81 – Regression, Distribution and Inference
Cholesky Normal Distribution in the Space of Symmetric Positive-Definite Matrices
Benoit Ahanda
Bradley University
Leif Ellingson
Texas Tech University
Daniel E. Osborne
Florida Agricultural and Mechanical University
The aim of this paper is to introduce a probability distribution in the space of symmetric positive definite (SPD) matrices called the Cholesky normal distribution. Because the space of SPD matrices is a non-Euclidean manifold, standard arithmetic and thusly standard statistical methods do not directly apply for data on this space. Instead, researchers typically either perform an intrinsic analysis by defining a Riemannian metric and then projecting the data onto a tangent space or an intrinsic analysis by embedding the space into the space of symmetric matrices. For both approaches, since there are not many probability distributions defined on the space of SPD matrices, researchers typically use nonparametric inference procedures, which may be too computationally expensive for practical use on large-scale data analyses. Following from Schwartzman (2015), we utilize the Cholesky metric on the space of SPD matrices to define the distribution, investigate some of its properties, and develop a parametric inference procedure for the mean of SPD matrices.