）之间的交叉激励是通过定期会议和团队方式实现的。PI鼓励和促进代表性不足的群体和初级教师和专业人员的参与。该项目的成果和发现由PI通过高知名度的会议报告、期刊出版物、网站帖子和研究生课程介绍来传播。

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.