# PointDensity

PointDensity[pdata]

estimates the point density function from point data pdata.

PointDensity[pdata,pmethod]

estimates the point density function with the partition method pmethod.

PointDensity[bdata,]

estimates the point density function from binned data bdata.

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 . 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:
•  proc a 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

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## Basic Examples(3)

Create a SpatialPointData:

Plot the points:

Estimate the point intensity:

Value at a point:

Visualize the intensity estimation:

Calculate PointDensity on the surface of the Earth:

Visualize the density function using random locations:

Create density for an InhomogeneousPoissonPointProcess from data:

Compute the point intensity function:

Define an InhomogeneousPoissonPointProcess with the computed point density:

Sample from this point process:

## Scope(7)

### Point Data(4)

Create a homogeneous univariate SpatialPointData:

Compute the point intensity function using different partition methods:

Visualize using a random point sample:

Create an inhomogeneous univariate SpatialPointData:

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

Visualize:

The point density of clustered data:

Compute the point density from data:

Visualize:

Point the density for a hardcore process:

Compute the point density:

Visualize:

### Point Processes(3)

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

The point density function for InhomogeneousPoissonPointProcess:

The point density function for BinomialPointProcess on a ball:

In 1D:

In 2D:

In 3D:

For a process defined on a square:

## Properties & Relations(1)

The density integrates to the number of points:

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.

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2024_pointdensity, author="Wolfram Research", title="{PointDensity}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PointDensity.html}", note=[Accessed: 10-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_pointdensity, organization={Wolfram Research}, title={PointDensity}, year={2020}, url={https://reference.wolfram.com/language/ref/PointDensity.html}, note=[Accessed: 10-September-2024 ]}