# StraussPointProcess

StraussPointProcess[μ,γ,rs,d]

represents a Strauss point process with constant intensity μ, interaction parameter γ and interaction radius rs in d.

# Details

• StraussPointProcess models point configurations with a constant repulsive pairwise interaction for points within radius rs of each other but that are otherwise uniformly distributed.
• The Strauss model is typically used when the process interaction has a constant penalty for points within radius rs, including for locations of plants, birds nests and biological cells.
•
• The Strauss 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 rs as follows:
•  pair potential pair interaction
• A point configuration from a Strauss point process StraussPointProcess[μ,γ,rs,d] in an observation region reg has density function proportional to , where with respect to PoissonPointProcess[1,d].
• The Papangelou conditional density for adding a point to a point configuration is where .
• StraussPointProcess allows μ, γ and to be any positive numbers such that , and d to be any positive integer.
• StraussPointProcess simplifies to HardcorePointProcess when and to PoissonPointProcess when . Smaller values of inhibit points being closer than .
• 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 StraussPointProcess are:
•  Automatic automatically choose the parameter estimator "MaximumPseudoLikelihood" maximize the pseudo-likelihood
• StraussPointProcess can be used with such functions as RipleyK and RandomPointConfiguration.

# Examples

open allclose all

## Basic Examples(2)

Sample from a Strauss point process:

Visualize the points in the sample:

Sample from a geo region:

## Scope(4)

Generate several realizations from a Strauss point process in :

Estimate the process:

Generate several realizations from a Strauss point process on the surface of the Earth:

Estimate the process:

Generate samples with an increasing interacting radius:

Generate samples with increasing interacting parameter γ:

## Options(4)

### Method(4)

Use the Markov chain Monte Carlo simulation method:

Specify the number of recursive calls to the sampler:

Specify the length of run:

Provide an initial state for the simulation:

Visualize the birth and death process at different stages:

Use coupling from the past for exact sampling:

## Applications(1)

Compute the average number of points in a unit disk for a Strauss point process:

## Possible Issues(2)

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:

The objective function for estimating the parameters is the pseudo log-likelihood:

The estimated process has a higher pseudo log-likelihood:

Both processes generate data that looks similar:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_strausspointprocess, author="Wolfram Research", title="{StraussPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/StraussPointProcess.html}", note=[Accessed: 16-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_strausspointprocess, organization={Wolfram Research}, title={StraussPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/StraussPointProcess.html}, note=[Accessed: 16-April-2024 ]}