University of South Carolina
The statistical analysis of covariance matrices occurs in many important applications, e.g. in diffusion tensor imaging or longitudinal data analysis. Methodology is discussed for estimating covariance matrices which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. We make connections with the use of Procrustes methods in shape analysis, and comparisons are made with other estimation techniques, including using the matrix logarithm, Riemannian metric, matrix square root and Cholesky decomposition. Our main application will be diffusion tensor imaging which is used to study the fiber structure in the white matter of the brain. Diffusion weighted MR images involve applying a set of gradients in a design of directions, and the recorded data are related to the Fourier transforms of the displacement of water molecules at each voxel. Under a multivariate Gaussian diffusion model a single tensor (3 x 3 covariance matrix) is fitted at each voxel. We discuss the statistical analysis of diffusion tensors, including construction of means, spatial interpolation, geodesics, and principal components analysis. This is joint work with Alexey Koloydenko and Diwei Zhou.