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 m=sum_(i!=j)Boole[TemplateBox[{{{p, _, i}, -, {p, _, j}}}, Norm]<r_s] with respect to PoissonPointProcess[1,d].
  • The Papangelou conditional density for adding a point to a point configuration is where m=sum_iBoole[TemplateBox[{{{p, _, i}, -, q}}, Norm]<r_s].
  • StraussPointProcess allows μ, γ and r_(s) 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 gamma inhibit points being closer than r_(s).
  • 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:
  • Automaticautomatically choose the parameter estimator
    "MaximumPseudoLikelihood"maximize the pseudo-likelihood
  • StraussPointProcess can be used with such functions as RipleyK and RandomPointConfiguration.

Examples

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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.

BibTeX

@misc{reference.wolfram_2020_strausspointprocess, author="Wolfram Research", title="{StraussPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/StraussPointProcess.html}", note=[Accessed: 15-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_strausspointprocess, organization={Wolfram Research}, title={StraussPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/StraussPointProcess.html}, note=[Accessed: 15-January-2021 ]}

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