Local linear Fréchet curve regression in manifolds

Abstract: Global Fr\'echet functional regression has been recently addressed from time correlated bivariate curve data evaluated in a manifold (see Torres et al. 2025). For this type of curve data sets, the present paper solves the problem of local linear approximation of the Fr\'echet conditional mean in an extrinsic and intrinsic way. The extrinsic local linear Fr\'echet functional regression predictor is obtained in the time varying tangent space by projection into an orthornormal basis of the ambient Hilbert space. The conditions assumed ensure the existence and uniqueness of this predictor, and its computation via exponential and logarithmic maps. A weighted Fr\'echet mean approach is adopted in the computation of an intrinsic local linear Fr\'echet functional regression predictor. The asymptotic optimality of this intrinsic local approximation is also proved. The performance of the empirical version of both, extrinsic and intrinsic functional predictors, and of a Nadaraya-Watson type Fr\'echet curve predictor is illustrated in the simulation study undertaken. The finite-sample size properties are also tested in a real-data application via cross-validation. Specifically, functional prediction of the magnetic vector field from the time-varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft is addressed.