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I have two point patterns, one is real data, the other one is simulated data from a model I constructed for the real data.

Now I want to see if the second order property of the model matches that of the real data.

What I plan on doing is to fit the pair correlation function for both real data and simulated data and take the ratio between them.

However, I do realize there are some issues with this approach in that my data is highly inhomogeneous, so misfit of the inhomogeneity in the model can cause differences in the ratio of pair correlation function.

But the main question is, assuming the first order inhomogeneity of model fits the data reasonably well, does comparing the pair correlation function correctly presents the differences in the second order property?

Any input will be helpful. Thank you very much!

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The standard procedure is to generate several (e.g. 99) simulations based on the fitted model and then use these to make envelopes of the estimated pair correlation function. This allows you to see the expected variability under the model and then compare with the estimated pair correlation function from your data. This can be interpreted formally as a Monte Carlo significance test and there is a lot of literature on the topic. For a software implementation see the envelope function in the R package spatstat (disclaimer: I'm a coauthor).

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  • $\begingroup$ Thanks for the reply. That is what I plan to do, although my apology for not emphasize on generating several simulation based on fitted model. However, I do have some question on interpreting the differences. For example, if I take the ratio between the pair correlation function of data and model, and I find that the ratio is say 1.3 at $r = 0.5$ distance, does this have the interpretation that the data is 1.3 times more likely in terms of probability to have two points at $r = 0.5$ distance compared to the model? Thanks again! $\endgroup$
    – NamelessGods
    Commented Feb 9, 2020 at 0:29
  • $\begingroup$ My thought is that the ratio between the pcf of data and model is a new ''pcf'' taking the model as a reference where the standard pcf is taking Poisson process as base reference. I just want to make sure my understanding is correct regarding this. $\endgroup$
    – NamelessGods
    Commented Feb 9, 2020 at 0:37
  • $\begingroup$ If you do simulation envelopes as suggested no ratios are involved. I think estimated ratios of pcfs are bound to be numerically very unstable (in particular at short ranges), so I would be hesitant to rely on these. $\endgroup$
    – Ege Rubak
    Commented Feb 9, 2020 at 21:05

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