Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
Ian L. Dryden, Alexey Koloydenko, and Diwei Zhou
Source: Ann. Appl. Stat. Volume 3, Number 3
(2009), 1102-1123.
Abstract
The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes Anisotropy is discussed.
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Keywords: Anisotropy; Cholesky; geodesic; matrix logarithm; principal components; Procrustes; Riemannian; shape; size; Wishart
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