Title: Statistical challenges for analysis of data in some non-standard spaces
Authors: Stephan Huckemann - University of Goettingen (Germany) [presenting]
Abstract: Various kinds of non-Euclidean data, spaces modeling such data, data descriptors, inferential methods and three statistical challenges in this context are introduced. 1) In spaces of globally nonpositive curvature, e.g. in Billera-Holmes-Vogtman (BHV) phylogenetic tree spaces, means of probability distributions are unique but there is the price of ``stickiness'' to pay: there may be no asymptotic distribution. In contrast, on compact manifolds (already on spheres) uniqueness may be lost, or asymptotic distributions may have arbitrary slow ``smeary'' rates. 2) In more general spaces, even local uniqueness of geodesics may be lost, such as tropical phylogenetic tree spaces, although they may feature less ``stickiness'' than BHV spaces. 3) In shape analysis, configurations are studied modulo group actions. Once the configurations are no longer in a Euclidean space (they are Euclidean for classical Kendall shapes), it is unclear how to model directions of internal correlations. Already on a two-torus, the only directions producing nonwinding geodesics are horizontal or vertical. We survey workarounds for higher dimensional torus data, as occur in biomolecule modeling or more general polysphere data, as arise in medical imaging.