Dynamical local Fr\'echet curve regression in manifolds

TL;DR

This paper introduces methods for local linear approximation of the Fréchet mean in manifolds, both extrinsic and intrinsic, with theoretical guarantees and practical applications to Earth's magnetic field prediction.

Contribution

It develops novel local linear Fréchet regression techniques on manifolds, providing theoretical properties and demonstrating their effectiveness on real satellite data.

Findings

Extrinsic and intrinsic predictors perform well in simulations.

Asymptotic optimality of the intrinsic predictor is established.

Application to Earth's magnetic field prediction shows practical utility.

Abstract

The present paper solves the problem of local linear approximation of the Fr\'echet 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 Fr\'echet functional regression in manifolds). The extrinsic local linear Fr\'echet 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 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 finite sample size performance of the empirical version of both, extrinsic and intrinsic local functional predictors, and of a Nadaraya-Watson type Fr\'echet 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.