Non-Euclidean statistical analysis of covariance matrices and diffusion tensors

TL;DR

This paper explores non-Euclidean statistical methods for analyzing covariance matrices and diffusion tensors, focusing on metrics, anisotropy, and interpolation techniques relevant to applications like diffusion tensor imaging.

Contribution

It introduces a framework for covariance matrix analysis using various metrics, including the Procrustes size-and-shape metric, and discusses regularization methods for tensors.

Findings

Procrustes metric is effective for near-deficient rank matrices.

Different metrics offer advantages depending on application context.

Regularization techniques improve tensor analysis robustness.

Abstract

The statistical analysis of covariance matrices occurs in many important applications, e.g. in diffusion tensor imaging and longitudinal data analysis. We consider the situation where it is of interest to estimate an average covariance matrix, describe its anisotropy, to carry out principal geodesic analysis and to interpolate between covariance matrices. There are many choices of metric available, each with its advantages. The particular choice of what is best will depend on the particular application. The use of the Procrustes size-and-shape metric is particularly appropriate when the covariance matrices are close to being deficient in rank. We discuss the use of different metrics for diffusion tensor analysis, and we also introduce certain types of regularization for tensors.