Estimation of Functionals of High Dimensional Covariance Matrices

高维协方差矩阵泛函的估计

• 批准号: 1209191
• 负责人: Huibin Zhou
• 金额: $ 30万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1209191&HistoricalAwards=false
• 关键词: Estimation; Functionals; Dimensional; Covariance; Matrices

In a wide range of applications involving covariance, people are interested in certain aspects, i.e., functionals, of the covariance structure. Despite recent progress on methodological work on covariance matrices estimation, there has been remarkably little fundamental theoretical and methodological studies on optimal estimation of functionals of high dimensional covariance matrices. The goal of this proposal is to develop a coherent theory to unveil the precision to which covariance matrix functionals can be estimated, to develop general methodologies for optimal estimation of functionals of covariance matrices, to establish the asymptotic equivalence between covariance matrices estimation and Gaussian sequence models, and to address applications that arise in finance, bioinformatics, genomics, and meteorology etc. The research presented in this proposal will significantly advance the theoretical understanding of estimating functionals of large scale covariance matrices. In particular, the asymptotic equivalence theory to be developed will help build a close and inspiring connection between matrices estimation and classical Gaussian sequence models which had been well studied in the past thirty years, and help carry over results and methodologies developed in Gaussian sequence models to matrices estimation. The proposed optimal estimation procedures for those fundamentally important statistics methodologies, for example, principal components analysis, graphical model, and linear and quadratic discriminant analysis, will provide more accurate estimation and prediction rules in a wide range of applications.With the emergence of high dimensional data from modern technologies, estimating large scale covariance matrices and their functionals is becoming a crucial problem in many fields including climate studies, genomics and proteomics, functional magnetic resonance imaging, portfolio allocation and risk management. The traditional sample covariance matrix estimator has been used frequently in practice when analyzing high dimensional data, which may result in poor performance and invalid conclusions. To overcome the difficulty associated with the high dimensionality, regularized methods have been developed in recent years. A central role of covariance matrix in statistical analysis and its wide range of important statistical applications ensure that the progress the investigator and his colleagues make towards their proposed objectives will have a great impact in the broad scientific community which includes astronomy, bioinformatics, finance, genomics, meteorology and clinical research. Research results from this proposal will be disseminated through research articles, workshops and seminar series to researchers in other disciplines. The project will integrate research and education by teaching monograph courses and organizing workshops and seminars to help graduate students and postdocs, particularly minority, women, and domestic students and young researchers, work on this topic.

在涉及协方差的广泛应用中，人们对某些方面感兴趣，即，协方差结构的泛函。尽管最近的协方差矩阵估计的方法学工作取得了进展，有显着的基本理论和方法的研究很少高维协方差矩阵的泛函的最佳估计。该提案的目标是发展一个连贯的理论，以揭示协方差矩阵泛函可以估计的精度，发展协方差矩阵泛函最优估计的一般方法，建立协方差矩阵估计和高斯序列模型之间的渐近等价性，并解决金融，生物信息学，基因组学，本文的研究将对大尺度协方差矩阵泛函估计的理论研究有重要的推动作用。特别是，渐近等价理论的发展将有助于建立一个密切的和鼓舞人心的联系之间的矩阵估计和经典的高斯序列模型，已在过去的三十年中得到了很好的研究，并帮助结转的结果和方法，高斯序列模型的矩阵估计。本文提出的主成分分析、图模型、线性判别分析和二次判别分析等重要统计方法的最优估计方法，将为更广泛的应用提供更精确的估计和预测规则。大尺度协方差矩阵及其泛函的估计在气候研究、基因组学和蛋白质组学、功能磁共振成像、投资组合分配和风险管理等领域都是一个非常重要的问题。传统的样本协方差矩阵估计在实际应用中经常被用于分析高维数据，这可能导致性能不佳和结论无效。为了克服与高维相关的困难，近年来已经开发了正则化方法。 协方差矩阵在统计分析中的核心作用及其广泛的重要统计应用确保了研究者及其同事在实现其拟议目标方面取得的进展将对广泛的科学界产生重大影响，包括天文学，生物信息学，金融，基因组学，气象学和临床研究。这项建议的研究成果将通过研究文章、讲习班和系列研讨会向其他学科的研究人员传播。该项目将通过教授专题课程和组织讲习班和研讨会，将研究与教育结合起来，以帮助研究生和博士后，特别是少数民族、妇女、国内学生和年轻研究人员从事这一专题的工作。