Variational approach for spatial point process intensity estimation

TL;DR

This paper presents a new, simple variational estimator for the intensity function of inhomogeneous spatial point processes, which is faster and easier to implement than existing methods, with proven consistency and normality.

Contribution

The paper introduces a novel variational estimator for spatial point process intensity that is simple, computationally efficient, and theoretically supported, extending estimation techniques in spatial statistics.

Findings

Estimator is strongly consistent and asymptotically normal.

Performs favorably compared to maximum likelihood in simulations.

Applicable in various dimensions and with partial covariate knowledge.

Abstract

We introduce a new variational estimator for the intensity function of an inhomogeneous spatial point process with points in the $d$-dimensional Euclidean space and observed within a bounded region. The variational estimator applies in a simple and general setting when the intensity function is assumed to be of log-linear form $\beta+{\theta }^{\top}z(u)$ where $z$ is a spatial covariate function and the focus is on estimating ${\theta }$. The variational estimator is very simple to implement and quicker than alternative estimation procedures. We establish its strong consistency and asymptotic normality. We also discuss its finite-sample properties in comparison with the maximum first order composite likelihood estimator when considering various inhomogeneous spatial point process models and dimensions as well as settings were $z$ is completely or only partially known.