THEORETICAL ANALYSIS OF NONPARAMETRIC FILAMENT ESTIMATION

• 成果类型: Article
• 署名作者: Qiao, Wanli; Polonik, Wolfgang
• 署名单位: University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1405
• 发表日期: 2016
• 页码: 1269-1297
• 关键词: principal curves; density-function; mean shift; uniform; Consistency; galaxies; gradient

摘要

This paper provides a rigorous study of the nonparametric estimation of filaments or ridge lines of a probability density f. Points on the filament are considered as local extrema of the density when traversing the support of f along the integral curve driven by the vector field of second eigenvectors of the Hessian of f. We parametrize points on the filaments by such integral curves, and thus both the estimation of integral curves and of filaments will be considered via a plug-in method using kernel density estimation. We establish rates of convergence and asymptotic distribution results for the estimation of both the integral curves and the filaments. The main theoretical result establishes the asymptotic distribution of the uniform deviation of the estimated filament from its theoretical counterpart. This result utilizes the extreme value behavior of nonstationary Gaussian processes indexed by manifolds M-h, h is an element of (0, 1] as h -> 0.