Local Fr\'echet regression with toroidal predictors

TL;DR

This paper introduces a novel regression framework for responses in metric spaces and predictors on a torus, with estimators respecting their geometries and demonstrating superior performance in simulations and real data.

Contribution

It is the first to develop a regression method accommodating metric space responses and toroidal predictors with intrinsic estimators respecting their geometries.

Findings

Proposed estimators are consistent and have established convergence rates.

Simulation studies show superior performance over existing methods.

Application to real data demonstrates practical effectiveness.

Abstract

We provide the first regression framework that simultaneously accommodates responses taking values in a general metric space and predictors lying on a general torus. We propose intrinsic local constant and local linear estimators that respect the underlying geometries of both the response and predictor spaces. Our local linear estimator is novel even in the case of scalar responses. We further establish their asymptotic properties, including consistency and convergence rates. Simulation studies, together with an application to real data, illustrate the superior performance of the proposed methodology.