21 research outputs found
Large sample theory of intrinsic and extrinsic sample means on manifolds--II
This article develops nonparametric inference procedures for estimation and
testing problems for means on manifolds. A central limit theorem for Frechet
sample means is derived leading to an asymptotic distribution theory of
intrinsic sample means on Riemannian manifolds. Central limit theorems are also
obtained for extrinsic sample means w.r.t. an arbitrary embedding of a
differentiable manifold in a Euclidean space. Bootstrap methods particularly
suitable for these problems are presented. Applications are given to
distributions on the sphere S^d (directional spaces), real projective space
RP^{N-1} (axial spaces), complex projective space CP^{k-2} (planar shape
spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space
\Sigma_3^4.Comment: Published at http://dx.doi.org/10.1214/009053605000000093 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Topological Data Analysis for Object Data
Statistical analysis on object data presents many challenges. Basic summaries
such as means and variances are difficult to compute. We apply ideas from
topology to study object data. We present a framework for using persistence
landscapes to vectorize object data and perform statistical analysis. We apply
to this pipeline to some biological images that were previously shown to be
challenging to study using shape theory. Surprisingly, the most persistent
features are shown to be "topological noise" and the statistical analysis
depends on the less persistent features which we refer to as the "geometric
signal". We also describe the first steps to a new approach to using topology
for object data analysis, which applies topology to distributions on object
spaces.Comment: 16 pages, 12 figure
Sticky central limit theorems on open books
Given a probability distribution on an open book (a metric space obtained by
gluing a disjoint union of copies of a half-space along their boundary
hyperplanes), we define a precise concept of when the Fr\'{e}chet mean
(barycenter) is sticky. This nonclassical phenomenon is quantified by a law of
large numbers (LLN) stating that the empirical mean eventually almost surely
lies on the (codimension and hence measure ) spine that is the glued
hyperplane, and a central limit theorem (CLT) stating that the limiting
distribution is Gaussian and supported on the spine. We also state versions of
the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on
the spine) and partly sticky (i.e., is, on the spine but not sticky).Comment: Published in at http://dx.doi.org/10.1214/12-AAP899 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org