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How to identify specific spatial patterns? (application: star distributions, cluster processes)

In particular, how do you assess whether a particular spatial process, based on one observation of the process in question, belongs to a particular family of stochastic spatial processes.

For instance, let's imagine that you observe star distributions. You build a theoretical model of cluster processes to model star distributions: for instance, a Poisson process to model centroids of star clusters, and at each point (centroid) of the parent process, a second process consisting of points (stars) randomly distributed in a circle of radius R around the star cluster's centroid (R being itself a random variable).

Which metric would you use to assess whether your observation (star distribution) fits with the theoretical model? Which simple functional uniquely identifies a specific spatial process? I'm looking at very simple metrics such as star counts within some regions, and distances between stars (mean, standard deviation, distribution).

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