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I'm very confused about why the Strauss model only allows 0<gamma<1 while it's okay to have gamma>1 in Strauss-Hard to represent the clustering behavior of points. What is the mathematical reason behind this difference? And why can't Stauss have gamma>1?

Eric Chen
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  • NB [CrossValidated](https://stats.stackexchange.com/) is the correct place to ask stats questions - Stack Overflow is for specific programming questions. – Andrea M Apr 09 '22 at 08:52
  • This question lies at the interface between software and computing and is not well-covered by either StackOverflow or CrossValidated. There is no established tag in CrossValidated for spatial point process theory. Many questions about spatial point processes are posed here in StackOverflow with tag 'spatstat'. – Adrian Baddeley Apr 11 '22 at 04:40

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This is explained on page 514-515 of the spatstat book.

A Strauss process with gamma > 1 does not exist because the density is not integrable in that case.

In the Strauss process, the probability of a point pattern x(actually, the probability density) is p = c beta^n gamma^s where n is the number of points and s is the number of pairs of points that are closer than the threshold distance r. If gamma > 1 then we can make p infinitely large by placing a lot of points very close together. This means that the formula does not lead to a well-defined point process.

In the Strauss-hard core process, the formula for p is the same as for the Strauss process provided that no pair of points comes closer than the hard core distance h, but otherwise, p is zero. For this model it is not possible to make p infinitely large, because if all the points must respect the hard core distance h, then the number of pairs that are closer than the interaction radius r must be bounded. So, the Strauss-hard core process exists for any gamma >= 0; there is no constraint on gamma.

Adrian Baddeley
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  • Thanks so much for your swift and detailed explanation! I've read your answer and page 514-515 of the book and have a follow-up question: I reckon the Strauss is a special case of Strauss-Hard with h=0. So with h getting smaller in Strauss-Hard, the upper bound of the num of pairs s that are between h and r is getting larger. Is there a very small h at which the upper bound of s is too large for gamma > 1 in Strauss-Hard? – Eric Chen Apr 08 '22 at 14:59
  • Strauss is the special case of StraussHard with `h=0`. If `h` is positive then `gamma` is pernitted to take any nonnegative value. If `h=0` then `gamma` must lie between 0 and 1 inclusive. – Adrian Baddeley Apr 09 '22 at 03:20