# NeymanScottPointProcess

NeymanScottPointProcess[μ,λ,rdist,d]

represents a NeymanScott point process with density function μ, cluster mean λ and radial cluster point distribution rdist in .

NeymanScottPointProcess[μ,λ,mdist,d]

uses a multivariate cluster point distribution mdist in .

# Details

• NeymanScottPointProcess is also known as the center-satellite process.
• NeymanScottPointProcess models clustered point configurations with centers placed according to an inhomogeneous Poisson point process and cluster points distributed around the centers according to a cluster distribution.
• Typical uses include herds of animals in the wild, clusters of seedlings around a parent tree, modeling bombing patterns and insect larvae patterns.
• Cluster centers are placed according to InhomogeneousPoissonPointProcess with density function in .
• The point count of a cluster is distributed according to PoissonDistribution with mean λ.
• Cluster points following an isotropic distribution are most easily specified using a radial distribution rdist.
• A general cluster distribution can be specified using a multivariate distribution mdist.
• NeymanScottPointProcess is a general Poisson cluster process; common Poisson cluster processes have dedicated functions and are easier and more efficient to use when applicable.
•  process radial distribution characteristic MaternPointProcess uniform cluster points ThomasPointProcess normal cluster points CauchyPointProcess heavy tail cluster points VarianceGammaPointProcess normal and gamma mixture cluster points
• NeymanScottPointProcess allows λ to be any positive real number and and d to be any positive integer.
• The following settings can be used for PointProcessEstimator for estimating NeymanScottPointProcess:
•  "FindClusters" use FindClusters function "MethodOfMoments" use a homogeinity measure to estimate the parameters
• NeymanScottPointProcess can be used with such functions as RipleyK, PointCountDistribution and RandomPointConfiguration.

# Examples

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## Basic Examples(4)

Sample from a NeymanScott point process with a radial cluster distribution:

Sample from a 3D NeymanScott point process over a unit ball with multivariate cluster distribution:

Sample over a geographical region:

Valid density functions are the same as for InhomogeneousPoissonPointProcess:

## Scope(2)

Sample over a valid region whose dimension is equal to its embedding dimension:

Sample from a NeymanScott point process in the region and visualize the points:

Simulate a point configuration from a NeymanScott point process:

Use the "FindClusters" method to estimate a point process model:

## Properties & Relations(1)

PointCountDistribution is known:

Mean and variance:

Plot the PDF:

Simulate the distribution:

The probability density histogram:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_neymanscottpointprocess, author="Wolfram Research", title="{NeymanScottPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/NeymanScottPointProcess.html}", note=[Accessed: 16-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_neymanscottpointprocess, organization={Wolfram Research}, title={NeymanScottPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/NeymanScottPointProcess.html}, note=[Accessed: 16-April-2024 ]}