15 research outputs found
M\"{o}bius deconvolution on the hyperbolic plane with application to impedance density estimation
In this paper we consider a novel statistical inverse problem on the
Poincar\'{e}, or Lobachevsky, upper (complex) half plane. Here the Riemannian
structure is hyperbolic and a transitive group action comes from the space of
real matrices of determinant one via M\"{o}bius transformations. Our
approach is based on a deconvolution technique which relies on the
Helgason--Fourier calculus adapted to this hyperbolic space. This gives a
minimax nonparametric density estimator of a hyperbolic density that is
corrupted by a random M\"{o}bius transform. A motivation for this work comes
from the reconstruction of impedances of capacitors where the above scenario on
the Poincar\'{e} plane exactly describes the physical system that is of
statistical interest.Comment: Published in at http://dx.doi.org/10.1214/09-AOS783 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Sticky Flavors
The Fr\'echet mean, a generalization to a metric space of the expectation of
a random variable in a vector space, can exhibit unexpected behavior for a wide
class of random variables. For instance, it can stick to a point (more
generally to a closed set) under resampling: sample stickiness. It can stick to
a point for topologically nearby distributions: topological stickiness, such as
total variation or Wasserstein stickiness. It can stick to a point for slight
but arbitrary perturbations: perturbation stickiness. Here, we explore these
and various other flavors of stickiness and their relationship in varying
scenarios, for instance on CAT() spaces, .
Interestingly, modulation stickiness (faster asymptotic rate than )
and directional stickiness (a generalization of moment stickiness from the
literature) allow for the development of new statistical methods building on an
asymptotic fluctuation, where, due to stickiness, the mean itself features no
asymptotic fluctuation. Also, we rule out sticky flavors on manifolds in
scenarios with curvature bounds
Confidence tubes for curves on SO(3) and identification of subject-specific gait change after kneeling
In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which might be a major risk factor for the development of knee osteoarthritis, we develop confidence tubes for curves following a perturbation model on SO(3) using the Gaussian kinematic formula which are equivariant under gait similarities and have precise coverage even for small sample sizes. Applying them to gait curves from eight volunteers undergoing kneeling tasks and adjusting for different walking speeds and marker replacement at different visits, allows us to identify at which phases of the gait cycle the gait pattern changed due to kneeling.Peer Reviewe
Stability of the cut locus and a Central Limit Theorem for Fréchet means of Riemannian manifolds
We obtain a central limit theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Linâs Omnibus central limit theorem for FrĂ©chet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case
Foundations of the Wald Space for Phylogenetic Trees
Evolutionary relationships between species are represented by phylogenetic
trees, but these relationships are subject to uncertainty due to the random
nature of evolution. A geometry for the space of phylogenetic trees is
necessary in order to properly quantify this uncertainty during the statistical
analysis of collections of possible evolutionary trees inferred from biological
data. Recently, the wald space has been introduced: a length space for trees
which is a certain subset of the manifold of symmetric positive definite
matrices. In this work, the wald space is introduced formally and its topology
and structure is studied in detail. In particular, we show that wald space has
the topology of a disjoint union of open cubes, it is contractible, and by
careful characterization of cube boundaries, we demonstrate that wald space is
a Whitney stratified space of type (A). Imposing the metric induced by the
affine invariant metric on symmetric positive definite matrices, we prove that
wald space is a geodesic Riemann stratified space. A new numerical method is
proposed and investigated for construction of geodesics, computation of
Fr\'echet means and calculation of curvature in wald space. This work is
intended to serve as a mathematical foundation for further geometric and
statistical research on this space.Comment: 42 pages, 15 figure