Spatio-temporal point process statistics: A review

Spatio-temporal point process data have been analysed quite a bit in specialised fields, with the aim of better understanding the inherent mechanisms that govern the temporal evolution of events placed in a planar region. In particular, in the last decade there has been an acceleration of methodological developments, accompanied by a broad collection of applications as spatio-temporally indexed data have become more widely available in many scientific fields. We present a self-contained review describing statistical models and methods that can be used to analyse patterns of points in space and time when the questions of scientific interest concern both their spatial and their temporal behaviour. We revisit moment characteristics that define summary statistics, as well as conditional intensities which uniquely characterise certain spatio-temporal point processes. We make use of these concepts to describe models and associated methods of inference for spatio-temporal point process data. Three new motivating real-data examples are described and analysed throughout the paper to illustrate the most relevant techniques, discussing the pros and cons of the different considered approaches.