Resumen de Nonparametric inference for first-order characteristics of spatial and spatio-temporal point processes
Isabel Fuentes Santos
A point process is a stochastic process that generates a random collection of events in some metric space. A spatial point process generates events on a planar bounded region. If in addition to the spatial location of the events we know the time of occurrence we have a spatio-temporal point pattern. Spatial and spatio-temporal point patterns arise in a wide variety of scientific contexts, including ecology, seismology, epidemiology, cosmology and geography. In particular, the spatial location and time of occurrence of wildfires, the main threat for forests around the world, can be seen as spatio-temporal point patterns. The first-order intensity function characterizes the structure of events and is needed to estimate the second-order characteristics, which describe interaction between events. For these reasons modeling the first-order intensity function is a main issue in the analysis of both spatial and spatio-temporal point processes, and kernel estimators with scalar bandwidth have been widely used to this purpose. In the spatial framework, this work focuses on the consistent kernel intensity estimator with full matrix bandwidth. We develop an effective smooth bootstrap procedure which allows to estimate consistently the MISE of the consistent kernel intensity estimator, and we suggest a procedure to select the optimal bandwidth matrix. First-order separability is commonly assumed to estimate the spatio-temporal intensity function without being formally tested. This work proposes nonparametric separability tests for the intensity function of spatio-temporal point processes with discrete temporal component, which can be seen as multitype spatial point processes, and with continuous temporal component. The different techniques developed in this work have been applied to the analysis of wildfires registered in Galicia along the period 1999-2008.
