Academic

A ridge in a data cloud is a low-dimensional geometric feature that generalizes the concept of local modes, in the sense that the density values on ridge points are local maxima in some constrained subspace. It has been used as a mathematical model for filamentary structures observed in, for example, cosmological data and GPS data, which exhibit network-like spatial patterns. In this talk I will present a nonparametric method of constructing confidence regions for ridges of probability density functions. In this method ridges are viewed as the intersections of level sets of the functions that define ridge points. The vertical variation of the plug-in kernel estimators for these functions constrained on the ridges is used as the measure of maximal deviation for ridge estimation. Our confidence regions for ridges are determined by the asymptotic distribution of this maximal deviation, which is established by utilizing the extreme value distribution of nonstationary chi-square fields indexed by manifolds.