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I have a point pattern which is clearly inhomogeneous. Furthermore, the inhomogeneity has two components: a large scale effect and a local scale effect. I have constructed a Markov point process model to capture these two first order intensity variations. However, it is clear that the empirical pair correlation function should not be used to determine the second order behavior of the point pattern. So I simply modelled the second order behavior as a Poisson process. Now I fitted the model, obtained simulations from fitted model and try to compare the descrepancies in second order behavior between the simulated data and the real data.

I don't want to use intensity reweighted second order summary statistics as it will very likely to give highly biased results due to potential misfit of the first order intensity from the model.

My question is, in this scenario, does it make sense to compare second order behavior between the simulated data and the real data using any kind of unweighted second order summary statistics? If so, which one would be better? NND distribution or PCF?

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I may have come up with a solution to this problem and want some opinion on the validity.

After fitting the model, I can simulate a batch of point patterns from the fitted model. Then I can calculate an average PCF from these simulated point patterns and compare this to the PCF of the real data. The ratio of PCF from real data to that of simulated data should in principle give me an adjusted PCF. Since, before, I am fitting a PCF with respect to a unit rate Poisson process but now I am fitting a PCF with respect to a point process that has a density of the fitted model.

Does this make sense?

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