Backward Nested Descriptors Asymptotics with Inference on Stem Cell Differentiation

TL;DR

This paper develops asymptotic theory for backward nested descriptors in manifold data, enabling inference on stem cell differentiation and providing tools for statistical analysis of complex geometric data.

Contribution

It introduces general asymptotic results for backward nested descriptors, including consistency and normality, applicable to principal nested spheres and geodesic analysis.

Findings

Asymptotic strong consistency of BNFDs established.

Joint asymptotic normality derived under general conditions.

Application to stem cell data identified differentiation change points.

Abstract

For sequences of random backward nested subspaces as occur, say, in dimension reduction for manifold or stratified space valued data, asymptotic results are derived. In fact, we formulate our results more generally for backward nested families of descriptors (BNFD). Under rather general conditions, asymptotic strong consistency holds. Under additional, still rather general hypotheses, among them existence of a.s. local twice differentiable charts, asymptotic joint normality of a BNFD can be shown. If charts factor suitably, this leads to individual asymptotic normality for the last element, a principal nested mean or a principal nested geodesic, say. It turns out that these results pertain to principal nested spheres (PNS) and principal nested great subsphere (PNGS) analysis by Jung et al. (2010) as well as to the intrinsic mean on a first geodesic principal component (IMo1GPC) for manifolds and Kendall's shape spaces. A nested bootstrap two-sample test is derived and illustrated with simulations. In a study on real data, PNGS is applied to track early human mesenchymal stem cell differentiation over a coarse time grid and, among others, to locate a change point with direct consequences for the design of further studies.