Topology-guided sampling of nonhomogeneous random processes

47. Konstantin Mischaikow, Thomas Wanner:
    Topology-guided sampling of nonhomogeneous random processes
    Annals of Applied Probability 20(3), pp. 1068-1097, 2010.

Abstract

Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications these structures can be expressed as nodal domains of real-valued functions and are obtained only through experimental observation or numerical simulations. In both cases, the data on which the topological measurements are based are derived via some form of finite sampling or discretization. In this paper we present a probabilistic approach to quantifying the number of components of generalized nodal domains of non-homogeneous random processes on the real line via finite discretizations, i.e., we consider excursion sets of a random process relative to a non-constant deterministic threshold function. Our results furnish explicit probabilistic a-priori bounds for the suitability of certain discretization sizes and also provide information for the choice of location of the sampling points in order to minimize the error probability. We illustrate our results for a variety of random processes, demonstrate how they can be used to sample the classical nodal domains of deterministic functions perturbed by additive noise, and discuss their relation to the density of zeros.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/1010.3128, while the published version of the paper can be found at https://doi.org/10.1214/09-AAP652.

Bibtex

@article{mischaikow:wanner:10a,
   author = {Konstantin Mischaikow and Thomas Wanner},
   title = {Topology-guided sampling of nonhomogeneous random processes},
   year = 2010,
   journal = {Annals of Applied Probability},
   volume = 20,
   number = 3,
   pages = {1068--1097},
   doi = {10.1214/09-AAP652}
   }