Geometric Means of Positive Definite Matrices and the Matrix-Variate Log-Normal Distribution

• Activity Number: 507
• Type: Invited
• Date/Time: Wednesday, August 7, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Armin Schwartzman*+
• Companies: Harvard School of Public Health
• Keywords: random matrix ; Riemannian manifold ; geometric mean ; intrinsic mean ; diffusion tensor imaging

Abstract

This work introduces a new lognormal family of distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. This family arises as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. PD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of PD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, helping resolve the choice of voxelwise average type for this form of PD matrix data.