# VarianceGammaPointProcess

VarianceGammaPointProcess[μ,λ,α,β,d]

represents a variance gamma cluster point process with density μ, cluster mean λ and shape parameters α and β in .

# Details  • VarianceGammaPointProcess models clustered point configurations with centers uniformly distributed over space and cluster points isotropically distributed with a flexible radial distribution.
• • Typical uses include cosmology and rainforest tree distribution.
• 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 in each cluster in are distributed according to VarianceGammaDistribution[α, ,0,0].
• The cluster points in are distributed according to MultinormalDistribution[DiagonalMatrix[{u,u,}]], with sampled from GammaDistribution[α,β] centered at a cluster center.
• • VarianceGammaPointProcess allows μ, λ, α and β to be any positive real numbers, and d to be any positive integer.
• The following settings can be used for PointProcessEstimator for estimating VarianceGammaPointProcess:
•  "FindClusters" use FindClusters function "MethodOfMoments" use a homogeneity measure to estimate the parameters
• VarianceGammaPointProcess can be used with such functions as RipleyK, PointCountDistribution and RandomPointConfiguration.

# Examples

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## Basic Examples(4)

Sample from a variance gamma point process over a unit disk:

Sample from a variance gamma point process over a unit ball:

Sample from a variance gamma point process over a geo region:

Pair correlation function of a variance gamma point process:

Visualize the function with given parameter values:

## Scope(2)

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

Check the region conditions:

Sample points:

Simulate a point configuration from a variance gamma point process:

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

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

## Properties & Relations(5)

The PointCountDistribution is known:

Mean and variance:

Plot the PDF:

Simulate the distribution:

The probability density histogram:

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

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

Compare to a Poisson point process:

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

Compare to a Poisson point process:

The pair correlation of a variance gamma point process is larger than 1:

Compare to a homogeneous Poisson point process: