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October 2018 Backward nested descriptors asymptotics with inference on stem cell differentiation
Stephan F. Huckemann, Benjamin Eltzner
Ann. Statist. 46(5): 1994-2019 (October 2018). DOI: 10.1214/17-AOS1609


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, Dryden and Marron [Biometrika 99 (2012) 551–568] 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.


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Stephan F. Huckemann. Benjamin Eltzner. "Backward nested descriptors asymptotics with inference on stem cell differentiation." Ann. Statist. 46 (5) 1994 - 2019, October 2018.


Received: 1 September 2016; Revised: 1 March 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964323
MathSciNet: MR3845008
Digital Object Identifier: 10.1214/17-AOS1609

Primary: 62G20 , 62G25
Secondary: 58C06 , 60D05 , 62H11

Keywords: asymptotic consistency and normality , dimension reduction on manifolds , flags of subspaces , Fréchet means , geodesic principal component analysis , Kendall’s shape spaces , principal nested spheres

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • October 2018
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