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ThomasPointProcess
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ThomasPointProcess

[Experimental]

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ThomasPointProcess[μ,λ,σ,d]

represents a Thomas cluster point process with density μ, cluster mean λ and scale parameter σ in R^(d).

Details

Examples

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Basic Examples  (4)Summary of the most common use cases

Sample from a Thomas point process over a unit disk:

In[1]:=1
✖
https://wolfram.com/xid/0d4jv629oujvur26-4mqsf
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0d4jv629oujvur26-d381by
Out[2]=2

Sample from a Thomas point process over a unit ball:

In[1]:=1
✖
https://wolfram.com/xid/0d4jv629oujvur26-dgk7lg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d4jv629oujvur26-svdcv0
Out[2]=2

Sample from a Thomas point process over a geo region:

In[1]:=1
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https://wolfram.com/xid/0d4jv629oujvur26-uf2aio
In[2]:=2
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https://wolfram.com/xid/0d4jv629oujvur26-uc5jvc
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0d4jv629oujvur26-uc869u
Out[3]=3

Pair correlation function of a Thomas point process:

In[1]:=1
✖
https://wolfram.com/xid/0d4jv629oujvur26-0puauw
Out[1]=1

Visualize the function with given parameter values:

In[2]:=2
✖
https://wolfram.com/xid/0d4jv629oujvur26-daj61m
Out[2]=2

Scope  (2)Survey of the scope of standard use cases

Sample from any valid RegionQ, whose RegionEmbeddingDimension is equal to its RegionDimension:

In[1]:=1
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https://wolfram.com/xid/0d4jv629oujvur26-qxrhco

Check the region conditions:

In[2]:=2
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https://wolfram.com/xid/0d4jv629oujvur26-pacet0
Out[2]=2

Sample points:

In[3]:=3
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https://wolfram.com/xid/0d4jv629oujvur26-ydq337
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0d4jv629oujvur26-h8cla8
Out[4]=4

Simulate a point configuration from a Thomas point process:

In[1]:=1
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https://wolfram.com/xid/0d4jv629oujvur26-8c7zax
In[2]:=2
✖
https://wolfram.com/xid/0d4jv629oujvur26-9g5xhe
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0d4jv629oujvur26-3o7emz
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0d4jv629oujvur26-m5qyr2
Out[4]=4

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/0d4jv629oujvur26-1vmg8n
In[2]:=2
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https://wolfram.com/xid/0d4jv629oujvur26-69w4w
Out[2]=2

Mean and variance:

In[3]:=3
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https://wolfram.com/xid/0d4jv629oujvur26-k29vdf
Out[3]=3

Plot the PDF:

In[4]:=4
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https://wolfram.com/xid/0d4jv629oujvur26-iw9fj7
Out[4]=4

Simulate the distribution:

In[5]:=5
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https://wolfram.com/xid/0d4jv629oujvur26-rl0tf0

The probability density histogram:

In[6]:=6
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https://wolfram.com/xid/0d4jv629oujvur26-57hv5g
Out[6]=6

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

In[1]:=1
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https://wolfram.com/xid/0d4jv629oujvur26-zht24f
In[2]:=2
✖
https://wolfram.com/xid/0d4jv629oujvur26-1gisn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0d4jv629oujvur26-myijw0
Out[3]=3

Visualize:

In[4]:=4
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https://wolfram.com/xid/0d4jv629oujvur26-26ar4l
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0d4jv629oujvur26-5pj0tq
Out[5]=5

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

In[1]:=1
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https://wolfram.com/xid/0d4jv629oujvur26-zglflc
In[2]:=2
✖
https://wolfram.com/xid/0d4jv629oujvur26-cp3zmu
In[3]:=3
✖
https://wolfram.com/xid/0d4jv629oujvur26-7f0ncr
Out[3]=3

In 2D and fixed density:

In[4]:=4
✖
https://wolfram.com/xid/0d4jv629oujvur26-7jdja1
Out[4]=4

Besag's of the Thomas point process is greater than of the Poisson point process:

In[1]:=1
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https://wolfram.com/xid/0d4jv629oujvur26-037cpr
In[2]:=2
✖
https://wolfram.com/xid/0d4jv629oujvur26-kdnk7q
In[3]:=3
✖
https://wolfram.com/xid/0d4jv629oujvur26-vk8oue
Out[3]=3

In 2D and fixed density:

In[4]:=4
✖
https://wolfram.com/xid/0d4jv629oujvur26-1avd72
Out[4]=4

The pair correlation function of a Thomas point process is greater than 1:

In[1]:=1
✖
https://wolfram.com/xid/0d4jv629oujvur26-tlke7q
Out[1]=1

Compare to the homogeneous Poisson point process:

In[2]:=2
✖
https://wolfram.com/xid/0d4jv629oujvur26-o75yfs
Out[2]=2
Wolfram Research (2020), ThomasPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ThomasPointProcess.html.
✖
Wolfram Research (2020), ThomasPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ThomasPointProcess.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_thomaspointprocess, author="Wolfram Research", title="{ThomasPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ThomasPointProcess.html}", note=[Accessed: 27-April-2025 ]}

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@misc{reference.wolfram_2025_thomaspointprocess, author="Wolfram Research", title="{ThomasPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ThomasPointProcess.html}", note=[Accessed: 27-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_thomaspointprocess, organization={Wolfram Research}, title={ThomasPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/ThomasPointProcess.html}, note=[Accessed: 27-April-2025 ]}

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@online{reference.wolfram_2025_thomaspointprocess, organization={Wolfram Research}, title={ThomasPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/ThomasPointProcess.html}, note=[Accessed: 27-April-2025 ]}