PenttinenPointProcess
✖
PenttinenPointProcess
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 allBasic Examples (2)Summary of the most common use cases
Sample from a Penttinen point process:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-2mnetb

https://wolfram.com/xid/0g33v44ab0z9rrb1e-cr2aru

https://wolfram.com/xid/0g33v44ab0z9rrb1e-tfb0w

Visualize the points in the sample:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-1pasjf

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

https://wolfram.com/xid/0g33v44ab0z9rrb1e-dm5er3

https://wolfram.com/xid/0g33v44ab0z9rrb1e-h6txtp


https://wolfram.com/xid/0g33v44ab0z9rrb1e-c96vjb

Scope (2)Survey of the scope of standard use cases
Generate three realizations from a Penttinen point process:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-y8f38b


https://wolfram.com/xid/0g33v44ab0z9rrb1e-kv2ge2


https://wolfram.com/xid/0g33v44ab0z9rrb1e-casa32

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

https://wolfram.com/xid/0g33v44ab0z9rrb1e-omonh

https://wolfram.com/xid/0g33v44ab0z9rrb1e-ccigf0

https://wolfram.com/xid/0g33v44ab0z9rrb1e-2efjv

Visualize the point configurations:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-nozzkz


https://wolfram.com/xid/0g33v44ab0z9rrb1e-e4zptm

Options (3)Common values & functionality for each option
Method (3)
Sample using the Markov chain Monte Carlo method:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-mr431h

https://wolfram.com/xid/0g33v44ab0z9rrb1e-q096j

Specify the number of recursive calls to the sampler:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-b28ma3


https://wolfram.com/xid/0g33v44ab0z9rrb1e-hxjlyn

Provide an initial state for the simulation:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-gx4n0o

https://wolfram.com/xid/0g33v44ab0z9rrb1e-ca6pqg


https://wolfram.com/xid/0g33v44ab0z9rrb1e-6rjhmz

https://wolfram.com/xid/0g33v44ab0z9rrb1e-bdh1bd

https://wolfram.com/xid/0g33v44ab0z9rrb1e-hxc47q

Visualize the points in the sample:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-scg73

Possible Issues (1)Common pitfalls and unexpected behavior
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:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-o98jqw

https://wolfram.com/xid/0g33v44ab0z9rrb1e-u8mwy4


Raise the number of recursive calls to the sampler:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-3b099d

Specify a larger length of run:

https://wolfram.com/xid/0g33v44ab0z9rrb1e-1mpfbh

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.
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.
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
Wolfram Language. (2020). PenttinenPointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PenttinenPointProcess.html
BibTeX
@misc{reference.wolfram_2025_penttinenpointprocess, author="Wolfram Research", title="{PenttinenPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PenttinenPointProcess.html}", note=[Accessed: 09-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_penttinenpointprocess, organization={Wolfram Research}, title={PenttinenPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/PenttinenPointProcess.html}, note=[Accessed: 09-July-2025
]}