Galaxies, trees, and influenza cases have something in common: they tend to occur in clusters. The issue of how to model clustered spatial patterns is thus of interest to a variety of scientific disciplines. For many people, the mention of spatial clumping brings to mind the negative binomial distribution. This is appropriate, because the negative binomial distribution is overdispersed, meaning its variance is greater than its mean (in contrast to the Poisson distribution, which has equal variance and mean). Unfortunately, there are some difficulties in using the negative binomial distribution in spatial modeling, which we attempt to explain in this post.
Suppose that you want to create a model that generates spatial locations of trees. You want the model to be stochastic, so that each time you generate a forest, you end up with a different configuration of trees. The relevant tool is a spatial point process; mathematical details are described in books such as Daley and Veres-Jones. Each forest that you generate is called a realization of the process.