Geometric Perspectives on the Correlation

在现代统计分析中，数据集往往包含大量具有复杂相关性结构的变量。这种情况在经济、工程、金融、遗传学、基因组学、神经科学等重要问题中尤为常见。衡量变量之间相关性的最重要的度量之一是相关系数，它描述了变量之间的线性相关性。在上述新范式中，理解相关变量的相关性和行为是一个至关重要的问题，并促使统计学家开发新的理论和方法。在这一挑战的激励下，PI建议通过新的几何角度来研究这种相关性。总体目标是(1)发展关于相关性的有用理论和方法，(2)在几何学和统计学之间建立更紧密的联系。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这种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)低等级相关结构的检测；(3)相关矩阵空间上的概率度量。在这些研究中，统计和几何洞察力的新颖整合是所提出的解决方案的特征，并促进了精确的概率陈述。拟议研究的完成将使人们对几何学和统计学之间的联系有一个全面的了解和更强的联系。该研究所还对研究生和本科生的教育以及向更广泛的科学界传播研究成果制定了全面的计划。

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.