On the CLT on low dimensional stratified spaces

Leif Ellingson, Harrie Hendriks, Vic Patrangenaru, Paul San Valentin

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Noncategorical observations, when regarded as points on a stratified space, lead to a nonparametric data analysis extending data analysis on manifolds. In particular, given a probability measure on a sample space with a manifold stratification, one may define the associated Fréchet function, Fréchet total variance, and Fréchet mean set. The sample counterparts of these parameters have a more nuanced asymptotic behaviors than in nonparametric data analysis on manifolds. This allows for the most inclusive data analysis known to date. Unlike the case of manifolds, Fréchet sample means on stratified spaces may stick to a lower dimensional stratum, a new dimension reduction phenomenon. The downside of stickiness is that it yields a less meaningful interpretation of the analysis. To compensate for this, an extrinsic data analysis, that is more sensitive to input data is suggested. In this paper one explores analysis of data on low dimensional stratified spaces, via simulations. An example of extrinsic analysis on phylogenetic tree data is also given.

Original languageEnglish
Title of host publicationTopics in Nonparametric Statistics - Proceedings of the 1st Conference of the International Society for Nonparametric Statistics
EditorsDimitris N. Politis, Michael G. Akritas, Soumendra N. Lahiri
PublisherSpringer New York LLC
Pages227-239
Number of pages13
ISBN (Electronic)9781493905683
DOIs
StatePublished - 2014
Event1st Conference of the International Society of Nonparametric Statistics, ISNPS 2012 - Chalkidiki, Greece
Duration: Jun 15 2012Jun 19 2012

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume74
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

Conference1st Conference of the International Society of Nonparametric Statistics, ISNPS 2012
CountryGreece
CityChalkidiki
Period06/15/1206/19/12

Keywords

  • Central limit theorem
  • Frechet means
  • Inrinsic mean
  • Intrinsic means
  • Stratified space

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  • Cite this

    Ellingson, L., Hendriks, H., Patrangenaru, V., & Valentin, P. S. (2014). On the CLT on low dimensional stratified spaces. In D. N. Politis, M. G. Akritas, & S. N. Lahiri (Eds.), Topics in Nonparametric Statistics - Proceedings of the 1st Conference of the International Society for Nonparametric Statistics (pp. 227-239). (Springer Proceedings in Mathematics and Statistics; Vol. 74). Springer New York LLC. https://doi.org/10.1007/978-1-4939-0569-0_21