On spatially-adaptive intensity estimation for spatial point processes

Abstract Spatial adaptation plays an important role in intensity estimation for spatial point processes, especially in the presence of heterogeneity, where the required degree of smoothing varies across the domain. In this paper, we propose a spatially-adaptive intensity estimation framework building on the resample-smoothed Voronoi intensity estimator. The framework combines resample smoothing with a location-dependent thinning mechanism, allowing the amount of smoothing to vary according to the local structure of the point pattern. The thinning probabilities are driven by a pilot estimate of the intensity and controlled by the strength of the adaptation. Expressions for the bias and variance are derived, providing insight into how the adaptive mechanism affects the local bias-variance trade-off. Simulation studies covering Poisson and log-Gaussian Cox point processes indicate that location-dependent thinning substantially improves estimation accuracy in the presence of spatial heterogeneity, particularly under stronger resample smoothing induced by lower retention probabilities, while maintaining performance comparable to the standard resample-smoothed Voronoi estimator in homogeneous settings. Moderate levels of spatial adaptation provide the most stable performance, whereas stronger adaptation can improve the recovery of sharp intensity contrasts in highly inhomogeneous settings. Further comparisons show that the proposed estimator consistently outperforms both the resample-smoothed Voronoi and the spatially-adaptive kernel-based estimators in terms of mean integrated squared error. Finally, the practical utility of the proposed framework is illustrated through applications to wildfire data from California, USA, and traffic accident data from Madrid, Spain.