Some properties and applications of local second-order characteristics for spatio-temporal point processes on networks

Point processes on linear networks are increasingly being considered to analyse events occurring on particular network-based structures. In this work, we extend Local Indicators of Spatio-Temporal Association (LISTA) functions to the non-Euclidean space of linear networks, allowing to obtain information on how events relate to nearby events. In particular, we propose the local version of two inhomogeneous second-order statistics for spatio-temporal point processes on linear networks, the K- and the pair correlation functions. We also show that these LISTA functions are useful for diagnostics of models specified on the networks, and can be helpful to assess the goodness-of-fit of different spatio-temporal models fitted to point patterns occurring on linear networks. Our methods do not rely on any particular model assumption on the data, and thus they can be applied for whatever is the underlying model of the process. Furthermore, we use the LISTA functions to build a test for the identification of differences in the local spatio-temporal structure of two point patterns: a pattern of interest and a background one, both occurring on the same linear network. We finally present a real data analysis of traffic accidents in Medellin (Colombia)