Abstract
Local Fréchet regression is a nonparametric regression method for metric space valued responses and Euclidean predictors, which can be utilized to obtain estimates of smooth trajectories taking values in general metric spaces from noisy metric space valued random objects. We derive uniform rates of convergence, which so far have eluded theoretical analysis of this method, for both fixed and random target trajectories, where we utilize tools from empirical processes. These results are shown to be widely applicable in metric space valued data analysis. In addition to simulations, we provide two pertinent examples where these results are important: The consistent estimation of the location of properly defined extrema in metric space valued trajectories, which we illustrate with the problem of locating the age of minimum brain connectivity as obtained from fMRI data; and time warping for metric space valued trajectories, illustrated with yearly age-at-death distributions for different countries.
Funding Statement
Research supported in part by NSF Grants DMS-1712864 and DMS-2014626. Funding sources for ADNI are as listed at http://adni.loni.usc.edu.
Acknowledgments
We wish to thank two anonymous referees, an Associate Editor, and the Editor for their helpful and constructive comments which led to numerous improvements in the paper. Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database. As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators and databases can be found at http://adni.loni.usc.edu
Citation
Yaqing Chen. Hans-Georg Müller. "Uniform convergence of local Fréchet regression with applications to locating extrema and time warping for metric space valued trajectories." Ann. Statist. 50 (3) 1573 - 1592, June 2022. https://doi.org/10.1214/21-AOS2163
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