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MaternPointProcess
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MaternPointProcess

[Experimental]

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Sample from a Matérn point process over a unit disk:

In[1]:=1
✖
https://wolfram.com/xid/0k7xsqtlgylde-4mqsf
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0k7xsqtlgylde-d381by
Out[2]=2

Sample from a Matérn point process over a unit ball:

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-dgk7lg
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0k7xsqtlgylde-svdcv0
Out[2]=2

Sample from a Matérn point process over a geo region:

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-uf2aio
In[2]:=2
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https://wolfram.com/xid/0k7xsqtlgylde-uc5jvc
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0k7xsqtlgylde-ng3ovh
Out[3]=3

Scope  (3)Survey of the scope of standard use cases

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

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-qxrhco

Check the region conditions:

In[2]:=2
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https://wolfram.com/xid/0k7xsqtlgylde-pacet0
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0k7xsqtlgylde-ydq337
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0k7xsqtlgylde-h8cla8
Out[4]=4

Simulate a point configuration from a Matérn point process:

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-8c7zax
In[2]:=2
✖
https://wolfram.com/xid/0k7xsqtlgylde-9g5xhe
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0k7xsqtlgylde-3o7emz
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0k7xsqtlgylde-m5qyr2
Out[4]=4

Pair correlation function of a Matérn point process:

In[1]:=1
✖
https://wolfram.com/xid/0k7xsqtlgylde-0puauw
Out[1]=1

Visualize the function with given parameter values:

In[2]:=2
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https://wolfram.com/xid/0k7xsqtlgylde-daj61m
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

PointCountDistribution is known:

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-pfh07m
In[2]:=2
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https://wolfram.com/xid/0k7xsqtlgylde-69w4w
Out[2]=2

Mean and variance:

In[3]:=3
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https://wolfram.com/xid/0k7xsqtlgylde-6765kl
Out[3]=3

Plot the PDF:

In[4]:=4
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https://wolfram.com/xid/0k7xsqtlgylde-iw9fj7
Out[4]=4

Simulate the distribution:

In[5]:=5
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https://wolfram.com/xid/0k7xsqtlgylde-k0gjl

The probability density histogram:

In[6]:=6
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https://wolfram.com/xid/0k7xsqtlgylde-nd9jc
Out[6]=6

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

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-zht24f
In[2]:=2
✖
https://wolfram.com/xid/0k7xsqtlgylde-rsadsn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0k7xsqtlgylde-tcihup
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0k7xsqtlgylde-5pj0tq
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-zglflc
In[2]:=2
✖
https://wolfram.com/xid/0k7xsqtlgylde-forzeq
Out[2]=2

Compare to the Poisson point process:

In[3]:=3
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https://wolfram.com/xid/0k7xsqtlgylde-cp3zmu
In[4]:=4
✖
https://wolfram.com/xid/0k7xsqtlgylde-9isgv3
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0k7xsqtlgylde-ch3e7z
In[2]:=2
✖
https://wolfram.com/xid/0k7xsqtlgylde-egoxn6
Out[2]=2

Compare to the Poisson point process:

In[3]:=3
✖
https://wolfram.com/xid/0k7xsqtlgylde-cgq77l
In[4]:=4
✖
https://wolfram.com/xid/0k7xsqtlgylde-sqgcvm
Out[4]=4

Pair correlation of Matérn point process is larger than 1:

In[1]:=1
✖
https://wolfram.com/xid/0k7xsqtlgylde-2w2wb7
Out[1]=1

Compare to homogeneous Poisson point process:

In[2]:=2
✖
https://wolfram.com/xid/0k7xsqtlgylde-ydohzs
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

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:

In[1]:=1
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https://wolfram.com/xid/0k7xsqtlgylde-w9v2sr

Estimate all the parameters:

In[4]:=4
✖
https://wolfram.com/xid/0k7xsqtlgylde-xjutht
Out[4]=4

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

In[3]:=3
✖
https://wolfram.com/xid/0k7xsqtlgylde-esn2pa
Out[3]=3
Wolfram Research (2020), MaternPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/MaternPointProcess.html.
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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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_maternpointprocess, author="Wolfram Research", title="{MaternPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/MaternPointProcess.html}", note=[Accessed: 04-June-2025 ]}

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@misc{reference.wolfram_2025_maternpointprocess, author="Wolfram Research", title="{MaternPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/MaternPointProcess.html}", note=[Accessed: 04-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_maternpointprocess, organization={Wolfram Research}, title={MaternPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/MaternPointProcess.html}, note=[Accessed: 04-June-2025 ]}

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@online{reference.wolfram_2025_maternpointprocess, organization={Wolfram Research}, title={MaternPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/MaternPointProcess.html}, note=[Accessed: 04-June-2025 ]}