Set estimation from reflected Brownian motion

TL;DR

This paper develops and analyzes methods for estimating a set's shape and boundary from trajectories of reflected Brownian motion, with applications in ecology for animal home range estimation.

Contribution

It provides the first theoretical framework for set estimation from reflected Brownian motion trajectories, including consistency and convergence rates.

Findings

Establishes consistency of the estimators.

Derives convergence rates for the estimators.

Validates methods on simulated and real data.

Abstract

We study the problem of estimating a compact set $S\subset \mathbb{R}^d$ from a trajectory of a reflected Brownian motion in $S$ with reflections on the boundary of $S$. We establish consistency and rates of convergence for various estimators of $S$ and its boundary. This problem has relevant applications in ecology in estimating the home range of an animal based on tracking data. There are a variety of studies on the habitat of animals that employ the notion of home range. This paper offers theoretical foundations for a new methodology that, under fairly unrestrictive shape assumptions, allows one to find flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.