Geometric Perspectives on the Correlation

相关性的几何视角

• 批准号: 1613112
• 负责人: Kai Zhang
• 金额: $ 12万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613112&HistoricalAwards=false
• 关键词: Geometric; Perspectives; Correlation

In modern statistical analysis, datasets often contain a large number of variables with complicated dependence structures. This situation is especially common in important problems in economics, engineering, finance, genetics, genomics, neurosciences, etc. One of the most important measures on the dependence between variables is the correlation coefficient, which describes their linear dependence. In the new paradigm described above, understanding the correlation and the behavior of correlated variables is a crucial problem and prompts statisticians to develop new theories and methods. Motivated by this challenge, the PI proposes to study the correlation through novel geometric perspectives. The overall objective is (1) to develop useful theories and methods on the correlation and (2) to build a stronger connection between geometry and statistics. The PI anticipates the achievement of his goals through an integration of research and education plans.The research agenda is to systematically investigate three fundamental aspects of the correlation: (1) the magnitude and distribution of the maximal spurious sample correlation; (2) the detection of a low-rank correlation structure; and (3) the probability measure over the space of correlation matrices. In these studies, the novel integration of statistical and geometric insights characterizes the proposed solutions and facilitates precise probability statements. Completion of the proposed research will provide a comprehensive understanding of the correlation and a stronger connection between geometry and statistics. The PI also has comprehensive plans on educating graduate and undergraduate students and on disseminating the research results to the broader scientific community.

在现代统计分析中，数据集通常包含大量具有复杂依赖结构的变量。这种情况在经济学、工程、金融、遗传学、基因组学、神经科学等重要问题中尤其常见。变量之间依赖性最重要的衡量标准之一是相关系数，它描述了它们的线性依赖性。在上述新范式中，理解相关性和相关变量的行为是一个关键问题，并促使统计学家开发新的理论和方法。受这一挑战的推动，PI 提议通过新颖的几何视角来研究相关性。总体目标是（1）开发有关相关性的有用理论和方法，以及（2）在几何和统计学之间建立更牢固的联系。 PI期望通过研究和教育计划的整合来实现他的目标。研究议程是系统地研究相关性的三个基本方面：（1）最大虚假样本相关性的大小和分布； (2)低秩相关结构的检测； (3) 相关矩阵空间上的概率测度。在这些研究中，统计和几何见解的新颖整合表征了所提出的解决方案，并促进了精确的概率陈述。完成拟议的研究将提供对几何和统计学之间的相关性和更强的联系的全面理解。 PI 还制定了教育研究生和本科生以及向更广泛的科学界传播研究成果的全面计划。

项目成果

Calibrated percentile double bootstrap for robust linear regression inference

用于稳健线性回归推理的校准百分位双引导

• DOI: 10.5705/ss.202016.0546
• 发表时间: 2018
• 期刊: Statistica sinica
• 影响因子: 1.4
• 作者: Daniel McCarthy, Kai Zhang