32,309 research outputs found
Intrinsic Inference on the Mean Geodesic of Planar Shapes and Tree Discrimination by Leaf Growth
For planar landmark based shapes, taking into account the non-Euclidean
geometry of the shape space, a statistical test for a common mean first
geodesic principal component (GPC) is devised. It rests on one of two
asymptotic scenarios, both of which are identical in a Euclidean geometry. For
both scenarios, strong consistency and central limit theorems are established,
along with an algorithm for the computation of a Ziezold mean geodesic. In
application, this allows to verify the geodesic hypothesis for leaf growth of
Canadian black poplars and to discriminate genetically different trees by
observations of leaf shape growth over brief time intervals. With a test based
on Procrustes tangent space coordinates, not involving the shape space's
curvature, neither can be achieved.Comment: 28 pages, 4 figure
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
Extrinsic local regression on manifold-valued data
We propose an extrinsic regression framework for modeling data with manifold
valued responses and Euclidean predictors. Regression with manifold responses
has wide applications in shape analysis, neuroscience, medical imaging and many
other areas. Our approach embeds the manifold where the responses lie onto a
higher dimensional Euclidean space, obtains a local regression estimate in that
space, and then projects this estimate back onto the image of the manifold.
Outside the regression setting both intrinsic and extrinsic approaches have
been proposed for modeling i.i.d manifold-valued data. However, to our
knowledge our work is the first to take an extrinsic approach to the regression
problem. The proposed extrinsic regression framework is general,
computationally efficient and theoretically appealing. Asymptotic distributions
and convergence rates of the extrinsic regression estimates are derived and a
large class of examples are considered indicating the wide applicability of our
approach
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