The Fr\'echet correlation coefficient for heterogeneous random objects

TL;DR

This paper introduces the Fréchet correlation coefficient (FCC), a new measure for assessing the explanatory strength of predictors in regression models with heterogeneous, non-Euclidean responses and predictors.

Contribution

The paper proposes the FCC, a model-free, directional correlation measure for heterogeneous objects, along with a novel partition-based estimator and a bootstrap test for dependence.

Findings

FCC is interpretable on a unit scale and ranges from zero to one.

The partition-based estimator improves computational efficiency.

Simulation studies demonstrate the effectiveness of the proposed methods.

Abstract

Modern regression analysis often involves responses and predictors taking values in the same or distinct metric spaces. To rank non-Euclidean heterogeneous predictors in regression by explanatory strength, analogous to the classical $R^2$, we introduce the Fr\'echet correlation coefficient (FCC), defined as the relative reduction in the Fr\'echet variance of the response after conditioning on a specific predictor. FCC is directional, model-free, and interpretable on a unit-scale, attaining one under almost sure functional dependence and zero when the Fr\'echet mean is invariant to conditioning. We propose a novel partition-based estimator that avoids explicit nonparametric estimation of the conditional Fr\'echet mean function, thereby improving both computational efficiency and flexibility. A tailored wild bootstrap algorithm is further developed for testing the Fr\'echet conditional mean dependence. We establish asymptotic theory and evaluate power through extensive simulations.