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PenttinenPointProcess[μ,γ,rp,d]

represents a Penttinen point process with constant intensity μ, interaction parameter γ and interaction radius rp in .

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
Possible Issues  
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PenttinenPointProcess

PenttinenPointProcess[μ,γ,rp,d]

represents a Penttinen point process with constant intensity μ, interaction parameter γ and interaction radius rp in .

Details

  • PenttinenPointProcess is also known as pairwise area interaction process.
  • PenttinenPointProcess models point configurations where points have a pairwise repulsion that is log-linear in the measure of the overlap between balls around the points of radius rp, which are otherwise uniformly distributed.
  • The Penttinen model is typically used when the process interaction depends on the amount of shared resources within radius rp, such as plants, trees and nests of animals.
  • The Penttinen point process can be defined as a GibbsPointProcess in terms of its intensity μ and the pair potential ϕ or pair interaction h, which are both parametrized by γ and rp as follows:
  • pair potential
    pair interaction
  • Here is the measure of overlapping balls:
  • overlapping area in
    overlapping volume in
    1-r TemplateBox[{{{d, /, 2}, +, 1}}, Gamma] TemplateBox[{{1, /, 2}, {{(, {1, -, d}, )}, /, 2}, {3, /, 2}, {{(, {r, ^, 2}, )}, /, {(, {4, , {{r, _, p}, ^, 2}}, )}}}, Hypergeometric2F1]/(sqrt(pi) r_p TemplateBox[{{{(, {d, +, 1}, )}, /, 2}}, Gamma])overlapping measure in
  • A point configuration from a Penttinen point process 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 to a point configuration is mu product_ih(TemplateBox[{{{p, _, i}, -, q}}, Norm]).
  • PenttinenPointProcess allows μ, γ and rp to be positive numbers such that , and d to be any positive integer.
  • PenttinenPointProcess simplifies to PoissonPointProcess when . Smaller values of gamma inhibit points within r_(p).
  • Possible Method settings in RandomPointConfiguration for StraussPointProcess are:
  • "MCMC"Markov chain Monte Carlo birth and death
    "Exact"coupling from the past
  • Possible PointProcessEstimator settings in EstimatedPointProcess for PenttinenPointProcess are:
  • Automaticautomatically choose the parameter estimator
    "MaximumPseudoLikelihood"maximize the pseudo-likelihood
  • PenttinenPointProcess can be used with such functions as RipleyK and RandomPointConfiguration.

Examples

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Basic Examples  (2)

Sample from a Penttinen point process:

Wolfram Language code: proc = PenttinenPointProcess[30, 0.1, 1., 2];
Wolfram Language code: reg = Rectangle[{0, 0}, {10, 10}];
Wolfram Language code: pts = RandomPointConfiguration[proc, reg]

Visualize the points in the sample:

Wolfram Language code: ListPlot[pts]

Sample from a Penttinen point process defined on the surface of the Earth:

Wolfram Language code: reg = Entity["Country", "Portugal"];
Wolfram Language code: pts = RandomPointConfiguration[PenttinenPointProcess[Quantity[0.01, 1/"Kilometers"^2], 0.3, Quantity[10, "Kilometers"], 2], reg]

Visualize the points:

Wolfram Language code: GeoListPlot[pts]

Scope  (2)

Generate three realizations from a Penttinen point process:

Wolfram Language code: pts = RandomPointConfiguration[ PenttinenPointProcess[100, .5, .05, 2], Rectangle[], 2]
Wolfram Language code: ListPlot[pts]

Estimate the parameters:

Wolfram Language code: Clear[μ, γ, rp, d]; EstimatedPointProcess[pts, PenttinenPointProcess[μ, γ, rp, d]]

Generate three realizations from a Penttinen point process on the surface of the Earth:

Wolfram Language code: μ = Quantity[.3, "Kilometers" ^ -2]; rp = Quantity[10., "Kilometers"]; γ = .5;
Wolfram Language code: proc = PenttinenPointProcess[μ, γ, rp, 2]; reg = GeoDisk[Entity["City", {"Lima", "Lima", "Peru"}], Quantity[30, "Kilometers"]];
Wolfram Language code: pts = RandomPointConfiguration[proc, reg, 2]

Visualize the point configurations:

Wolfram Language code: GeoListPlot[pts]

Estimate the parameters:

Wolfram Language code: Clear[a, b, c, d]; EstimatedPointProcess[pts, PenttinenPointProcess[a, b, c, d]]

Options  (3)

Method  (3)

Sample using the Markov chain Monte Carlo method:

Wolfram Language code: proc = PenttinenPointProcess[100, .2, .01, 2]; reg = Disk[];
Wolfram Language code: RandomPointConfiguration[proc, reg, Method -> "MCMC"]

Specify the number of recursive calls to the sampler:

Wolfram Language code: RandomPointConfiguration[proc, reg, Method -> {"MCMC", MaxRecursion -> 6}]

Specify the length of run:

Wolfram Language code: RandomPointConfiguration[proc, reg, Method -> {"MCMC", "LengthOfRun" -> 5 * 10 ^ 4}]

Provide an initial state for the simulation:

Wolfram Language code: proc = PenttinenPointProcess[50, 0.3, .1, 2]; reg = Disk[];
Wolfram Language code: RandomPointConfiguration[proc, reg, Method -> {"MCMC", "InitialState" -> RandomPoint[reg, 100]}]

Sample using an exact method:

Wolfram Language code: proc = PenttinenPointProcess[30, .5, .3, 2];
Wolfram Language code: reg = Rectangle[];
Wolfram Language code: pts = RandomPointConfiguration[proc, reg, Method -> "Exact"]

Visualize the points in the sample:

Wolfram Language code: ListPlot[pts]

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:

Wolfram Language code: proc = PenttinenPointProcess[10 ^ 3, .2, .01, 2]; reg = Disk[];
Wolfram Language code: RandomPointConfiguration[proc, reg]

Raise the number of recursive calls to the sampler:

Wolfram Language code: RandomPointConfiguration[proc, reg, Method -> {"MCMC", MaxRecursion -> 6}]

Specify a larger length of run:

Wolfram Language code: RandomPointConfiguration[proc, reg, Method -> {"MCMC", "LengthOfRun" -> 5 * 10 ^ 4}]
Wolfram Research (2020), PenttinenPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/PenttinenPointProcess.html.

Text

Wolfram Research (2020), PenttinenPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/PenttinenPointProcess.html.

CMS

Wolfram Language. 2020. "PenttinenPointProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PenttinenPointProcess.html.

APA

Wolfram Language. (2020). PenttinenPointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PenttinenPointProcess.html

BibTeX

@misc{reference.wolfram_2026_penttinenpointprocess, author="Wolfram Research", title="{PenttinenPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PenttinenPointProcess.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_penttinenpointprocess, organization={Wolfram Research}, title={PenttinenPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/PenttinenPointProcess.html}, note=[Accessed: 13-June-2026]}