Local Fr\'echet functional regression in manifolds from time-correlated bivariate curve data

Abstract

Under mild conditions, a least-squares local linear Fr\'echet curve predictor is derived for a response and a regressor evaluated in a separable Hilbert space. The conditions that allow the implementation of the local linear Fr\'echet functional predictor in the ambient L2-space of vector functions, with values in the time-varying tangent space of a compact Riemannian manifold, are established. An intrinsic local linear Fr\'echet curve predictor on such a manifold is then proposed, based on a weighted Fr\'echet mean approach. Its asymptotic optimality is proved. Simulations and a real-data application are considered to 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 usingobservations through time of the geocentric latitude and longitude of the NASA MAGSAT spacecraft.