Resample-smoothing of Voronoi intensity estimators

TL;DR

This paper introduces a resample-smoothing method for Voronoi intensity estimators that improves their accuracy by balancing over- and under-smoothing through repeated thinning, validated by simulations and real data applications.

Contribution

It proposes a novel resample-smoothing technique for Voronoi intensity estimators using repeated thinning, enhancing estimation accuracy in arbitrary metric spaces.

Findings

Resample-smoothing significantly improves intensity estimation accuracy.

The method is effective for both planar and linear network point patterns.

Proposed data-driven approach aids in optimal smoothing parameter selection.

Abstract

Voronoi intensity estimators, which are non-parametric estimators for intensity functions of point processes, are both parameter-free and adaptive; the intensity estimate at a given location is given by the reciprocal size of the Voronoi/Dirichlet cell containing that location. Their major drawback, however, is that they tend to under-smooth the data in regions where the point density of the observed point pattern is high and over-smooth in regions where the point density is low. To remedy this problem, i.e. to find some middle-ground between over- and under-smoothing, we propose an additional smoothing technique for Voronoi intensity estimators for point processes in arbitrary metric spaces, which is based on repeated independent thinnings of the point process/pattern. Through a simulation study we show that our resample-smoothing technique improves the estimation significantly. In addition, we study statistical properties such as unbiasedness and variance, and propose a rule-of-thumb and a data-driven cross-validation approach to choose the amount of thinning/smoothing to apply. We finally apply our proposed intensity estimation scheme to two datasets: locations of pine saplings (planar point pattern) and motor vehicle traffic accidents (linear network point pattern).