Concurrent Object Regression

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 response) and a vector in $\reals^p$ (as 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. We develop generalized versions of both global least squares regression and locally weighted least squares smoothing in the context of concurrent regression for responses which 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 mortality data and resting state functional Magnetic Resonance Imaging data (fMRI) as responses.