Séance Séminaire

Séminaire de Probabilités et Statistique

Estimating the intensity function of spatial point processes outside the observation window

We aim to estimate the intensity function of a point process in windows where it has not been observed, conditional to its realization in an observed window. We define a predictor as the best linear unbiased combination of the point pattern. We show that the weight function associated to the predictor is the solution of a Fredholm equation of second kind. Both the kernel and the source term of the Fredholm equation are related to the second order characteristics of the point process through the pair correlation function. We proposed two approximations to solve the Fredholm equation in order to obtain practical solutions and restrict the solution space to that generated by linear combinations of basis functions: step functions and elementary functions of a finite element basis, which provide a continuous approximation. Results are presented and illustrated on simulations and real data in different situations: for stationary and nonstationary processes, using covariates or the realization of an additional point process.