Global Fr\'echet regression from time correlated bivariate curve data in manifolds

Abstract

Global Fr\'echet regression is addressed from the observation of a strictly stationary bivariate curve process, evaluated in a finite--dimensional compact differentiable Riemannian manifold, with bounded positive smooth sectional curvature. The involved univariate curve processes respectively define the functional response and regressor, having the same Fr\'echet functional mean. The supports of the marginal probability measures of the regressor and response processes are assumed to be contained in a ball, whose radius ensures the injectivity of the exponential map. This map has time--varying origin at the common marginal Fr\'echet functional mean. A weighted Fr\'echet mean approach is adopted in the definition of the theoretical loss function. The regularized Fr\'echet weights are computed, in the time--varying tangent space from the log--mapped regressors. Under these assumptions, and some Lipschitz regularity sample path conditions, when a unique minimizer exists, the uniform weak--consistency of the empirical Fr\'echet curve predictor is obtained, under mean--square ergodicity of the log--mapped regressor process in the first two moments. A simulated example in the sphere illustrates the finite sample size performance of the proposed Fr\'echet predictor. Predictions in time of the spherical coordinates of the magnetic field vector are obtained from the time--varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft in the real--data example analyzed.