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HardcorePointProcess

HardcorePointProcess[μ,r_h,d]

represents a hard-core point process with constant intensity μ and hard-core radius r_h in .

Details

HardcorePointProcess models point configurations where the points cannot be within a radius r_h 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 r_h as follows:
pair potential

pair interaction
A point configuration from a hard-core point process HardcorePointProcess[μ,r_h,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 r_h 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, r_h].
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.

Examples

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Basic Examples (2)

Sample from a hard-core point process in :

Visualize the points in the sample:

Sample from a hard-core point process defined on the surface of the Earth:

Visualize the points:

Scope (3)

Generate three realizations from a hard-core point process in :

Estimate the parameters:

Generate three realizations from a hard-core point process on the surface of the Earth:

Visualize the point configurations:

Estimate the parameters:

Generate samples with increasing hard-core radius:

Plot samples with the repulsion disks:

Check that the hard-core constraint is obeyed:

Options (4)

Method (4)

Simulate using the Markov chain Monte Carlo method:

Specify the number of recursive calls to the sampler:

Specify the length of run:

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:

Use coupling from the past for exact sampling:

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:

Based on optimal packing :

The packing density:

For 3D:

Based on optimal packing :

The packing density:

Compute the average number of points in a unit disk for a hard-core point process:

Possible Issues (1)

By default, the simulation will run until the number of points converges to a steady state, or until the default number of iterations is reached:

Raise the number of recursive calls to the sampler:

Increase the length of run:

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