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Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres. (arXiv:1705.06226v1 [math.ST])
来源于:arXiv
Functional data analysis on nonlinear manifolds has drawn recent interest. We
propose an intrinsic principal component analysis for smooth Riemannian
manifold-valued functional data and study its asymptotic properties. The
proposed Riemannian functional principal component analysis (RFPCA) is carried
out by first mapping the manifold-valued data through Riemannian logarithm maps
to tangent spaces around the time-varying Fr\'echet mean function, and then
performing a classical multivariate functional principal component analysis on
the linear tangent spaces. Representations of the Riemannian manifold-valued
functions and the eigenfunctions on the original manifold are then obtained by
mapping back with exponential maps. The tangent-space approximation through
functional principal component analysis is shown to be well-behaved in terms of
controlling the residual variation if the Riemannian manifold has nonnegative
curvature. We derive uniform convergence rates for the model components, 查看全文>>