Edge correction for intensity estimation of 3D heterogeneous point processes from replicated data

A common task in the analysis of point patterns is to estimate the intensity of the underlying process, i.e., the expected number of points per unit area or volume at each position over space. In biological studies, a specific feature of point patterns and processes is their confinement within bounded domains of Rd because of the organization of biological systems into various compartments (tissues, cells, nuclei, etc.). This induces a systematic negative bias in intensity estimates at the boundary of the confinement domain. Here, we address this edge effect issue in the context of intensity estimation from multiple realizations, which is particularly relevant to biological studies because of experimental replications. We introduce an edge correction method for a recently proposed maximum likelihood intensity estimator based on distance statistics that requires no additional parameter in the estimation method. We describe corrections relying on locally planar or spherical shape approximations of the domain boundary. Based on corrected estimators, we propose a strategy for the statistical mapping of a 3D bounded process that adapts to the shape of the domain boundary. We demonstrate quantitatively the robustness of the correction method using point patterns simulated over domains of Rd with various shapes. We illustrate the practical usefulness of the method by analyzing the 3D spatial organization of compartments within plant cells. The method should be useful for the statistical analysis of biological structures at different scales.