The Annals of Statistics

Fréchet regression for random objects with Euclidean predictors

Alexander Petersen and Hans-Georg Müller

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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.

Article information

Ann. Statist., Volume 47, Number 2 (2019), 691-719.

Received: July 2016
Revised: June 2017
First available in Project Euclid: 11 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62J99: None of the above, but in this section 62G08: Nonparametric regression

Least squares regression random objects metric spaces local linear regression functional connectivity densities as objects


Petersen, Alexander; Müller, Hans-Georg. Fréchet regression for random objects with Euclidean predictors. Ann. Statist. 47 (2019), no. 2, 691--719. doi:10.1214/17-AOS1624.

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Supplemental materials

  • Proofs of theoretical results. The supplement includes four sections of proofs. The first section contains proofs of propositions verifying that our theoretical assumptions hold for the examples included in Section 3. The other three contain proofs for each of Sections 3–5.