Jesper Moller: 25 years more with point patterns in Euclidean space and beyond


Abstract: 
On occasion of the 25 years anniversary of the Danish Society of Theoretical Statistics in 1996, I was invited 
to give a talk on log Gaussian Cox processes, and when celebrating the society’s 50 anniversary in 2021, I was 
invited to give the present talk. It covers some of the exciting developments in statistical methodology for 
the analysis of spatial point patterns within the last 25 years. The focus will be on some selected topics 
related to point patterns in  Euclidean and more general metric spaces, including the sphere and linear networks 
(such as road networks and dendrite networks), where technical details will be omitted and some application examples 
will be presented. I start with some background material, in particular a discussion of stationarity versus spatial 
inhomogeneity. Next I study different spatial point process models specified by a covariance function: log Gaussian 
Cox processes, determinantal point processes, and independent thinnings using correlated thinning probabilities. 
Third I discuss perfect simulation and doubly intractable distributions, exemplified with Bayesian analysis of mixture 
models with Gibbs or determinantal point process priors for the locations of components. Finally, a number of other 
topics will be briefly discussed.