Some properties of local weighted second-order statistics for spatio-temporal point processes

Abstract

Diagnostics of goodness-of-fit in the theory of point processes are often considered through the transformation of data into residuals as a result of a thinning or a rescaling procedure. We alternatively consider here second-order statistics coming from weighted measures. Motivated by Adelfio and Schoenberg (2009) for the temporal and spatial cases, we consider an extension to the spatio-temporal context in addition to focussing on local characteristics. In particular, our proposed method assesses goodness-of-fit of spatio-temporal models by using local weighted secondorder statistics, computed after weighting the contribution of each observed point by the inverse of the conditional intensity function that identifies the process. Weighted second-order statistics directly apply to data without assuming homogeneity nor transforming the data into residuals, eliminating thus the sampling variability due to the use of a transforming procedure. We provide some characterisations and show a number of simulation studies. Keywords K-function • Local properties • Residual analysis • Second-order characteristics • Spatio-temporal point patterns

Local spatial log-Gaussian Cox processes for seismic data

“…This section reviews some methods widely used for diagnostics of global spatial point processes models, here proposed as diagnostic methods for spatial models in the local framework. As stated in some previous papers (Adelfio et al 2019;Adelfio and Schoenberg 2009), the main problem when dealing with residual analysis for point processes is to find a correct definition of residuals, since the one used in dependence models cannot be used for point processes. Two of the mostly used methods for diagnostics of spatial point processes are the inhomogeneous K-function ( 13)…”

“…Indeed, when a model is fitted to a set of random points, diagnostic measures are necessary to assess the goodness-of-fit and to evaluate the ability of that model to describe the random point pattern behaviour (Adelfio et al 2019). However, tools for checking or criticizing the fitted model are quite limited.…”

Self-exciting point process modelling of crimes on linear networks

“…As stated in some previous papers (Adelfio et al, 2020; Adelfio and Schoenberg, 2009), the main problem when dealing with residual analysis for point processes is to find a correct definition of residuals, since the one used in dependence models cannot be used for point processes.…”

“…First, superthinned residuals (Clements et al, 2012), which have the disadvantage of being less straightforwardly applicable, as they require a tuning parameter to be chosen. Another possible option could have been using the weighted spatio-temporal second order statistics, as in Adelfio et al (2020), which do not require any transformation of the data. However, we opted for diagnostic procedures separated in space and time, due to the separable specification of the models employed in the article.…”

On Some Statistical Properties of the Spatio-Temporal Product Density

“…A complete simulation exercise to check the normality property for the secondorder characteristics is given in Adelfio et al (2020). Therefore we take advantage of the computational results reported in this work which go in the same direction of the product density to omit presenting those results here.…”