Single Index Fr\'echet Regression

TL;DR

This paper introduces a novel single index regression model for metric space-valued responses, enabling parameter estimation and hypothesis testing, with applications demonstrated on fMRI data.

Contribution

It develops the first asymptotic theory for single index models with non-Euclidean responses, extending Fréchet regression to include interpretable parameters.

Findings

Asymptotic distribution derived for model parameters.

Simulation studies show good finite sample performance.

Application to fMRI data demonstrates practical utility.

Abstract

Single index models provide an effective dimension reduction tool in regression, especially for high dimensional data, by projecting a general multivariate predictor onto a direction vector. We propose a novel single-index model for regression models where metric space-valued random object responses are coupled with multivariate Euclidean predictors. The responses in this regression model include complex, non-Euclidean data that lie in abstract metric spaces, including covariance matrices, graph Laplacians of networks, and univariate probability distribution functions. While Fr\'echet regression has proved useful for modeling the conditional mean of such random objects given multivariate Euclidean vectors, it does not provide for regression parameters such as slopes or intercepts, since the metric space-valued responses are not amenable to linear operations. As a consequence, distributional results for Fr\'echet regression have been elusive. We show here that for the case of multivariate Euclidean predictors, the parameters that define a single index and projection vector can be used to substitute for the inherent absence of parameters in Fr\'echet regression. Specifically, we derive the asymptotic distribution of suitable estimates of these parameters, which then can be utilized to test linear hypotheses for the parameters, subject to an identifiability condition. We demonstrate the finite sample performance of estimation and inference for the proposed single index Fr\'echet regression model through simulation studies. The method is illustrated for resting-state functional Magnetic Resonance Imaging (fMRI) data from the ADNI study.