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EmptySpaceF

[Experimental]

✖
EmptySpaceF[pdata,r]

estimates the empty space function for point data pdata at radius r.

✖
EmptySpaceF[pproc,r]

computes for point process pproc.

✖
EmptySpaceF[bdata,r]

computes for binned data bdata.

✖
EmptySpaceF[pspec]

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

Details and Options

  • EmptySpaceF is also known as spherical contact distribution function.
  • The function gives the probability of finding a point within distance of an arbitrary location which is typically not a point of pdata.
  •     
  • When comparing with a Poisson point process, the results are:
  •     
  • The radius r can be a single value or a list of values. With no radius r specified, EmptySpaceF returns a PointStatisticFunction that can be used to evaluate the function repeatedly.
  • The point data 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 by discretizing the observation region and assumes constant point intensity.
  • 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 empty space function at a given distance:

In[1]:=1
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https://wolfram.com/xid/0d6gcrzw5m-7xwnvr
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-4wykf
Out[3]=3

Estimate the empty space function within a range of distances:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-ygbnac
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-bmyu9a
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-ee0oxv

Visualize the result with ListPlot:

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-ef5jdl
Out[4]=4

Empty space function of a cluster point process:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-yksgj0
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-daj61m
Out[2]=2

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

Point Data  (4)

Estimate the empty space function for some data at distance 0.1:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-e2gx6r
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-bf4bxp
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-kyzwq
Out[4]=4

Create a PointStatisticFunction for future use:

In[1]:=1
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https://wolfram.com/xid/0d6gcrzw5m-js3gk8
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-odkmbl
Out[2]=2

Compute a value at a given radius:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-vx5qgr
Out[3]=3

Estimate the empty space function without explicitly providing the observation region:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-f7t1ko
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-7rwaj3
Out[2]=2

Observation region generated by Ripley–Rasson estimator:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-hc1hyh
Out[3]=3

Estimated function at distance 0.05:

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-h12d14
Out[4]=4

Use EmptySpaceF with GeoPosition:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-1vo729
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-xpkh6o
Out[2]=2

Plot the point statistics function:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-7m1ksc
Out[3]=3

Point Processes  (3)

The empty space function for PoissonPointProcess has closed form:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-3hltk1
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-m6387j
Out[2]=2

Visualize for fixed intensity and varying dimension:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-8g4nyq
Out[3]=3

The empty space function for a cluster process ThomasPointProcess with specified dimension:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-7ib6aq
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-qpvl0c
Out[2]=2

In 3D:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-sehhpq
In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-kghzua
Out[4]=4

The empty space function for a cluster process MaternPointProcess with specified dimension:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-oi9wov
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-ofqh4k
Out[2]=2

In 3D:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-yqw7j7
In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-1q0vyt
Out[4]=4

Options  (3)Common values & functionality for each option

SpatialBoundaryCorrection  (2)

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

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-1aqzq
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-88aqlu
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-eholvz
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0d6gcrzw5m-dlt8hb
Out[5]=5

"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
✖
https://wolfram.com/xid/0d6gcrzw5m-g51708
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0d6gcrzw5m-e9hv8s
Out[7]=7

Compare the different edge correction methods:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-cse7si

Estimate the values of the empty space function with three different methods:

In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-ihhmaw
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-lflu49

Visualize the results:

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-8edxf9
In[5]:=5
✖
https://wolfram.com/xid/0d6gcrzw5m-pw5mgr
Out[5]=5

Method  (1)

Discretization setting can be provided under Method as suboptions:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-uj3zyb

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

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-cpbs9d
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-kd4lvk
Out[4]=4

Applications  (3)Sample problems that can be solved with this function

The empty space function for spatially random data:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-3j6zid

Visualize the results:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-e4j1vb
In[5]:=5
✖
https://wolfram.com/xid/0d6gcrzw5m-9eo6ef
Out[5]=5

The empty space function for hardcore data:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-887yj9

Estimate the values of the empty space function with given data:

In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-qpbqmc

Visualize the results:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-p2nje6
Out[3]=3

On Monday, May 20, 2019, the Severe Prediction Center highlighted a small corridor from the northeastern part of the Texas Panhandle to central Oklahoma in a 45 percent probability of EF2–EF5 tornadoes to occur within 25 miles of a given location, which defines a value of EmptySpaceF:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-q64sed

Assuming uniform intensity over the region, use the Poisson point process as a tornado pattern model:

In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-q8tcl2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-d56no8
Out[3]=3

Define the Poisson point process:

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-d3ujrz
Out[4]=4

Compute probabilities of a tornado within a given radius of a location:

In[5]:=5
✖
https://wolfram.com/xid/0d6gcrzw5m-rilcg6
Out[5]=5

Simulate a possible tornado spread over the state of Oklahoma:

In[6]:=6
✖
https://wolfram.com/xid/0d6gcrzw5m-8lp3a2
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0d6gcrzw5m-f3mm3b
Out[7]=7

Properties & Relations  (4)Properties of the function, and connections to other functions

The empty space function behaves like a CDF:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-5a1gl8

Visualize the results:

In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-i96592
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-usfklo
Out[3]=3

The empty space function for a PoissonPointProcess is equivalent to SurvivalFunction at 0 of PointCountDistribution on a disk of radius :

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-nh7u6b
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-kdmuk4
Out[2]=2

Define the point count distribution for the process on a ball of radius :

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-02en7s

Compare its survival function at 0 with the empty space function for fixed dimensions:

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-waz3gk
Out[4]=4

In 3D:

In[5]:=5
✖
https://wolfram.com/xid/0d6gcrzw5m-h5d2e0
Out[5]=5

In random dimension:

In[6]:=6
✖
https://wolfram.com/xid/0d6gcrzw5m-bjwdco
Out[6]=6

The empty space and the nearest neighbor functions of a PoissonPointProcess are identical:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-ezl23u
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-peicch
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-s1mw7o
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-iuhnsy
Out[4]=4

In 1D they are both equivalent to the CDF of an ExponentialDistribution:

In[5]:=5
✖
https://wolfram.com/xid/0d6gcrzw5m-tk0pll
In[6]:=6
✖
https://wolfram.com/xid/0d6gcrzw5m-ynkq7r
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0d6gcrzw5m-9i3ozc
Out[7]=7

EmptySpaceF is often compared with NearestNeighborG, which estimates the probability of finding another point within distance r from a point in the point collection:

In[1]:=1
✖
https://wolfram.com/xid/0d6gcrzw5m-eqvqul
In[2]:=2
✖
https://wolfram.com/xid/0d6gcrzw5m-o9rtf6

Visualize the result with ListPlot:

In[3]:=3
✖
https://wolfram.com/xid/0d6gcrzw5m-pj4yjt
Out[3]=3

Compare the estimates between EmptySpaceF and NearestNeighborG for point data generated by HardcorePointProcess:

In[4]:=4
✖
https://wolfram.com/xid/0d6gcrzw5m-wwblhc
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0d6gcrzw5m-h8iraa
In[6]:=6
✖
https://wolfram.com/xid/0d6gcrzw5m-9axykf
Out[6]=6
Wolfram Research (2020), EmptySpaceF, Wolfram Language function, https://reference.wolfram.com/language/ref/EmptySpaceF.html.
✖
Wolfram Research (2020), EmptySpaceF, Wolfram Language function, https://reference.wolfram.com/language/ref/EmptySpaceF.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_emptyspacef, author="Wolfram Research", title="{EmptySpaceF}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/EmptySpaceF.html}", note=[Accessed: 07-June-2025 ]}

✖
@misc{reference.wolfram_2025_emptyspacef, author="Wolfram Research", title="{EmptySpaceF}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/EmptySpaceF.html}", note=[Accessed: 07-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_emptyspacef, organization={Wolfram Research}, title={EmptySpaceF}, year={2020}, url={https://reference.wolfram.com/language/ref/EmptySpaceF.html}, note=[Accessed: 07-June-2025 ]}

✖
@online{reference.wolfram_2025_emptyspacef, organization={Wolfram Research}, title={EmptySpaceF}, year={2020}, url={https://reference.wolfram.com/language/ref/EmptySpaceF.html}, note=[Accessed: 07-June-2025 ]}