Spatio-temporal modelling of a Cox point process sampled by a curve, filtering and Inference

summary:The paper deals with Cox point processes in time and space with Lévy based driving intensity. Using the generating functional, formulas for theoretical characteristics are available. Because of potential applications in biology a Cox process sampled by a curve is discussed in detail. The filtering of the driving intensity based on observed point process events is developed in space and time for a parametric model with a background driving compound Poisson field delimited by special test sets. A hierarchical Bayesian model with point process densities yields the posterior. Markov chain Monte Carlo "Metropolis within Gibbs" algorithm enables simultaneous filtering and parameter estimation. Posterior predictive distributions are used for model selection and a numerical example is presented. The new approach to filtering is related to the residual analysis of spatio-temporal point processes