  Because most natural phenomena exhibit dependence at multiple scales like
locations of earthquakes or forest fire occurrences, spatio-temporal
single-scale point process models are unrealistic in many applications. This
motivates us to construct generalizations of classical Gibbs models. In this
paper, we extend the Geyer saturation point process model to the
spatio-temporal multi-scale framework. The simulation process is carried out
through a birth-death Metropolis-Hastings algorithm. In a simulation study, we
compare two common methods for statistical inference in Gibbs models: the
pseudo-likelihood and logistic likelihood approaches that we tailor to this
model. Finally, we illustrate this new model on forest fire occurrences
modeling in Southern France.
