Lognormal Distributions and Geometric Averages of Positive Definite Matrices

TL;DR

This paper defines a lognormal distribution family for positive definite matrices, explores their geometric averages under different metrics, and applies these concepts to diffusion tensor imaging data analysis.

Contribution

It introduces a formal lognormal family for PD matrices and derives their distributions as limits of geometric averages under two geometries.

Findings

Large-sample confidence regions for matrix means

Comparison of averaging methods in diffusion imaging

Distributional limits for geometric matrix averages

Abstract

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained 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, permitting a comparison between the various average types from the point of view of their sampling variability.