Uniform convergence of local Fr\'echet regression, with applications to locating extrema and time warping for metric space valued trajectories

TL;DR

This paper establishes uniform convergence rates for local Fréchet regression on metric space valued data, enabling reliable estimation of extrema and alignment of trajectories in complex data settings.

Contribution

It provides the first theoretical analysis of uniform convergence for local Fréchet regression, extending its applicability to metric space valued data analysis.

Findings

Uniform convergence rates derived for fixed and random trajectories.

Applications demonstrated in brain connectivity and demographic data.

Method enables consistent extremum estimation and trajectory alignment.

Abstract

Local Fr\'echet regression is a nonparametric regression method for metric space valued responses and Euclidean predictors, which can be utilized to obtain estimates of smooth trajectories taking values in general metric spaces from noisy metric space valued random objects. We derive uniform rates of convergence, which so far have eluded theoretical analysis of this method, for both fixed and random target trajectories, where we utilize tools from empirical processes. These results are shown to be widely applicable in metric space valued data analysis. In addition to simulations, we provide two pertinent examples where these results are important: The consistent estimation of the location of properly defined extrema in metric space valued trajectories, which we illustrate with the problem of locating the age of minimum brain connectivity as obtained from fMRI data; Time warping for metric space valued trajectories, illustrated with yearly age-at-death distributions for different countries.