Classification of points in superpositions of Strauss and Poisson processes

Consider a realisation of a point process which is formed as a superposition of a regular point process, here a Strauss process, and some Poisson noise. The aim of the current work is to decide which of the two processes each point belongs to. We construct an MCMC algorithm which estimates the parameters of the superposition model and obtains posterior probabilities for each point of being a Strauss point. The algorithm is evaluated in a simulation study. Finally, it is applied to our motivating data set containing the locations of air bubbles, some of which are noise, in an Antarctic ice core.