# BesagL

BesagL[pdata,r]

estimates Besag's function for point data pdata at radius r.

BesagL[pproc,r]

computes for the point process pproc.

BesagL[bdata,r]

computes for binned data bdata.

BesagL[pspec]

generates the function that can be applied repeatedly to different radii r.

# Details and Options    • BesagL is a transformation of the RipleyK function that makes it easier to compare to a completely spatially random reference process.
• BesagL is defined as , where is RipleyK, is the spatial dimension, and is the volume of a unit ball in .
• • BesagL measures spatial homogeneity of a point collection within distance r. Comparing with a Poisson point process gives:
• more dispersed than Poisson like Poisson, i.e. complete spatial randomness more clustered than Poisson
• • The radius r can be a single value or a list of values. With no radius r specified, BesagL returns a PointStatisticFunction that can be used to evaluate the function repeatedly.
• The points pdata can have the following forms:
•  {p1,p2,…} points pi GeoPosition[…],GeoPositionXYZ[…],… geographic points SpatialPointData[…] spatial point collection {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 point process pproc can have the following forms:
•  proc a point process proc {proc,reg} a point process proc and observation region reg
• The observation region reg should be parameter free and SpatialObservationRegionQ.
• The binned data bdata is from SpatialBinnedPointData and is treated as an InhomogeneousPoissonPointProcess with a piecewise constant density function.
• For pdata, is computed by counting distinct pairs of points within distance r of each other.
• For pproc, is computed by using exact formulas or by simulation to generate point data.
• The following options can be given:
•  Method Automatic what methods to use SpatialBoundaryCorrection Automatic what boundary correction to use
• The following settings can be used for SpatialBoundaryCorrection:
•  Automatic automatically determined boundary correction None no boundary correction "BorderMargin" use interior margin for observation region "Ripley" uses weights depending on the point distance to boundary

# Examples

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## Basic Examples(3)

Estimate Besag's function at a given distance:

Estimate Besag's function within a range of distances:

Visualize the result with ListPlot:

Besag's function of a cluster point process:

Visualize the function with given parameter values:

## Scope(10)

### Point Data(5)

Estimate Besag's function at distance 0.2:

Obtain empirical estimates of Besag's function from a list of given distances:

Use BesagL with SpatialPointData:

Create a PointStatisticFunction for future use:

Estimate Besag's function without explicitly providing the observation region:

Observation region generated by RipleyRasson estimator:

Estimated function at distance 0.3:

Use BesagL with GeoPosition:

Plot the point statistics function:

### Point Processes(5)

Besag's function for PoissonPointProcess does not depend on the density or the dimension:

Besag's function for a cluster process ThomasPointProcess with specified dimension:

In 3D:

Besag's function for a cluster process MaternPointProcess with specified dimension:

In 3D:

Besag's function for a cluster process CauchyPointProcess:

Besag's function for a cluster process VarianceGammaPointProcess:

## Options(2)

### SpatialBoundaryCorrection(2)

The BesagL estimator without boundary correction is biased and should not be used unless with a large point set:

The default method "BorderMargin" only considers the points that are distance from the boundary:

Boundary correction method "Ripley" weights each pair of points to make the estimator unbiased:

Compare different edge correction methods:

Estimate the values of Besag's function with three different methods:

Visualize the results and compare to the theoretical value:

## Applications(5)

Besag's function is cumulative in the distance and hence monotone increasing:

Besag's function for complete spatial randomness:

Compute Besag's function for few dimensions:

Visualize the results:

Points in a hardcore point process cannot be closer than the hard-core radius :

Estimate the values of Besag's function:

Visualize the results:

Find hard-core radii estimates for the three samples:

Besag's of clustered data is higher than complete spatially random data. Sample from a cluster process:

Generate a control sample from a Poisson point process with the same intensity:

Compare the Besag's functions:

Use Besag's function to estimate PairCorrelationG:

Pair correlation from data:

Compute Besag's function:

Estimate pair correlation:

Compare the estimate with the pair correlation computed from the data:

## Properties & Relations(1)

BesagL is the variance-stabilized RipleyK: , where is Ripley's function, is the spatial dimension and is the volume of a unit ball in :

## Possible Issues(1)

Empirical BesagL with border correction may not be increasing (especially for smaller sets):

The uncorrected BesagL is increasing: