On the CLT on Low Dimensional Stratified Spaces

Abstract

AbstractNoncategorical 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.KeywordsCentral limit theoremStratified spaceFrechet meansIntrinsic meansInrinsic means