SpatialBoundaryCorrection

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`SpatialBoundaryCorrection`

[Experimental]

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SpatialBoundaryCorrection

is an option to various spatial statistics functions that control how to correct for boundary effects of observation regions.

Details

SpatialBoundaryCorrection is also known as edge correction or border correction.
Boundary correction attempts to correct the bias introduced when spatial statistics is computed over a bounded region near the boundary.
A typical statistic will count the number points within distance r of a location. If that location is closer to the boundary than r, the resulting count will be too low and hence introduce a bias.
Possible general settings include:
Automatic automatically determined boundary correction

None no boundary correction

"BorderMargin" creates an interior boundary margin
The "BorderMargin" for a given radius will drop points closer to the boundary than .
Possible special setting include:

	"Ripley"	uses weights depending on the point distance to boundary
	"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

Different functions may document additional settings available for that particular function.

Examples

open allclose all

Basic Examples (2)Summary of the most common use cases

Use specific edge correction method:

In[1]:=1

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-cse7si

In[3]:=3

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-qbvr50

Out[3]=3

Turn off border correction:

In[1]:=1

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-u59h4x

In[2]:=2

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-8ylo1j

Out[2]=2

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

With no boundary correction, the estimator of RipleyK is biased and should not be used unless with a large point set:

In[1]:=1

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-ch7be2

In[3]:=3

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-ds1aaa

In[4]:=4

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-88aqlu

Out[4]=4

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

In[5]:=5

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-g1v0t0

Out[6]=6

In[7]:=7

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-eholvz

Out[7]=7

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

In[8]:=8

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-dlt8hb

Out[8]=8

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

In[1]:=1

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-murbz9

In[2]:=2

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-iu66tv

Out[2]=2

The uncorrected RipleyK is increasing:

In[3]:=3

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https://wolfram.com/xid/01k6gi6rc33zo1xgi-oy3eb9

Out[3]=3

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