HistogramPointDensity

HistogramPointDensity[pdata]

estimates the histogram point density function for point data pdata.

HistogramPointDensity[pdata,bspec]

estimates the histogram point density function with histogram bins specified by bspec.

HistogramPointDensity[bdata,,]

estimates the histogram point density function for binned data bdata.

HistogramPointDensity[pproc,,]

computes the histogram point density function for the point process pproc.

Details and Options

  • Point density is also known as point intensity.
  • HistogramPointDensity 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 .
  • HistogramPointDensity is a partition-based estimator of the point density, where the bin specification bspec is used to control the smoothing.
  • Histogram point density is typically used to define an inhomogeneous Poisson process or a measure of inhomogeneity.
  • HistogramPointDensity returns a PointDensityFunction that can be used to evaluate the density function repeatedly.
  • The points pdata can have the following forms:
  • {p1,p2,}points pi
    GeoPosition[],GeoPositionXYZ[],geographic points
    SpatialPointData[]spatial point collection with observation region
    {pts,reg}point collection pts and observation region reg
  • If the observation region reg is not given, a region is automatically computed using RipleyRassonRegion.
  • The binned data bdata is assumed to come in the form of SpatialBinnedPointData.
  • 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 following geometric and geographic bin specifications bspec can be given:
  • MeshRegion[]
  • explicit MeshRegion with cells
  • "ObservationMesh"discretization of observation region
    {reg1,reg2, ...}
  • explicit list of disjoint regions
  • shapegeometric shape depending on the dimension
    {shape,"Count"n}aggregated bins to approximately n bins
    {shape,"Measure"ν}aggregated bins of approximate measure ν
    {shape,"Diameter"d}aggregated bins of approximate diameter d
  • Possible settings for shape include:
  • "Triangle"triangle bins in 2D
    "Square"square bins in 2D
    "Hexagon"hexagonal bins in 2D
  • The geometric bins can also be created by providing bspec coordinate-wise:
  • nuse n bins
    {w}use bins of width w
    {min,max,w}use bins of width w from min to max
    {{b1,b2,}}use bins [b1,b2),[b2,b3),
    Automaticdetermine bin widths automatically
    "name"use a named binning method
    fwapply fw to get an explicit bin specification {b1,b2,}
    {xspec,yspec,}give different x, y, etc. specifications
  • Possible named binning methods include:
  • "FreedmanDiaconis"twice the interquartile range divided by the cube root of sample size
    "Knuth"balance likelihood and prior probability of a piecewise uniform model
    "Scott"asymptotically minimize the mean square error
    "Sturges"compute the number of bins based on the length of data
    "Wand"one-level recursive approximate Wand binning
  • All the bins are trimmed to intersect with the observation region.

Examples

open allclose all

Basic Examples  (2)

Create a SpatialPointData:

Estimate the point density:

Value at a point:

Visualize the density estimation:

Compute histogram point density for a list of geographic points:

Visualize the density:

Scope  (5)

Create a homogeneous univariate SpatialPointData:

Compute point density function using different bin shapes:

Visualize using values at random locations:

Histogram point density of clustered data:

Compute histogram point density from data:

Visualize:

Histogram point density for a hardcore process:

Compute histogram point density from data for various bin diameters:

Visualize:

Compare bin shape specifications:

Define a mesh region and polygon composite region:

Compute densities:

Visualize the densities:

Allowed bin specifications on the surface of the Earth:

Create the list of administrative polygons:

Compute densities:

Visualize:

Properties & Relations  (1)

HistogramPointDensity is related to Histogram with bin height specification "Intensity":

Specifying bin width in computation of histogram point density:

Compare the density function to the intensity Histogram with the same bin width:

For 2D data:

Compute histogram point intensity:

Compare plot of the density function with the Histogram3D of the data:

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

Text

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

BibTeX

@misc{reference.wolfram_2020_histogrampointdensity, author="Wolfram Research", title="{HistogramPointDensity}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HistogramPointDensity.html}", note=[Accessed: 16-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_histogrampointdensity, organization={Wolfram Research}, title={HistogramPointDensity}, year={2020}, url={https://reference.wolfram.com/language/ref/HistogramPointDensity.html}, note=[Accessed: 16-January-2021 ]}

CMS

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

APA

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