represents a hard-core point process with constant intensity μ and hard-core radius rh in .
- HardcorePointProcess models point configurations where the points cannot be within a radius rh of each other but otherwise are uniformly distributed with intensity μ points per volume unit.
- The hard-core model is typically used when the underlying points behave like a collection of hard marbles, including things like gas molecules, metal deposits, sintered material and biological cells.
- The hard-core 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 rh as follows:
pair potential pair interaction
- A point configuration from a hard-core point process HardcorePointProcess[μ,rh,d] in an observation region reg has density function proportional to with respect to PoissonPointProcess[1,d].
- The Papangelou conditional density for adding a point to a point configuration is .
- HardcorePointProcess allows μ and rh to be any positive numbers, and d to be any positive integer.
- HardcorePointProcess is a special case of GibbsPointProcess and is equivalent to StraussPointProcess[μ, 0, rh].
- Possible Method settings in RandomPointConfiguration for HardcorePointProcess are:
"MCMC" MCMC birth and death "Exact" coupling from the past
- Possible PointProcessEstimator settings in EstimatedPointProcess for HardcorePointProcess are:
Automatic automatically choose the parameter estimator "MaximumPseudoLikelihood" maximize the pseudo-likelihood
- HardcorePointProcess can be used with such functions as RipleyK and RandomPointConfiguration.
Examplesopen allclose all
Basic Examples (2)
Generate three realizations from a hard-core point process in :
Generate three realizations from a hard-core point process on the surface of the Earth:
Visualize the point configurations:
Generate samples with increasing hard-core radius:
Plot samples with the repulsion disks:
Simulate using the Markov chain Monte Carlo method:
Specify the number of recursive calls to the sampler:
Provide an initial state for the simulation:
The initial point must have nonzero density to ensure that the result is a valid configuration:
Check if the minimal distance between the points is smaller than the hard-core radius:
Visualize the birth and death process at different stages:
Properties & Relations (3)
For the large intensity μ, the samples saturate:
The number of points saturates at a density that is significantly lower than the theoretical maximum packing:
Compute the average number of points in a unit disk for a hard-core point process:
Wolfram Research (2020), HardcorePointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/HardcorePointProcess.html.
Wolfram Language. 2020. "HardcorePointProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HardcorePointProcess.html.
Wolfram Language. (2020). HardcorePointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HardcorePointProcess.html