UNIFORM CONVERGENCE OF LOCAL FRECHET REGRESSION WITH APPLICATIONS TO LOCATING EXTREMA AND TIME WARPING FOR METRIC SPACE VALUED TRAJECTORIES

• 成果类型: Article
• 署名作者: Chen, Yaqing; Muller, Hans-Georg
• 署名单位: University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2163
• 发表日期: 2022
• 页码: 1573-1592
• 关键词: extrinsic sample means; Nonparametric Regression; functional connectivity; brain connectivity; alzheimers-disease; Consistency; density; STABILITY; MANIFOLDS; alignment

摘要

Local Frechet 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.