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September 2009 Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
Ian L. Dryden, Alexey Koloydenko, Diwei Zhou
Ann. Appl. Stat. 3(3): 1102-1123 (September 2009). DOI: 10.1214/09-AOAS249

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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Ian L. Dryden. Alexey Koloydenko. Diwei Zhou. "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

Information

Published: September 2009
First available in Project Euclid: 5 October 2009

zbMATH: 1196.62063
MathSciNet: MR2750388
Digital Object Identifier: 10.1214/09-AOAS249

Keywords: Anisotropy , Cholesky , Geodesic , matrix logarithm , principal components , Procrustes , Riemannian , shape , size , Wishart

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.3 • No. 3 • September 2009
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