# RipleyK

RipleyK[pdata,r]

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

RipleyK[pproc,r]

computes for the point process pproc.

RipleyK[bdata,r]

computes for binned data bdata.

RipleyK[pspec]

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

# Details and Options    • The product , where is the mean density, gives the expected number of points within distance r of a typical point, not counting the point itself.
• • RipleyK measures spatial homogeneity of a point collection within distance r. In comparing with a Poisson point process, the results are:
• more dispersed than Poisson like Poisson, i.e. complete spatial randomness more clustered than Poisson
• Here, is the volume of a unit ball in .
• • The radius r can be a single value or a list of values. With no radius r specified, RipleyK 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 a parameter-free 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 Ripley's function at a given radius:

Estimate Ripley's function within a range of distances:

Visualize the result with ListPlot:

Ripley's function of a cluster point process:

Visualize the function with given parameter values:

## Scope(10)

### Point Data(5)

Estimate Ripley's function at distance 0.2:

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

Use RipleyK with SpatialPointData:

Create a PointStatisticFunction for future use:

Find the value of the function at a given radius:

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

Observation region generated by the RipleyRasson estimator:

Estimated function at distance 0.3:

Use RipleyK with GeoPosition:

Plot the point statistics function:

### Point Processes(5)

Ripley's function for PoissonPointProcess has a closed form that does not depend on the intensity:

The function is proportional to :

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

It is always greater than the two-dimensional PoissonPointProcess of the same density:

In 3D:

Compare with the corresponding Poisson point process:

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

In 3D:

Ripley's function for a cluster process CauchyPointProcess:

Ripley's function for a cluster process VarianceGammaPointProcess:

## Options(2)

### SpatialBoundaryCorrection(2)

The RipleyK 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:

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

Compare different edge correction methods:

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

Visualize the results and compare to the theoretical value:

## Applications(6)

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

Ripley's function for complete spatial randomness:

Compute Ripley's function for a few dimensions:

Visualize the results:

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

Estimate the values of Ripley's function:

Visualize the results:

Find hardcore radii estimates for the three samples:

Ripley'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 RipleyK functions:

Earthquakes of magnitude 4 or greater in California for the years 20002015:

Extract the earthquakes' positions:

Compute RipleyK:

Mean point density of the data:

The expected number of earthquakes in within a radius of 2 miles of a typical point in the data:

Use Ripley's function to estimate PairCorrelationG:

Pair correlation from data:

Compute Ripley'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 :

Compute both statistics:

Check the formula:

## Possible Issues(1)

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

The uncorrected RipleyK is increasing: