Dynamical local Fréchet curve regression in manifolds

Abstract

Under mild conditions, this paper derives a least-squares local linear Fréchet curve predictor for response and regressor evaluated in a separable Hilbert space. We obtain the conditions allowing the implementation of this local linear Fréchet functional predictor in the ambient L^{2}-space of vector functions, with values in the time-varying tangent space on a compact Riemannian manifold. An intrinsic local linear Fréchet curve predictor evaluated in such a manifold is secondly proposed, based on a weighted Fréchet mean approach. Its asymptotical optimality is proved. The simulation study and real-data application analyze the finite-sample performance of the empirical versions of both predictors, compared with a geodesic Nadaraya-Watson-type curve predictor. In the real-data application, the functional prediction of the time-varying spherical coordinates of the Earth's magnetic field is addressed, from the observation of the geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft.