Dynamical local Fréchet curve regression in manifolds

The present paper solves the problem of local linear approximation of the Fréchet conditional mean in an extrinsic and intrinsic way from time correlated bivariate curve data evaluated in a manifold (see Torres et al, 2025, on global Fréchet functional regression in manifolds). The extrinsic local linear Fréchet functional regression predictor is obtained in the time-varying tangent space by projection into an orthornormal eigenfunction basis in 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 Fréchet mean approach is adopted in the computation of an intrinsic local linear Fréchet functional regression predictor. The asymptotic optimality of this intrinsic local approximation is also proved. The finite sample size performance of the empirical version of both, extrinsic and intrinsic local functional predictors, and of a Nadaraya-Watson type Fréchet curve predictor is illustrated in the simulation study undertaken. As motivating real data application, we consider the prediction problem of the Earth's magnetic field from the time-varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft.