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SpatialJ
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SpatialJ

[Experimental]

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SpatialJ[pdata,r]

estimates the function for point data pdata at radius r.

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SpatialJ[pproc,r]

computes for the point process pproc.

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SpatialJ[bdata,r]

computes for binned data bdata.

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SpatialJ[pspec]

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

Details and Options

  • The function is the ratio of the probability of finding no point within distance of a typical point to the probability of finding no point within distance of any location. It is given as , where is NearestNeighborG and is EmptySpaceF.
  • SpatialJ measures spatial homogeneity of a point collection within distance . In comparing with a Poisson point process:
  • 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, SpatialJ 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 parameter free and SpatialObservationRegionQ.
  • The binned data bdata is from SpatialBinnedPointData and is treated as an InhomogeneousPoissonPointProcess with a piecewise constant intensify function.
  • For pdata, is computed from the points pi by combining the estimates of and . The estimation assumes the point pattern is stationary in space.
  • For pproc, is computed by using exact formulas or by simulation to generate point data.
  • The following options can be given:
  • Method Automaticwhat 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
    "Hanisch"drops points for which the distance to the nearest neighbor is greater than the distance to boundary
    "KaplanMeier"SurvivalDistribution method: the point distance to its nearest neighbor is censored by its distance to the region boundary
    "NelsonAalen"SurvivalDistribution method: the point distance to its nearest neighbor is censored by its distance to the region boundary
  • The setting Method->{"Discretization"->opts} allows for adjusting the discretization method in the estimation. Here opts can be any valid options for DiscretizeRegion.

Examples

open allclose all

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

Estimate the function at a given distance:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-7xwnvr
In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-4wykf
Out[3]=3

Estimate the function within a range of distances:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-c9nijx
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-bmyu9a
In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-ee0oxv

Visualize the result with ListPlot:

In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-ef5jdl
Out[4]=4

Empty space function of a cluster point process:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-suhjpp
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-daj61m
Out[2]=2

Scope  (7)Survey of the scope of standard use cases

Point Data  (4)

Estimate the function for some data at distance 0.1:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-e2gx6r
In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-bf4bxp
Out[3]=3

Obtain empirical estimates of the function for a list of given distances:

In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-kyzwq
Out[4]=4

Create a PointStatisticFunction for future use:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-js3gk8
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-odkmbl
Out[2]=2

Compute a value at a given radius:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-vx5qgr
Out[3]=3

Estimate the function without explicitly providing the observation region:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-f7t1ko
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-7rwaj3
Out[2]=2

Observation region generated by Ripley–Rasson estimator:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-hc1hyh
Out[3]=3

Estimated function at distance 0.05:

In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-h12d14
Out[4]=4

Use SpatialJ with GeoPosition:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-1vo729
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-xpkh6o
Out[2]=2

Plot the point statistics function:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-7m1ksc
Out[3]=3

Point Processes  (3)

The function function for PoissonPointProcess is always identity:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-3hltk1
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-m6387j
Out[2]=2

This is so because EmptySpaceF and NearestNeighborG of a PoissonPointProcess are identical:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-peicch
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-s1mw7o
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0b8b6f7g1d-iuhnsy
Out[5]=5

The function for a cluster process ThomasPointProcess with specified dimension:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-7ib6aq
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-qpvl0c
Out[2]=2

In 3D:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-sehhpq
In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-kghzua
Out[4]=4

The function for a cluster process MaternPointProcess with specified dimension:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-oi9wov
In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-ofqh4k
Out[2]=2

In 3D:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-yqw7j7
In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-1q0vyt
Out[4]=4

Options  (2)Common values & functionality for each option

SpatialBoundaryCorrection  (1)

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

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-1aqzq
In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-88aqlu
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-eholvz
Out[4]=4

The "Hanisch" method weights each point in the observation region to make the estimated values unbiased:

In[5]:=5
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https://wolfram.com/xid/0b8b6f7g1d-dlt8hb
Out[5]=5

The "KaplanMeier" and "NelsonAalen" methods are estimators used in SurvivalDistribution. The distance of each point to its nearest neighbor point is censored by the distance of each point to the boundary of the observation region:

In[6]:=6
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https://wolfram.com/xid/0b8b6f7g1d-g51708
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0b8b6f7g1d-e9hv8s
Out[7]=7

Method  (1)

Discretization setting can be provided under Method as suboptions:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-uj3zyb

Estimate the function at the same radius with different values of MaxCellMeasure:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-cpbs9d
Out[3]=3

Use different discretization methods to estimate the function at the same radius:

In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-kd4lvk
Out[4]=4

Applications  (2)Sample problems that can be solved with this function

Compare empirical estimates and the theoretical function under complete spatial randomness:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-kinnz

Estimate the values of the function with given data:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-dbm3in

Visualize the results:

In[5]:=5
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https://wolfram.com/xid/0b8b6f7g1d-c9d4xs
Out[5]=5

Compare the function estimate of Matérn point processes with different cluster radius:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-887yj9

Estimate the values of the function with given data:

In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-c3jnbl

Visualize the results:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-o9l70r
Out[3]=3

Properties & Relations  (3)Properties of the function, and connections to other functions

SpatialJ measures clustering of a point collection. indicates the points are clustered within distance , while indicates the points are dispersed within distance . Generate samples from lattice points, a Poisson point process and a Thomas point process:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-cm1a2f
In[4]:=4
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https://wolfram.com/xid/0b8b6f7g1d-fnjfs5
Out[4]=4

Estimate from each sample and compare the results:

In[5]:=5
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https://wolfram.com/xid/0b8b6f7g1d-tg3u8
In[9]:=9
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https://wolfram.com/xid/0b8b6f7g1d-fmmdvv
Out[9]=9

Visualize how NearestNeighborG impacts SpatialJ:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-eqvqul
Out[1]=1

NearestNeighborG estimates the probability of finding another point within distance r from a point in the point collection:

In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-o9rtf6

Visualize and :

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-pj4yjt
Out[3]=3

Visualize how EmptySpaceF impacts SpatialJ:

In[1]:=1
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https://wolfram.com/xid/0b8b6f7g1d-5beqeu
Out[1]=1

EmptySpaceF estimates the probability of finding another point within distance r from a reference point:

In[2]:=2
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https://wolfram.com/xid/0b8b6f7g1d-c4kn8c

Visualize the result:

In[3]:=3
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https://wolfram.com/xid/0b8b6f7g1d-l9fyqo
Out[3]=3
Wolfram Research (2020), SpatialJ, Wolfram Language function, https://reference.wolfram.com/language/ref/SpatialJ.html.
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Wolfram Research (2020), SpatialJ, Wolfram Language function, https://reference.wolfram.com/language/ref/SpatialJ.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_spatialj, author="Wolfram Research", title="{SpatialJ}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/SpatialJ.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_spatialj, author="Wolfram Research", title="{SpatialJ}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/SpatialJ.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_spatialj, organization={Wolfram Research}, title={SpatialJ}, year={2020}, url={https://reference.wolfram.com/language/ref/SpatialJ.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_spatialj, organization={Wolfram Research}, title={SpatialJ}, year={2020}, url={https://reference.wolfram.com/language/ref/SpatialJ.html}, note=[Accessed: 26-March-2025 ]}