Local summary statistics for spatial point process intensity estimation

Common practice in spatial point process modelling dictates that formal analysis begins with intensity estimation, which is carried out by exploiting external covariates, when available. Using this intensity estimate, one usually proceeds by obtaining non-parametric summary statistics estimates, in order to assess which model best fits the data. The state of the art in parametric intensity modelling is employing the Poisson likelihood function, but this underperforms when the data come from a more complex model, with some kind of interaction among points. Hence, to address this shortcoming, we propose a method that incorporates local second-order characteristics to account for spatial dependencies in the model fitting procedure. Our method relies on a locally weighted Poisson log-likelihood, which avoids making explicit assumptions about the type and degree of spatial interaction. We are therefore able to include external covariates while exploiting the non-parametric methods' advantages, flexibly including second-order characteristics. We further propose a non-parametric test for the detection of interaction between points. Simulation studies demonstrate that the proposed method outperforms standard approaches in capturing diverse spatial interaction behaviours. An application to real forestry data further highlights the model's flexibility in the presence of locally varying point interaction structures.