FRECHET REGRESSION FOR RANDOM OBJECTS WITH EUCLIDEAN PREDICTORS

• 成果类型: Article
• 署名作者: Petersen, Alexander; Mueller, Hans-Georg
• 署名单位: University of California System; University of California Santa Barbara; University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1624
• 发表日期: 2019
• 页码: 691-719
• 关键词: state functional connectivity; Nonparametric Regression; spherical regression; MANIFOLDS; splines

摘要

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 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.