August 2023 Single index Fréchet regression
Satarupa Bhattacharjee, Hans-Georg Müller
Author Affiliations +
Ann. Statist. 51(4): 1770-1798 (August 2023). DOI: 10.1214/23-AOS2307

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, including covariance matrices, graph Laplacians of networks and univariate probability distribution functions, among other complex objects that lie in abstract metric spaces. While Fréchet 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 Fréchet 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 Fréchet 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. Consistent estimation of the link function of the single index Fréchet regression model is obtained through local linear Fréchet regression. We demonstrate the finite sample performance of estimation and inference for the proposed single index Fréchet regression model through simulation studies, including the special cases where responses are probability distributions and graph adjacency matrices. The method is illustrated for resting-state functional Magnetic Resonance Imaging (fMRI) data from the ADNI study.

Funding Statement

The research is supported in part by NSF Grant DMS-2014626 and a NIH ECHO grant. Funding sources for ADNI are as listed at http://adni.loni.usc.edu.

Acknowledgments

We thank four referees and an Associate Editor for helpful comments that led to many improvements. Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database. As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this paper. A complete listing of ADNI investigators and databases can be found at http://adni.loni.usc.edu.

Citation

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Satarupa Bhattacharjee. Hans-Georg Müller. "Single index Fréchet regression." Ann. Statist. 51 (4) 1770 - 1798, August 2023. https://doi.org/10.1214/23-AOS2307

Information

Received: 1 December 2022; Revised: 1 June 2023; Published: August 2023
First available in Project Euclid: 19 October 2023

Digital Object Identifier: 10.1214/23-AOS2307

Subjects:
Primary: 62G10 , 62R20
Secondary: 62G05 , 62G20

Keywords: Dimension reduction , inference , M-estimator , non-Euclidean data , random objects

Rights: Copyright © 2023 Institute of Mathematical Statistics

Vol.51 • No. 4 • August 2023
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