Weighted Poisson intensity modelling for marked point processes with real-valued marks

Real-valued marks in spatial point processes are often analyzed separately from first-order intensity estimation or through assumptions specific to their generating mechanism. We propose a Weighted Poisson Intensity approach for incorporating real-valued marks into first-order intensity modelling without requiring a full probabilistic specification of their marginal distribution. The proposed framework represents the intensity as a baseline Poisson component combined with a non-negative mark-related contribution. Since marks are observed as attributes of observed events, we consider two modelling views: a Parametric Marked Poisson model, in which each observed event is represented by a location–mark pair on the joint space and the mark enters as an explicit intensity argument, and a Semi-parametric Marked Poisson model, in which the observed marks are interpreted as spatially varying values sampled at event locations, which are reconstructed to form a mark-related adjustment over the ground domain. These models allow us to investigate how real-valued marks are associated with the fitted first-order intensity under these two alternative interpretations. Their performance is evaluated through a simulation study based on Poisson and clustered point processes under different mark-generating mechanisms, and their practical usefulness is illustrated through applications to real data.