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:
  • proca 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:
  • MethodAutomaticwhat methods to use
    SpatialBoundaryCorrection Automaticwhat boundary correction to use
  • The following settings can be used for SpatialBoundaryCorrection:
  • Automaticautomatically determined boundary correction
    Noneno 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 Ripley–Rasson 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 2000–2015:

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:

Wolfram Research (2020), RipleyK, Wolfram Language function, https://reference.wolfram.com/language/ref/RipleyK.html.

Text

Wolfram Research (2020), RipleyK, Wolfram Language function, https://reference.wolfram.com/language/ref/RipleyK.html.

CMS

Wolfram Language. 2020. "RipleyK." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RipleyK.html.

APA

Wolfram Language. (2020). RipleyK. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RipleyK.html

BibTeX

@misc{reference.wolfram_2024_ripleyk, author="Wolfram Research", title="{RipleyK}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/RipleyK.html}", note=[Accessed: 23-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_ripleyk, organization={Wolfram Research}, title={RipleyK}, year={2020}, url={https://reference.wolfram.com/language/ref/RipleyK.html}, note=[Accessed: 23-April-2024 ]}