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DiggleGrattonPointProcess

represents a Diggle–Gratton point process with constant intensity μ, interaction parameter κ, hard-core interaction radius δ and interaction radius ρ in .

Details

DiggleGrattonPointProcess is also known as hardcore Diggle process.
DiggleGrattonPointProcess models point configurations where the points cannot be within a radius δ of each other, have a decreasing repulsive pairwise interaction for points between radius δ and ρ of each other, and are otherwise uniformly distributed.

The Diggle–Gratton point process can be defined as a GibbsPointProcess in terms of its intensity μ and the pair potential or pair interaction , which are both parametrized by κ, δ and ρ as follows:
pair potential

pair interaction
A point configuration from a Diggle–Gratton point process DiggleGrattonPointProcess[μ,κ,δ,ρ,d] in an observation region reg has density function proportional to $mu^n product_(i!=j)h(TemplateBox[{{{p, _, i}, -, {p, _, j}}}, Norm])$ , with respect to PoissonPointProcess[1,d].
The Papangelou conditional density for adding a point q to a point configuration is $mu product_ih(TemplateBox[{{{p, _, i}, -, q}}, Norm])$ .
DiggleGrattonPointProcess allows μ, κ, δ and ρ to be positive numbers such that , and d to be any positive integer.
DiggleGrattonPointProcess simplifies to HardcorePointProcess when and to PoissonPointProcess when and . Higher values of make the process more repulsive within radius ρ.
Possible Method settings in RandomPointConfiguration for DiggleGrattonPointProcess are:
"MCMC" Markov chain Monte Carlo birth and death

"Exact" coupling from the past
Possible PointProcessEstimator settings in EstimatedPointProcess for DiggleGrattonPointProcess are:
Automatic automatically choose the parameter estimator

"MaximumPseudoLikelihood" maximize the pseudo-likelihood
DiggleGrattonPointProcess can be used with such functions as RipleyK and RandomPointConfiguration.