Perfect simulation for marked point processes
Some recently proposed exact simulation methods are extended to the case of marked point processes. Four families of algorithms are considered: coupling from the past, the clan of ancestors technique, the Gibbs sampler, and a Metropolis--Hastings algorithm based on birth and death proposals. Form a theoretical point of view, conditions are given under which the algorithms yield unbiased samples in finite time; for practical application, a C++ library for marked point processes developed at CWI is described. The various algorithms are tested on several models, including the Widom-Rowlinson mixture model, multi-type pairwise interaction processes, and the Candy line segment model. A simulation study is carried out in order to analyse the proposed methods in terms of speed of convergence in relation to the parameters of the model. For the range of models investigated, the clan of ancestors algorithm using the incompatibility index is the fastest method among the ones analysed, while coupling from the past is applicable to the widest range of parameter values, and usually faster than the Metropolis--Hastings sampler. The latter two methods tend to be cumbersome if the underlying model is neither attractive nor repulsive. If one is prepared to approximate by discretisation, a proper choice of Gibbs sampler makes it possible to obtain samples from models that lack monotonicity or have such a high local stability bound as to rule out coupling from the past or clan of ancestor approaches in practice.