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.
"Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging." Ann. Appl. Stat. 3 (3) 1102 - 1123, September 2009. https://doi.org/10.1214/09-AOAS249