# Attractive pairwise interaction point process

I'm doing some spatial pattern analysis. After looking over some work on Markov Point Processes, I'm finding that all of the pairwise interaction processes are 'repulsive'. In "Statistical Inference and Simulation for Spatial Point Processes" by Moller, he states simply that 'the attractive case is not well defined'. I was wondering if there are special cases in which it is well defined? How can I model an attractive pairwise interaction?

I suppose the alternative would be to move away from pairwise interaction and try to infer the intensity function for an inhomogeneous process?

## 1 Answer

The issue with attractive pairwise interactions is that the integral of the intensity of the process is infinite as each new point always increases the intensity for others in the neighbourhood.

If your process is mildly attractive on medium distances and strongly repulsive on very short ones (like physical particles), you can consider a pairwise interaction like Lennard-Jones or similar. See Van Lieshout (2000) Markov point processes and their applications or in a more applied fashion using R: Baddeley (2010) Analysing spatial point patterns in 'R'.

Else, the best way out is to use higher-order interactions such as area-interaction process (e.g. Baddeley and van Lieshout, 1995) or one of the processes in Geyer (1999) Likelihood inference for spatial point processes. Higher order interactions allow for both attractive and repulsive behaviour, e.g. points could be pairwise attractive, but triplets of points could be repulsive.

Depending on your application, the inhomogeneity of space could play a very important role. However, whether the intensity depends on the properties of underlying space or on the presence of other points should be dictated by theory. Even if these effects potentially could be hard to tell apart empirically.

• (+1) Welcome to the site, Sergiy! – cardinal Sep 17 '12 at 22:57
• I am trying to figure out how to calculate the expected value of the are interaction point process, do you have any advice about how should i proceed? since for me is confusing because the conditional intensity function uses a "predefined point pattern" $\int_{W} \lambda(x_j,\textbf{x})$ – LFRC May 15 '17 at 20:57