# 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 overlapping measure in
• A point configuration from a Penttinen point process 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 .
• 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 inhibit points within .
• 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:
•  Automatic automatically choose the parameter estimator "MaximumPseudoLikelihood" maximize the pseudo-likelihood
• PenttinenPointProcess can be used with such functions as RipleyK and RandomPointConfiguration.

# Examples

open allclose all

## Basic Examples(2)

Sample from a Penttinen point process:

Visualize the points in the sample:

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

Visualize the points:

## Scope(2)

Generate three realizations from a Penttinen point process:

Estimate the parameters:

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

Visualize the point configurations:

Estimate the parameters:

## Options(3)

### Method(3)

Sample 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:

Sample using an exact method:

Visualize the points in the sample:

## 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:

Specify a larger length of run:

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_2023_penttinenpointprocess, author="Wolfram Research", title="{PenttinenPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PenttinenPointProcess.html}", note=[Accessed: 16-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_penttinenpointprocess, organization={Wolfram Research}, title={PenttinenPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/PenttinenPointProcess.html}, note=[Accessed: 16-April-2024 ]}