TY - GEN

T1 - On the CLT on low dimensional stratified spaces

AU - Ellingson, Leif

AU - Hendriks, Harrie

AU - Patrangenaru, Vic

AU - Valentin, Paul San

N1 - Publisher Copyright:
© Springer Science+Business Media New York 2014.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Central limit theorem

KW - Frechet means

KW - Inrinsic mean

KW - Intrinsic means

KW - Stratified space

UR - http://www.scopus.com/inward/record.url?scp=84920036457&partnerID=8YFLogxK

U2 - 10.1007/978-1-4939-0569-0_21

DO - 10.1007/978-1-4939-0569-0_21

M3 - Conference contribution

AN - SCOPUS:84920036457

T3 - Springer Proceedings in Mathematics and Statistics

SP - 227

EP - 239

BT - Topics in Nonparametric Statistics - Proceedings of the 1st Conference of the International Society for Nonparametric Statistics

A2 - Politis, Dimitris N.

A2 - Akritas, Michael G.

A2 - Lahiri, Soumendra N.

PB - Springer New York LLC

Y2 - 15 June 2012 through 19 June 2012

ER -