## The Annals of Statistics

### Backward nested descriptors asymptotics with inference on stem cell differentiation

#### 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, 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

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

Dates
Revised: March 2017
First available in Project Euclid: 17 August 2018

https://projecteuclid.org/euclid.aos/1534492826

Digital Object Identifier
doi:10.1214/17-AOS1609

Mathematical Reviews number (MathSciNet)
MR3845008

Zentralblatt MATH identifier
06964323

#### Citation

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. https://projecteuclid.org/euclid.aos/1534492826

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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.