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Abstract
In many applications involving point pattern data, the Poisson process assumption
is unrealistic, with the data exhibiting a more regular spread. Such a repulsion between
events is exhibited by trees for example, because of competition for light and nutrients.
Other examples include the locations of biological cells and cities, and the times
of neuronal spikes. Given the many applications of repulsive point processes, there
is a surprisingly limited literature developing flexible, realistic and interpretable
models, as well as efficient inferential methods. We address this gap by developing
a modelling framework around the Mat\'ern type-III repulsive process. We consider
a number of extensions of the original Mat\'ern type-III process for both the homogeneous
and inhomogeneous cases. We also derive the probability density of this generalized
Mat\'ern process. This allows us to characterize the posterior distribution of the
various latent variables, and leads to a novel and efficient Markov chain Monte Carlo
algorithm involving data augmentation with a Poisson process. We apply our ideas to
two datasets involving the spatial locations of trees.