# CauchyPointProcess

CauchyPointProcess[μ,λ,b,d]

represents a Cauchy cluster point process with density μ, cluster mean λ and scale parameter b in .

# Details  • CauchyPointProcess models clustered point configurations with centers uniformly distributed over space and cluster points isotropically distributed with a heavy-tail radial distribution.
• • Typical uses include forestry to model locations of trees such as long-leaf pines, where the seeds can occasionally be dispersed a long distance from their source.
• 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 CauchyDistribution[0,b].
• The cluster points in are distributed according to MultivariateTDistribution[DiagonalMatrix[{b2,b2,},1] centered at a cluster center.
• • CauchyPointProcess allows μ, λ and b to be any positive real numbers, and d to be any positive integer.
• The following settings can be used for PointProcessEstimator for estimating CauchyPointProcess:
•  "FindClusters" use FindClusters function "MethodOfMoments" use a homogeneity measure to estimate the parameters
• CauchyPointProcess can be used with such functions as RipleyK, PointCountDistribution and RandomPointConfiguration.

# Examples

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## Basic Examples(3)

Sample from a Cauchy point process over a unit disk:

Sample from a Cauchy point process over a unit ball:

Sample from a Cauchy point process over a geo region:

## Scope(3)

Sample from a valid region whose dimension is equal to its embedding dimension:

Sample from a Cauchy point process in the region and visualize the points:

Simulate and estimate:

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

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

Pair correlation function of a Cauchy point process:

Visualize the function with given parameter values:

## Properties & Relations(6)

PointCountDistribution is known:

Mean and variance:

Plot the PDF:

Simulate the distribution:

The probability density histogram:

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

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

Compare to the Poisson point process:

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

Compare to the Poisson point process:

Pair correlation of a Cauchy point process is larger than 1:

Compare to a homogeneous Poisson point process:

Empty space function of a Cauchy point process in 3D: