Christopher Barr

Voronoi-type estimators for spatial intensity

A wide range of methods use Voronoi diagrams to estimate conditional intensity of an inhomogeneous Poisson point process. The inverse cell area (herein referred to as the Voronoi estimator) has been used as a simple, fully non-parametric estimator in neuroscience and astrophysics. Voronoi diagrams have also been used to build flexible prior distributions, and develop optimal quadrature approximations for psuedo-likelihood based approaches. The present work systematically investigates fundamental properties of the Voronoi estimator for inhomogeneous intensity. Known to be unbiased in the homogeneous case, we prove the Voronoi estimator is also approximately ratio unbiased in the inhomogeneous case, and that its bias goes to zero exponentially as conditional intensity increases. Simulation studies show the sampling distribution is well approximated by the inverse gamma model, but generally has high variance. Two additional Voronoi-type estimators (one based on the centroidal Voronoi diagram, the other using k-means clustering) are presented and offer more stable results.