Applying Backward Nested Subspace Inference to Tori and Polyspheres

Abstract

For data with non-Euclidean geometric structure, hypothesis testing is challenging because most statistical tools, for example principal component analysis (PCA), are specific for linear data with a Euclidean structure. In the last 15 years, the subject has advanced through the emerging development of central limit theorems, first for generalizations of means, then also for geodesics and more generally for lower dimensional subspaces. Notably, there are data spaces, even geometrically very benign, such as the torus, where this approach is statistically not feasible, unless the geometry is changed, to that of a sphere, say. This geometry is statistically so benign that nestedness of Euclidean PCA, which is usually not given for the above general approaches, is also naturally given through principal nested great spheres (PNGS) and even more flexible than Euclidean PCA through principal nested (small) spheres (PNS). In this contribution we illustrate applications of bootstrap two-sample tests for the torus and its higher dimensional generalizations, polyspheres.

Acknowledging the Niedersachsen Vorab of the Volkswagen Foundation.