Random sets on manifolds under an infinite–dimensional Log–Gaussian Cox process approach.

Abstract

A new framework is introduced in this paper for modeling and statistical analysis of point sets in a manifold, that randomly arise through time. Specifically, in the characterization of these sets, the random counting measure is assumed to belong to the family of Cox processes driven by a L2(M)–valued Log–Gaussian intensity, where M denotes here a compact two–point homogeneous space. The associated family of temporal covariance operators on L2(M) characterizes the n–order product density under stationarity in time. In particular, the pair correlation functional, the reduced second order moment measure or K function can also be constructed from this covariance operator family. Some functional summary statistics of interest are introduced, analyzing their asymptotic properties in the simulation study undertaken.