(Semi-)Intrinsic Statistical Analysis on Non-Euclidean Spaces

Abstract

Often, applications from biology and medical imaging lead to data on non-Euclidean spaces. On such spaces the Euclidean concept of a mean forks into several canonical generalizations of non-Euclidean means. More involved data descriptors, for instance principal components generalize into even more complicated concepts. (Semi)-intrinsic statistical analysis allows to study inference on descriptors that can be represented as elements of another non-Euclidean space. We give examples for geodesic principal components on shape spaces, concentric small circles on spheres and configurations on rotation groups. In particular, with respect to the statistical inference via central limit theorems, due to the geometry of the spaces, there are curious non-Euclidean phenomena.