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RipleyK
✖
RipleyK

[Experimental]

✖
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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Estimate Ripley's function at a given radius:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-7xwnvr
In[3]:=3
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https://wolfram.com/xid/0btn6u8fgphe-4wykf
Out[3]=3

Estimate Ripley's function within a range of distances:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-6p0x9e
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn6u8fgphe-sezo3
In[3]:=3
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https://wolfram.com/xid/0btn6u8fgphe-kknpl5

Visualize the result with ListPlot:

In[4]:=4
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https://wolfram.com/xid/0btn6u8fgphe-ef5jdl
Out[4]=4

Ripley's function of a cluster point process:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-0puauw
Out[1]=1

Visualize the function with given parameter values:

In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-daj61m
Out[2]=2

Scope  (10)Survey of the scope of standard use cases

Point Data  (5)

Estimate Ripley's function at distance 0.2:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-e2gx6r
In[3]:=3
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https://wolfram.com/xid/0btn6u8fgphe-bf4bxp
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0btn6u8fgphe-kyzwq
Out[4]=4

Use RipleyK with SpatialPointData:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-c48w4w
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn6u8fgphe-g7lbop
Out[2]=2

Create a PointStatisticFunction for future use:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-js3gk8
In[2]:=2
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https://wolfram.com/xid/0btn6u8fgphe-odkmbl
Out[2]=2

Find the value of the function at a given radius:

In[3]:=3
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https://wolfram.com/xid/0btn6u8fgphe-vx5qgr
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-f7t1ko
In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-7rwaj3
Out[2]=2

Observation region generated by the Ripley–Rasson estimator:

In[3]:=3
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https://wolfram.com/xid/0btn6u8fgphe-hc1hyh
Out[3]=3

Estimated function at distance 0.3:

In[4]:=4
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https://wolfram.com/xid/0btn6u8fgphe-h12d14
Out[4]=4

Use RipleyK with GeoPosition:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-1vo729
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn6u8fgphe-xpkh6o
Out[2]=2

Plot the point statistics function:

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-7m1ksc
Out[3]=3

Point Processes  (5)

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

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-3hltk1
In[2]:=2
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https://wolfram.com/xid/0btn6u8fgphe-m6387j
Out[2]=2

The function is proportional to :

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-8g4nyq
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-7ib6aq
In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-qpvl0c
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-7fxefx
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-0utcju
Out[4]=4

In 3D:

In[5]:=5
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https://wolfram.com/xid/0btn6u8fgphe-sehhpq
In[6]:=6
✖
https://wolfram.com/xid/0btn6u8fgphe-kghzua
Out[6]=6

Compare with the corresponding Poisson point process:

In[7]:=7
✖
https://wolfram.com/xid/0btn6u8fgphe-jna70l
Out[7]=7
In[8]:=8
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https://wolfram.com/xid/0btn6u8fgphe-mqowd
Out[8]=8

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

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-oi9wov
In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-ofqh4k
Out[2]=2

In 3D:

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-yqw7j7
In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-1q0vyt
Out[4]=4

Ripley's function for a cluster process CauchyPointProcess:

In[1]:=1
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https://wolfram.com/xid/0btn6u8fgphe-td7557
In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-g2fzj0
Out[2]=2

Ripley's function for a cluster process VarianceGammaPointProcess:

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-x1xqiz
In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-2cthww
Out[2]=2

Options  (2)Common values & functionality for each option

SpatialBoundaryCorrection  (2)

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

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-ch7be2
In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-ds1aaa
In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-88aqlu
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0btn6u8fgphe-g1v0t0
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0btn6u8fgphe-eholvz
Out[7]=7

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

In[8]:=8
✖
https://wolfram.com/xid/0btn6u8fgphe-dlt8hb
Out[8]=8

Compare different edge correction methods:

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-cse7si

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

In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-ihhmaw
In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-lflu49

Visualize the results and compare to the theoretical value:

In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-y0awku
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0btn6u8fgphe-pw5mgr
Out[5]=5

Applications  (6)Sample problems that can be solved with this function

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

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-b46qiv
In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-2cenh1
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-l40i3k
Out[4]=4

Ripley's function for complete spatial randomness:

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-kinnz

Compute Ripley's function for a few dimensions:

In[2]:=2
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https://wolfram.com/xid/0btn6u8fgphe-dbm3in

Visualize the results:

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-hsw68q
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-887yj9
In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-lj8rzu
Out[2]=2

Estimate the values of Ripley's function:

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-c3jnbl

Visualize the results:

In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-o9l70r
Out[4]=4

Find hardcore radii estimates for the three samples:

In[5]:=5
✖
https://wolfram.com/xid/0btn6u8fgphe-301t76
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-bpafvv

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

In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-cn3kgx
In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-baqyty
Out[3]=3

Compare the RipleyK functions:

In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-ultu1
In[5]:=5
✖
https://wolfram.com/xid/0btn6u8fgphe-kvvrg8
Out[5]=5

Earthquakes of magnitude 4 or greater in California for the years 2000–2015:

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-g9myqh
In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-0jjm5q
Out[2]=2

Extract the earthquakes' positions:

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-rs8dbf

Compute RipleyK:

In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-ma1dze
Out[4]=4

Mean point density of the data:

In[5]:=5
✖
https://wolfram.com/xid/0btn6u8fgphe-wvq11l

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

In[6]:=6
✖
https://wolfram.com/xid/0btn6u8fgphe-2hahxk
Out[6]=6

Use Ripley's function to estimate PairCorrelationG:

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-b7gwyb

Pair correlation from data:

In[2]:=2
✖
https://wolfram.com/xid/0btn6u8fgphe-m1i3f5
In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-5kciue

Compute Ripley's function:

In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-g5qjzu
Out[4]=4

Estimate pair correlation:

In[5]:=5
✖
https://wolfram.com/xid/0btn6u8fgphe-2j0s8j

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

In[6]:=6
✖
https://wolfram.com/xid/0btn6u8fgphe-m3yi06
Out[6]=6

Properties & Relations  (1)Properties of the function, and connections to other functions

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

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-dcqx9o

Compute both statistics:

In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-dau18n

Check the formula:

In[5]:=5
✖
https://wolfram.com/xid/0btn6u8fgphe-ww6824
Out[5]=5

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0btn6u8fgphe-murbz9
In[3]:=3
✖
https://wolfram.com/xid/0btn6u8fgphe-iu66tv
Out[3]=3

The uncorrected RipleyK is increasing:

In[4]:=4
✖
https://wolfram.com/xid/0btn6u8fgphe-oy3eb9
Out[4]=4
Wolfram Research (2020), RipleyK, Wolfram Language function, https://reference.wolfram.com/language/ref/RipleyK.html.
✖
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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_ripleyk, author="Wolfram Research", title="{RipleyK}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/RipleyK.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_ripleyk, author="Wolfram Research", title="{RipleyK}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/RipleyK.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ripleyk, organization={Wolfram Research}, title={RipleyK}, year={2020}, url={https://reference.wolfram.com/language/ref/RipleyK.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_ripleyk, organization={Wolfram Research}, title={RipleyK}, year={2020}, url={https://reference.wolfram.com/language/ref/RipleyK.html}, note=[Accessed: 29-March-2025 ]}