# MaternPointProcess

MaternPointProcess[μ,λ,rm,d]

represents a Matérn cluster point process with density μ, cluster mean λ and radius rm in .

# Details

• MaternPointProcess models clustered point configurations with cluster centers uniformly distributed over space and cluster points isotropically distributed with a uniform radial distribution.
•
• Typical uses include things like plants or trees as centers with seedlings as the points of the cluster.
• The cluster centers are placed according to PoissonPointProcess with density μ.
• The point count of a cluster is distributed according to PoissonDistribution with mean λ.
• The cluster points are uniformly distributed in a ball of radius rm around the cluster center.
•
• MaternPointProcess allows μ, λ and rm to be any positive real numbers and d to be any positive integer.
• The following settings can be used for PointProcessEstimator for estimating MaternPointProcess:
•  "FindClusters" use FindClusters function "MethodOfMoments" use a homogeneity measure to estimate the parameters
• MaternPointProcess can be used with such functions as RipleyK, PointCountDistribution and RandomPointConfiguration.

# Examples

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## Basic Examples(3)

Sample from a Matérn point process over a unit disk:

Sample from a Matérn point process over a unit ball:

Sample from a Matérn point process over a geo region:

## Scope(3)

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

Check the region conditions:

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

Simulate a point configuration from a Matérn point process:

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

Compare the Ripley measure between the original process and the estimated model:

Pair correlation function of a Matérn point process:

Visualize the function with given parameter values:

## Properties & Relations(5)

PointCountDistribution is known:

Mean and variance:

Plot the PDF:

Simulate the distribution:

The probability density histogram:

Ripley's and Besag's for Matérn point process in 2D:

Ripley's of Matérn point process is larger than for a Poisson point process:

Compare to the Poisson point process:

Besag's of Matérn point process is larger than for a Poisson point process:

Compare to the Poisson point process:

Pair correlation of Matérn point process is larger than 1:

Compare to homogeneous Poisson point process:

## Possible Issues(1)

The estimation algorithm splitting the point data into clusters may find a different cluster radius than in a model used to create a given point collection:

Estimate all the parameters:

Therefore, specifying a smaller cluster radius than is inferred from the data will result in the failure of finding a point process model:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_maternpointprocess, author="Wolfram Research", title="{MaternPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/MaternPointProcess.html}", note=[Accessed: 04-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_maternpointprocess, organization={Wolfram Research}, title={MaternPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/MaternPointProcess.html}, note=[Accessed: 04-August-2024 ]}