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PointDensity
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PointDensity

[Experimental]

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PointDensity[pdata]

estimates the point density function from point data pdata.

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PointDensity[pdata,pmethod]

estimates the point density function with the partition method pmethod.

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PointDensity[bdata,…]

estimates the point density function from binned data bdata.

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PointDensity[pproc,…]

computes the density function for point process pproc.

Details and Options

  • Point density is also known as point intensity.
  • The point density gives a function that describes how the number of points varies per length, area and volume in the observation region ℛ. The integral over the region is the total number of points .
  • PointDensity gives a partition-based estimator that adapts to the point collection in how the partition is formed, effectively using a cell per point.
  • The resulting point density function typically looks very noisy. To get a smoothed version of the point density, use HistogramPointDensity or SmoothPointDensity.
  • Point density is typically used to define an inhomogeneous Poisson process or a measure of inhomogeneity.
  • PointDensity returns a PointDensityFunction that can be used to evaluate the density function repeatedly.
  • The point data pdata can have the following forms:
  • {p1,p2,…}points pi
    GeoPosition[…],GeoPositionXYZ[…],…geographic points
    SpatialPointData[…]spatial point collection
    {pts,reg}point collection pts and observation region reg
  • If the observation region reg is not given, a region automatically computed using RipleyRassonRegion.
  • For the point intensity estimation from the binned data bdata, which is assumed to be SpatialBinnedPointData, the points are simulated uniformly in each bin.
  • The point process pproc can have the following forms:
  • proca point process proc with exact formulas
    {proc,reg}a point process proc and observation region reg based on simulation
  • The observation region reg should be a parameter-free, full-dimensional and bounded region as tested by SpatialObservationRegionQ.
  • The partition method pmethod can be used:
  • "Delaunay"Delaunay cells from point data; gives a piecewise linear density function (default)
    "Voronoi"Voronoi cells from point data; gives a piecewise-constant density function
    {"Voronoi",n}aggregate cells so that approximately n cells remain, based on the smallest cells first

Examples

open allclose all

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

Create a SpatialPointData:

In[6]:=6
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https://wolfram.com/xid/0d6f1yjjbr-05vj9t
Out[6]=6

Plot the points:

In[7]:=7
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https://wolfram.com/xid/0d6f1yjjbr-hjiyok
Out[7]=7

Estimate the point intensity:

In[8]:=8
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https://wolfram.com/xid/0d6f1yjjbr-b6a7b6
Out[8]=8

Value at a point:

In[9]:=9
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https://wolfram.com/xid/0d6f1yjjbr-vsuaf7
Out[9]=9

Visualize the intensity estimation:

In[10]:=10
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https://wolfram.com/xid/0d6f1yjjbr-17dhwf
Out[10]=10

Calculate PointDensity on the surface of the Earth:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-xqho8
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-birkq
In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-b67l3b
Out[3]=3

Visualize the density function using random locations:

In[4]:=4
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https://wolfram.com/xid/0d6f1yjjbr-lrkas8
In[5]:=5
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https://wolfram.com/xid/0d6f1yjjbr-0g2wpb
Out[5]=5

Create density for an InhomogeneousPoissonPointProcess from data:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-dg84fb
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-re6qm

Compute the point intensity function:

In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-ciog7o
Out[3]=3

Define an InhomogeneousPoissonPointProcess with the computed point density:

In[4]:=4
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https://wolfram.com/xid/0d6f1yjjbr-fehvrd

Sample from this point process:

In[5]:=5
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https://wolfram.com/xid/0d6f1yjjbr-foin3g
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0d6f1yjjbr-d7cvoq
Out[6]=6

Scope  (7)Survey of the scope of standard use cases

Point Data  (4)

Create a homogeneous univariate SpatialPointData:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-f7pdx3
Out[2]=2

Compute the point intensity function using different partition methods:

In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-fs63hf
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0d6f1yjjbr-c8mnfp
Out[4]=4

Visualize using a random point sample:

In[5]:=5
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https://wolfram.com/xid/0d6f1yjjbr-4fg35r
In[8]:=8
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https://wolfram.com/xid/0d6f1yjjbr-9q2xxf
Out[8]=8

Create an inhomogeneous univariate SpatialPointData:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-s8csrb
Out[1]=1

Compare the point density function with the smooth kernel density method:

In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-g6rshi
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-ghxscm
Out[3]=3

Visualize:

In[4]:=4
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https://wolfram.com/xid/0d6f1yjjbr-fjhoxl
Out[4]=4

The point density of clustered data:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-148izt
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-lrvzo4
Out[2]=2

Compute the point density from data:

In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-q20vmn
Out[3]=3

Visualize:

In[4]:=4
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https://wolfram.com/xid/0d6f1yjjbr-o4f0lh
Out[4]=4

Point the density for a hardcore process:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-y8f38b
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-k4qcyq
Out[2]=2

Compute the point density:

In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-i8vkwq
Out[3]=3

Visualize:

In[4]:=4
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https://wolfram.com/xid/0d6f1yjjbr-fbtenf
Out[4]=4

Point Processes  (3)

The point density function for PoissonPointProcess is constant in every dimension:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-3hltk1
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-m6387j
Out[2]=2

The point density function for InhomogeneousPoissonPointProcess:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-5jd3j6
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-hydsc
Out[2]=2

The point density function for BinomialPointProcess on a ball:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-7zm1eo
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-mcpeoc

In 1D:

In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-bmq95c
Out[3]=3

In 2D:

In[4]:=4
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https://wolfram.com/xid/0d6f1yjjbr-0s6qk7
Out[4]=4

In 3D:

In[5]:=5
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https://wolfram.com/xid/0d6f1yjjbr-uenv15
Out[5]=5

For a process defined on a square:

In[6]:=6
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https://wolfram.com/xid/0d6f1yjjbr-emjb8a
In[7]:=7
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https://wolfram.com/xid/0d6f1yjjbr-wp02k9
Out[7]=7

Applications  (1)Sample problems that can be solved with this function

Get the data of rat sightings in New York City:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-iqwxi6
Out[1]=1

List column keys to find locations column:

In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-niivws
Out[2]=2

Extract geo locations, delete missing and create SpatialPointData:

In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-s2adap
Out[3]=3

Compute point density:

In[8]:=8
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https://wolfram.com/xid/0d6f1yjjbr-sel3hj
Out[8]=8

Visualize the areas of prevalence of rat sightings in NYC on a sample of the locations:

In[9]:=9
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https://wolfram.com/xid/0d6f1yjjbr-mcpazh
In[11]:=11
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https://wolfram.com/xid/0d6f1yjjbr-n3nsl1
Out[11]=11

Properties & Relations  (1)Properties of the function, and connections to other functions

The density integrates to the number of points:

In[1]:=1
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https://wolfram.com/xid/0d6f1yjjbr-rdyzv
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0d6f1yjjbr-cnnhk2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0d6f1yjjbr-cl8c1x
Out[3]=3
Wolfram Research (2020), PointDensity, Wolfram Language function, https://reference.wolfram.com/language/ref/PointDensity.html.
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Wolfram Research (2020), PointDensity, Wolfram Language function, https://reference.wolfram.com/language/ref/PointDensity.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_pointdensity, author="Wolfram Research", title="{PointDensity}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PointDensity.html}", note=[Accessed: 08-July-2025 ]}

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@misc{reference.wolfram_2025_pointdensity, author="Wolfram Research", title="{PointDensity}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PointDensity.html}", note=[Accessed: 08-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_pointdensity, organization={Wolfram Research}, title={PointDensity}, year={2020}, url={https://reference.wolfram.com/language/ref/PointDensity.html}, note=[Accessed: 08-July-2025 ]}

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@online{reference.wolfram_2025_pointdensity, organization={Wolfram Research}, title={PointDensity}, year={2020}, url={https://reference.wolfram.com/language/ref/PointDensity.html}, note=[Accessed: 08-July-2025 ]}