CauchyPointProcess[μ,λ,b,d]
represents a Cauchy cluster point process with density μ, cluster mean λ and scale parameter b in 
.
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
open all close allBasic Examples (3)
Sample from a Cauchy point process over a unit disk:
pts = RandomPointConfiguration[CauchyPointProcess[10, 20, .01, 2], Disk[]]Show[RegionPlot[pts["ObservationRegion"]], ListPlot[pts]]Sample from a Cauchy point process over a unit ball:
pts = RandomPointConfiguration[CauchyPointProcess[30, 20, .01, 3], Ball[]]Show[RegionPlot3D[pts["ObservationRegion"], PlotStyle -> Opacity[.2], Boxed -> False], ListPointPlot3D[pts]]Sample from a Cauchy point process over a geo region:
proc = CauchyPointProcess[.006, .3, .1, 2];pts = RandomPointConfiguration[proc, Entity["Country", "Poland"]]GeoListPlot[pts]Scope (3)
Sample from a valid region whose dimension is equal to its embedding dimension:
ℛ = ImplicitRegion[x ^ 2 - 2y ^ 2 <= 1, {{x, -3, 3}, {y, -4, 4}}];{RegionQ[ℛ], RegionEmbeddingDimension[ℛ] == RegionDimension[ℛ]}Sample from a Cauchy point process in the region and visualize the points:
pts = RandomPointConfiguration[CauchyPointProcess[4, 15, 0.05, 2], ℛ]Show[RegionPlot[pts["ObservationRegion"]], ListPlot[pts]]proc = CauchyPointProcess[20, 30, 1, 2];
points = RandomPointConfiguration[proc, Rectangle[]];PointValuePlot[points]Use the "FindClusters" method to estimate a point process model:
est = EstimatedPointProcess[points, CauchyPointProcess[a, b, c, d], PointProcessEstimator -> "FindClusters"]Compare the Ripley 
measure between the original process and the estimated model:
DiscretePlot[{RipleyK[proc, r], RipleyK[est, r]}, {r, 0.1, .5, .005}, PlotLegends -> {"original process", "estimated model"}]Pair correlation function of a Cauchy point process:
pcf = PairCorrelationG[CauchyPointProcess[μ, λ, b, 3], r]Visualize the function with given parameter values:
Plot[pcf /. {μ -> 3, λ -> 2, b -> 1}, {r, 0, 3}]Properties & Relations (6)
PointCountDistribution is known:
proc = CauchyPointProcess[4, 15, .1, 2];dist = PointCountDistribution[proc, Disk[]]{Mean[dist], Variance[dist]}DiscretePlot[PDF[dist, x], {x, 50, 400, 5}]sample = RandomVariate[dist, 10 ^ 4];The probability density histogram:
Histogram[sample, 30, "PDF"]Ripley's 
and Besag's 
for Cauchy point process in 2D:
proc = CauchyPointProcess[μ, λ, b, 2];f1 = RipleyK[proc, r]f2 = BesagL[proc, r]Block[{μ = 20, λ = 10, b = .5}, Plot[{f1, f2}, {r, 0, 1}, PlotLegends -> {"Ripley's K", "Besag's L"}]]Ripley's 
of a Cauchy point process is larger than for a Poisson point process:
cproc = CauchyPointProcess[μ, λ, b, d];RipleyK[cproc][r]Compare to the Poisson point process:
pproc = PoissonPointProcess[μ, d];RipleyK[pproc][r]Besag's 
of a Cauchy point process is larger than for a Poisson point process:
cproc = CauchyPointProcess[μ, λ, b, d];BesagL[cproc][r]//TogetherCompare to the Poisson point process:
pproc = PoissonPointProcess[μ, d];BesagL[pproc][r]Pair correlation of a Cauchy point process is larger than 1:
PairCorrelationG[CauchyPointProcess[μ, λ, b, d], r]Compare to a homogeneous Poisson point process:
PairCorrelationG[PoissonPointProcess[μ, d], r]Empty space function of a Cauchy point process in 3D:
proc = CauchyPointProcess[4, 15, 0.05, 3];DiscretePlot[EmptySpaceF[proc, r], {r, 0.02, .5, 0.01}]Text
Wolfram Research (2020), CauchyPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/CauchyPointProcess.html.
CMS
Wolfram Language. 2020. "CauchyPointProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CauchyPointProcess.html.
APA
Wolfram Language. (2020). CauchyPointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CauchyPointProcess.html