Concurrent Object Regression

TL;DR

This paper introduces a novel concurrent regression model for analyzing time-varying relationships between complex object data in metric spaces and Euclidean predictors, extending regression techniques to non-Euclidean data.

Contribution

It develops the first concurrent regression framework for general metric space responses, including global and local methods, with consistency proofs and real data applications.

Findings

Successfully modeled human mortality data over time.

Applied to resting state fMRI data for brain connectivity analysis.

Demonstrated consistency and convergence of estimators.

Abstract

Modern-day problems in statistics often face the challenge of exploring and analyzing complex non-Euclidean object data that do not conform to vector space structures or operations. Examples of such data objects include covariance matrices, graph Laplacians of networks, and univariate probability distribution functions. In the current contribution a new concurrent regression model is proposed to characterize the time-varying relation between an object in a general metric space (as a response) and a vector in $\reals^p$ (as a predictor), where concepts from Fr\'echet regression is employed. Concurrent regression has been a well-developed area of research for Euclidean predictors and responses, with many important applications for longitudinal studies and functional data. However, there is no such model available so far for general object data as responses. We develop generalized versions of both global least squares regression and locally weighted least squares smoothing in the context of concurrent regression for responses that are situated in general metric spaces and propose estimators that can accommodate sparse and/or irregular designs. Consistency results are demonstrated for sample estimates of appropriate population targets along with the corresponding rates of convergence. The proposed models are illustrated with human mortality data and resting state functional Magnetic Resonance Imaging data (fMRI) as responses.