The Annals of Statistics

Backward nested descriptors asymptotics with inference on stem cell differentiation

Stephan F. Huckemann and Benjamin Eltzner

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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.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 1994-2019.

Received: September 2016
Revised: March 2017
First available in Project Euclid: 17 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties 62G25
Secondary: 62H11: Directional data; spatial statistics 58C06: Set valued and function-space valued mappings [See also 47H04, 54C60] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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


Huckemann, Stephan F.; Eltzner, Benjamin. Backward nested descriptors asymptotics with inference on stem cell differentiation. Ann. Statist. 46 (2018), no. 5, 1994--2019. doi:10.1214/17-AOS1609.

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Supplemental materials

  • Supplement to “Backward nested descriptors asymptotics with inference on stem cell differentiation”. The supplement contains five appendices. Enumeration of equations, tables and images continues consecutively into the Appendices. Appendix A shows that the relevant assumptions hold true for principal nested spheres (PNS) analysis. Appendix B shows the same for principal components for Kendall’s shape spaces. Appendix C contains full proofs of the theorems. Appendix D consists of a numerical study of the convergence rate of PNS estimators and finally, Appendix E discusses a possible alternative treatment of the data application and its shortcomings.