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Suppose we have two spatial point process on the same space. One has $n_1$ points, the other $n_2$ points.

I have been reading about statistically testing for independence, attraction or repulsion between the two point patterns.

One common suggestion is to randomly permute the $n=n_1+n_2$ markers and compute some statistic, say the cross k-function. Apparently comparing the original pattern statistic with the resulting distribution gives an p-value for independence.

Can anyone elaborate?

In particular if we have say three clusters of the two point patterns, suggesting attraction between the points, surely the majority of the permutations of the markers will give a similar statistic to the observed. So how is this testing for independence?

If anyone is interested, or wants more detail, I am trying to learn from

https://www.seas.upenn.edu/~ese502/NOTEBOOK/Part_I/5_Comparative_Analyses.pdf

section 5.6

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