A Study of Random Matrix Problems Related to Statistics

统计相关随机矩阵问题的研究

• 批准号: 0308151
• 负责人: Tiefeng Jiang
• 金额: $ 10.03万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0308151&HistoricalAwards=false
• 关键词: Study; Random; Matrix; Problems; Related

Within the general area of random matrix problems, the PI will especially focus on problems relevant to the following four types of matrices: (i) orthogonal and unitary matrices; (ii) sample correlation matrices; (iii) matrices with matrix-t distribution; (iv) Toeplitz matrices. Based on the PI's and other authors' work on orthogonal matrices, Diaconis has posed an open problem on how a typical random orthogonal matrix can be approximated by a matrix with independent standard normal random variables as entries. This is the motivation to study (i). Part (ii) comes from a statistical hypothesis testing problem when the dimension of a multivariate population distribution and the sample sizes of data from this population are large. By using Principal Component analysis, the maximum eigenvalue of the sample correlation matrix has to be studied. Part (iii) arises from a statistical study on a problem from Image Analysis. The largest entry of matrices with a matrix-t distribution is of central interest. Part (iv) is an unsolved problem in RTM. This type of matrix arises in time series analysis.The problems studied come from trading markets, engineering and science. The solutions can bring researchers and practitioners from different fields together to exchange ideas: the study helps practitioners by providing new techniques for use and researchers by obtaining motivation and real problems for solution. Matrices are always behind databases. Random matrix theories may give a clean understanding of databases in a certain sense. For example, the largest eigenvalue of a correlation matrix, which is one of the four proposed problems, can tell if multiple quantities depend on each other or not. Further, this work may help graduate students gain a better understanding of this subject.

在随机矩阵问题的一般领域内，PI将特别关注与以下四种类型的矩阵相关的问题：（i）正交和酉矩阵;（ii）样本相关矩阵;（iii）矩阵t分布矩阵;（iv）Toeplitz矩阵。基于PI和其他作者对正交矩阵的研究，Diaconis提出了一个公开问题，即如何用一个具有独立标准正态随机变量的矩阵来近似一个典型的随机正交矩阵。这就是学习的动机（一）。第（ii）部分来自一个统计假设检验问题，当一个多变量总体分布的维数和来自这个总体的数据的样本量是大的。利用主成分分析方法，研究样本相关矩阵的最大特征值。第（iii）部分来自图像分析问题的统计研究。具有矩阵-t分布的矩阵的最大项是中心兴趣。第（iv）部分是RTM中未解决的问题。这种类型的矩阵出现在时间序列分析中。研究的问题来自交易市场，工程和科学。这些解决方案可以将来自不同领域的研究人员和从业人员聚集在一起交流思想：研究通过提供新的技术来帮助从业人员使用，研究人员通过获得动力和真实的问题来解决。矩阵总是在数据库后面。随机矩阵理论在一定意义上可以给数据库一个清晰的理解。例如，相关矩阵的最大特征值，这是四个提出的问题之一，可以告诉如果多个量相互依赖或不。此外，这项工作可以帮助研究生更好地了解这个问题。