Theoretical Analysis of Nonparametric Filament Estimation
Open Access Publications from the University of California

## Published Web Location

https://arxiv.org/pdf/1510.07105.pdf
No data is associated with this publication.
Abstract

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 non-stationary Gaussian processes indexed by manifolds $M_h, h \in(0,1]$ as $h \to 0$.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Item not freely available? Link broken?
Report a problem accessing this item