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Abstract:
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The aim of this paper is to introduce a probability distribution in the space of symmetric positive de?nite (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 de?ning 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 de?ned 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 de?ne the distribution, investigate some of its properties, and develop a parametric inference procedure for the mean of SPD matrices.
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