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April 2019 Fréchet regression for random objects with Euclidean predictors
Alexander Petersen, Hans-Georg Müller
Ann. Statist. 47(2): 691-719 (April 2019). DOI: 10.1214/17-AOS1624

Abstract

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 Fréchet 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 Fréchet mean to the notion of a conditional Fréchet 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 Fréchet regression for demographic and brain imaging data. Local Fréchet regression is also illustrated via a simulation with response data which lie on the sphere.

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Alexander Petersen. Hans-Georg Müller. "Fréchet regression for random objects with Euclidean predictors." Ann. Statist. 47 (2) 691 - 719, April 2019. https://doi.org/10.1214/17-AOS1624

Information

Received: 1 July 2016; Revised: 1 June 2017; Published: April 2019
First available in Project Euclid: 11 January 2019

zbMATH: 07033148
MathSciNet: MR3909947
Digital Object Identifier: 10.1214/17-AOS1624

Subjects:
Primary: 62G05
Secondary: 62G08 , 62J99

Keywords: densities as objects , Functional connectivity , least squares regression , local linear regression , metric spaces , random objects

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 2 • April 2019
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