标题： 一种用于 pooled 协方差估计的分层特征模型 显示英文标题

标题： A hierarchical eigenmodel for pooled covariance estimation

摘要： 虽然对应于不同群体的一组协方差矩阵不太可能完全相等，但它们仍然可能表现出高度的相似性。 例如，一些变量对在大多数群体中可能呈正相关，而另一些变量对之间的相关性则可能始终为负。 在这种情况下，协方差矩阵之间的许多相似性可以通过主轴（由协方差矩阵的特征向量定义的轴）的相似性来描述。 估计跨群体特征向量的异质性程度对于多种估计任务可能会有帮助。 可以将特征向量矩阵汇集起来形成一组中心主轴，并且在这些轴相似的程度上，可以通过将小样本量群体的主轴向总体中心收缩来稳定协方差估计。 为此，本文开发了一种分层模型和估计程序，用于汇集多个群体的主轴。 群组间异质性的模型基于一种矩阵值的反对称 Bingham 分布，该分布能够灵活地描述正交矩阵群体的“中心”和“分散”概念。 显示英文摘要

摘要： While a set of covariance matrices corresponding to different populations are unlikely to be exactly equal they can still exhibit a high degree of similarity. For example, some pairs of variables may be positively correlated across most groups, while the correlation between other pairs may be consistently negative. In such cases much of the similarity across covariance matrices can be described by similarities in their principal axes, the axes defined by the eigenvectors of the covariance matrices. Estimating the degree of across-population eigenvector heterogeneity can be helpful for a variety of estimation tasks. Eigenvector matrices can be pooled to form a central set of principal axes, and to the extent that the axes are similar, covariance estimates for populations having small sample sizes can be stabilized by shrinking their principal axes towards the across-population center. To this end, this article develops a hierarchical model and estimation procedure for pooling principal axes across several populations. The model for the across-group heterogeneity is based on a matrix-valued antipodally symmetric Bingham distribution that can flexibly describe notions of ``center'' and ``spread'' for a population of orthonormal matrices.