MaternPointProcess[μ,λ,rm,d]
represents a Matérn cluster point process with density μ, cluster mean λ and radius rm in 
.
MaternPointProcess
MaternPointProcess[μ,λ,rm,d]
represents a Matérn cluster point process with density μ, cluster mean λ and radius rm in 
.
Details
- MaternPointProcess models clustered point configurations with cluster centers uniformly distributed over space and cluster points isotropically distributed with a uniform radial distribution.
- Typical uses include things like plants or trees as centers with seedlings as the points of the cluster.
- 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 are uniformly distributed in a ball of radius rm around the cluster center.
- MaternPointProcess allows μ, λ and rm to be any positive real numbers and d to be any positive integer.
- The following settings can be used for PointProcessEstimator for estimating MaternPointProcess:
-
"FindClusters" use FindClusters function "MethodOfMoments" use a homogeneity measure to estimate the parameters - MaternPointProcess can be used with such functions as RipleyK, PointCountDistribution and RandomPointConfiguration.
Examples
open all close allBasic Examples (3)
Sample from a Matérn point process over a unit disk:
pts = RandomPointConfiguration[MaternPointProcess[10, 20, .1, 2], Disk[]]Show[RegionPlot[pts["ObservationRegion"]], ListPlot[pts]]Sample from a Matérn point process over a unit ball:
pts = RandomPointConfiguration[MaternPointProcess[30, 20, .1, 3], Ball[]]Show[RegionPlot3D[pts["ObservationRegion"], PlotStyle -> Opacity[.2], Boxed -> False], ListPointPlot3D[pts["Points"]]]Sample from a Matérn point process over a geo region:
proc = MaternPointProcess[.01, .5, .1, 2];pts = RandomPointConfiguration[proc, Entity["Country", "CzechRepublic"]]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 Matérn point process in the region and visualize the points:
pts = RandomPointConfiguration[MaternPointProcess[4, 15, .3, 2], ℛ]Show[RegionPlot[pts["ObservationRegion"]], ListPlot[pts]]Simulate a point configuration from a Matérn point process:
proc = MaternPointProcess[20, 30, 1, 2];
points = RandomPointConfiguration[proc, Rectangle[]];PointValuePlot[points]Use the "FindClusters" method to estimate a point process model:
est = EstimatedPointProcess[points, MaternPointProcess[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 Matérn point process:
pcf = PairCorrelationG[MaternPointProcess[μ, λ, R, d], r]Visualize the function with given parameter values:
Plot[Table[pcf /. {μ -> 3, λ -> 2, R -> 1}, {d, 1, 4}]//Evaluate, {r, 0, 3}, PlotLegends -> {"1D", "2D", "3D", "4D"}]Properties & Relations (5)
PointCountDistribution is known:
proc = MaternPointProcess[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 Matérn point process in 2D:
proc = MaternPointProcess[μ, λ, R, 2];RipleyK[proc, r]BesagL[proc, r]Block[{μ = 20, m = 10, R = .5}, Plot[{RipleyK[proc][r], BesagL[proc][r]}, {r, 0, 1}, PlotLegends -> {"Ripley's K", "Besag's L"}]]Ripley's 
of Matérn point process is larger than for a Poisson point process:
mproc = MaternPointProcess[μ, λ, R, d];RipleyK[mproc][r]Compare to the Poisson point process:
pproc = PoissonPointProcess[μ, d];RipleyK[pproc][r]Besag's 
of Matérn point process is larger than for a Poisson point process:
mproc = MaternPointProcess[μ, λ, R, d];BesagL[mproc][r]Compare to the Poisson point process:
pproc = PoissonPointProcess[μ, d];BesagL[pproc][r]Pair correlation of Matérn point process is larger than 1:
pc = PairCorrelationG[MaternPointProcess[μ, λ, R, d], r]Compare to homogeneous Poisson point process:
PairCorrelationG[PoissonPointProcess[μ, d], r]Possible Issues (1)
The estimation algorithm splitting the point data into clusters may find a different cluster radius than in a model used to create a given point collection:
pts = RandomPointConfiguration[MaternPointProcess[10, 20, 0.1, 2], Disk[]];FindPointProcessParameters[pts, MaternPointProcess[μ, λ, r, d]]Therefore, specifying a smaller cluster radius 
than is inferred from the data will result in the failure of finding a point process model:
EstimatedPointProcess[pts, MaternPointProcess[μ, λ, .1, d]]
Text
Wolfram Research (2020), MaternPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/MaternPointProcess.html.
CMS
Wolfram Language. 2020. "MaternPointProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MaternPointProcess.html.
APA
Wolfram Language. (2020). MaternPointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MaternPointProcess.html