Fréchet regression for random objects with Euclidean predictors

By Alexander Petersen, H. Muller. Annals of Statistics, 2016

256 citations recorded by the source corpus.

Increasingly, statisticians are faced with the task of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space. To address the need for statistical methods for such data, we introduce the concept of Frechet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in $\mathcal{R}^p$, achieved by extending the classical concept of a Frechet mean to the notion of a conditional Frechet mean. We develop generalized versions of both global least squares regression and local weighted least squares smoothing. The target quantities are appropriately defined population versions of global and local regression for response objects in a metric space. We derive asymptotic rates of convergence for the corresponding fitted regressions using observed data to the population targets under suitable regularity conditions by applying empirical process methods. For the special case of random objects that reside in a Hilbert space, such as regression models with vector predictors and functional data as responses, we obtain a limit distribution. The proposed methods have broad applicability. Illustrative examples include responses that consist of probability distributions and correlation matrices, and we demonstrate both global and local Frechet regression for demographic and brain imaging data. Local Frechet regression is also illustrated via a simulation with response data which lie on the sphere.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9d291eb066631ce3eb4a7dbafef0ce447cdeef04

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Citations in the corpus

Principal component analysis for functional data on Riemannian manifolds and spheres (2017, background): “…lie in an intrinsically infinite dimensional linear or nonlinear space, dimension reduction techniques, in particular functional principal component analysis, play a central role in representing the random functions (Petersen and Müller 2016) and in other supervised/unsupervised learning tasks.”
Fréchet analysis of variance for random objects (2017, methodology): “To420 quantify intra-hub connectivity, following Petersen & Müller (2016), we analyze the distribution of the correlations between the signal at the seed voxel of the PCP hub and the signals of all other voxels within an 11× 11× 11 cube of voxels that is centered at the seed voxel.”
Distribution-on-Distribution Regression via Optimal Transport Maps (2021, methodology): “By comparison, Chen, Lin and M¨uller [2] introduce (linear) regression in Wasserstein space by means of a geometric approach, that is in a sense a linear model between tangent spaces.”