University of Washington - Department of Statistics
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 other pairs may be consistently negative. In such cases the similarities 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. Similar eigenvector matrices can be pooled to form a central set of principal axes, and covariance estimation for populations having small sample sizes can be stabilized by shrinking estimates of their population-specific 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 â€œcenterâ€ and â€œspreadâ€ for a population of orthonormal matrices.