A Study of Random Matrix Problems Related to Statistics

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)正交和酉阵；(Ii)样本相关矩阵；(Iii)具有矩阵-t分布的矩阵；(Iv)Toeplitz矩阵。基于PI和其他作者关于正交阵的工作，Diaconis提出了一个公开的问题，即一个典型的随机正交阵如何被一个以独立的标准正态随机变量为表项的矩阵逼近。这就是学习的动机(一)。部分(Ii)来自一个统计假设检验问题，当多变量总体分布的维度和来自该总体的数据的样本量较大时。利用主成分分析对样本相关矩阵的最大特征值进行了研究。第三部分是对图像分析中的一个问题进行统计研究。具有矩阵-t分布的矩阵的最大条目是中心感兴趣的。第四部分是RTM中尚未解决的问题。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实践者聚集在一起交流思想：该研究通过提供新的使用技术来帮助实践者，并通过获得解决实际问题的动机来帮助研究人员。矩阵始终位于数据库之后。随机矩阵理论可以在一定意义上对数据库有一个清晰的理解。例如，相关矩阵的最大特征值可以判断多个量是否相互依赖，这是所提出的四个问题之一。此外，这项工作可能会帮助研究生更好地理解这门学科。

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.