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BinnedVariogramList
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BinnedVariogramList

[Experimental]

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BinnedVariogramList[{loc1val1,loc2val2,…}]

computes a variogram using binned values.

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BinnedVariogramList[{loc1,loc2,…}{val1,val2,…}]

generates the same result.

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allows binning spec to be specified as given in HistogramList.

Details and Options

  • BinnedVariogramList is also known as empirical variogram or sample variogram.
  • BinnedVariogramList is typically used to get an initial assessment of the spatial data dependence in data. It is also used as a first stage in estimating high-quality EstimatedVariogramModel.
  • The variogram for a spatial process at locations and is given by . It is a measure of how quickly the process varies spatially.
  • When a process is weakly stationary, then the variogram depends only the difference of locations, i.e. . And when the process is isotropic, it only depends on the distance between locations where h=TemplateBox[{{{p, _, 1}, -, {p, _, 2}}}, Norm].
  • The value of for is computed as 1/(2 TemplateBox[{S}, Abs])sum_({p,q} in S)(x(p)-x(q))^2, where S={{p,q}|h-delta<=TemplateBox[{{p, -, q}}, Norm]<h+delta}. The result is a binned variogram:
  • The resulting binned variogram is typically not a valid variogram. It needs to be conditionally negative definite sum_(i=1)^nsum_(j=1)^nw_i w_j gamma(TemplateBox[{{{p, _, i}, -, {p, _, j}}}, Norm])<=0 for all weights wi such that and locations pi. However, it can be used to fit a variogram model that will be a valid variogram, as is done in EstimatedVariogramModel.
  • From the binned variogram, one can detect whether there is a trend in the data, which will result in an unbounded variogram.
  • The following options can be given:
  • DistanceFunction Automaticspecify the function to compute distance
    Method "NonLattice"specify the shape of the locations for binning
    SpatialTrendFunction Nonespecify the global trend model
  • The following settings can be used for Method:
  • "NonLattice" locations are given as a collection
  • The following Method options can be used:
  • "BinCenter""Centroid"how to compute bin centers
    "MaxDistanceRatio"1/3data pairs for which the ratio of their distance to the max pair distance is greater are dropped
    "MinPairs"30bins containing fewer pairs are dropped
    "ScaleEstimator""Cressie"which scale estimator to use
  • The following settings can be used for "ScaleEstimator":
  • "Cressie"use the fourth moment of square root
    "Matheron"use the second moment
    "Qn"use QnDispersion
    "Sn"use SnDispersion
  • BinnedVariogram returns two-dimensional WeightedData, with weights being the number of pairs for each distance bin.

Examples

open allclose all

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

Compute BinnedVariogramList from data:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-kzrafo
In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-2wu3l0
Out[3]=3

Compute binned variogram:

In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-42710s
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-k30hr9
Out[5]=5

Compute BinnedVariogramList from geo data:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-n28g7w
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-6nnfof
Out[2]=2

Specify values via annotation key:

In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-njr5ys
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-b6spmm
Out[4]=4

Scope  (3)Survey of the scope of standard use cases

Basic Uses  (3)

Compute the binned variogram for random locations:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-ygctlt
In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-e4om3q
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-zwssuw
Out[4]=4

Visualize the binned variogram with weights being the number of pairs for each distance bin:

In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-q6m0c1
Out[5]=5

Use HistogramList specs for binning:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-rtchvv

Plot the data:

In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-d652ab
Out[2]=2

Compute binned variogram with Automatic bin specification:

In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-1bg6xn

Use fixed number of bins:

In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-qnewlo

Use bins of given width:

In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-xgzc86

Use a named binning method (Scott is default):

In[6]:=6
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https://wolfram.com/xid/0dbk58a0yb8dqducai-whu5ai

Plot all binned variograms:

In[7]:=7
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https://wolfram.com/xid/0dbk58a0yb8dqducai-51f4g5
Out[7]=7

Compute BinnedVariogramList for geographical data:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-5256rf
In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-6y0v91
In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-f7f4pn
Out[3]=3


Binned variogram:

In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-lox6sw
Out[4]=4

Visualize:

In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-uckt2d
Out[5]=5

Options  (7)Common values & functionality for each option

DistanceFunction  (1)

For non-geographical locations, different DistanceFunction can be used:

In[8]:=8
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https://wolfram.com/xid/0dbk58a0yb8dqducai-5udbai
In[10]:=10
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https://wolfram.com/xid/0dbk58a0yb8dqducai-fzvmxc
Out[10]=10

EuclideanDistance is the default distance function for Cartesian coordinates:

In[11]:=11
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https://wolfram.com/xid/0dbk58a0yb8dqducai-kgb35g
Out[11]=11

Use the p-norm directly:

In[12]:=12
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https://wolfram.com/xid/0dbk58a0yb8dqducai-cainfi
Out[12]=12
In[24]:=24
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https://wolfram.com/xid/0dbk58a0yb8dqducai-1xhjxh
Out[24]=24

Method  (5)

Method  (1)

The default method "NonLattice" is designed to work for lists of locations and values:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-7c8k4k
In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-t5nt2d
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-o9jq3u
Out[3]=3

BinCenter  (1)

Compute BinnedVariogramList for various "BinCenter" specifications:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-h3hbh6
In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-ocy4r6
Out[2]=2

Specify "BinCenter" settings:

In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-ghygb2
In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-iatwtq
Out[4]=4

Visualize:

In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-36x1j8
Out[5]=5

MaxDistanceRatio  (1)

Compute BinnedVariogramList for various "MaxDistanceRatio" specifications:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-szk7ex
In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-1n3829
Out[2]=2

Specify "MaxDistanceRatio" settings:

In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-60jff
In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-in0v2n
Out[4]=4

Visualize:

In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-83igi8
Out[5]=5

MinPairs  (1)

Compute BinnedVariogramList for various "MinPairs" specifications:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-x8i69i
In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-r1jwuc
Out[2]=2

Specify "MinPairs" settings:

In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-m0tyn3
In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-z7xrmc
Out[4]=4

Visualize:

In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-dh6dbz
Out[5]=5

ScaleEstimator  (1)

Compute BinnedVariogramList for various "ScaleEstimator" specifications:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-wh2n1e
In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-qe6va7
Out[2]=2

Specify "ScaleEstimator" settings:

In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-0njujv
In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-gagxel
Out[4]=4

Visualize:

In[5]:=5
✖
https://wolfram.com/xid/0dbk58a0yb8dqducai-20r1nc
Out[5]=5

SpatialTrendFunction  (1)

By default, BinnedVariogramList assumes no trend, but the data can be automatically detrended:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-cj3fud
In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-xl8hkl
Out[4]=4

Specify trend settings using SpatialTrendFunction and compute binned variogram:

In[8]:=8
✖
https://wolfram.com/xid/0dbk58a0yb8dqducai-4ongt6
Out[9]=9

The plot shows that data has a trend of at least first order:

In[10]:=10
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https://wolfram.com/xid/0dbk58a0yb8dqducai-nv1tqm
Out[10]=10

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

Binned variogram can be used to get initial visual shape idea for EstimatedVariogramModel:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-bhwow0
Out[1]=1

Compute binned variogram for values specified by the annotation key:

In[2]:=2
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https://wolfram.com/xid/0dbk58a0yb8dqducai-162664
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dbk58a0yb8dqducai-82vfii
Out[3]=3

Fit a few models with slow initial variation:

In[7]:=7
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https://wolfram.com/xid/0dbk58a0yb8dqducai-6xlag2
In[8]:=8
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https://wolfram.com/xid/0dbk58a0yb8dqducai-kn5g8c
In[9]:=9
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https://wolfram.com/xid/0dbk58a0yb8dqducai-0q6xl
Out[9]=9

BinnedVariogramList can be used to indicate the presence of trend in the data:

In[1]:=1
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https://wolfram.com/xid/0dbk58a0yb8dqducai-o3tkam
In[2]:=2
✖
https://wolfram.com/xid/0dbk58a0yb8dqducai-rdzywb
Out[2]=2

Compute binned variogram with no trend specification:

In[3]:=3
✖
https://wolfram.com/xid/0dbk58a0yb8dqducai-z7c1dd
Out[3]=3

The plot shows that data has a trend:

In[4]:=4
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https://wolfram.com/xid/0dbk58a0yb8dqducai-b0p83y
Out[4]=4

Compute binned variogram with linear trend:

In[5]:=5
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https://wolfram.com/xid/0dbk58a0yb8dqducai-on72xc
Out[5]=5

Compare the linearly detrended binned variogram with the default:

In[6]:=6
✖
https://wolfram.com/xid/0dbk58a0yb8dqducai-ikk7ya
Out[6]=6

Possible Issues  (1)Common pitfalls and unexpected behavior

BinnedVariogramList will fail if there is not enough data to meet the minimum number of pairs per bin requirement:

In[17]:=17
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https://wolfram.com/xid/0dbk58a0yb8dqducai-tmbe81
In[19]:=19
✖
https://wolfram.com/xid/0dbk58a0yb8dqducai-z1bxqr
Out[19]=19
Wolfram Research (2021), BinnedVariogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/BinnedVariogramList.html.
✖
Wolfram Research (2021), BinnedVariogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/BinnedVariogramList.html.

Text

Wolfram Research (2021), BinnedVariogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/BinnedVariogramList.html.

✖
Wolfram Research (2021), BinnedVariogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/BinnedVariogramList.html.

CMS

Wolfram Language. 2021. "BinnedVariogramList." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BinnedVariogramList.html.

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Wolfram Language. 2021. "BinnedVariogramList." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BinnedVariogramList.html.

APA

Wolfram Language. (2021). BinnedVariogramList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BinnedVariogramList.html

✖
Wolfram Language. (2021). BinnedVariogramList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BinnedVariogramList.html

BibTeX

@misc{reference.wolfram_2025_binnedvariogramlist, author="Wolfram Research", title="{BinnedVariogramList}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/BinnedVariogramList.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_binnedvariogramlist, author="Wolfram Research", title="{BinnedVariogramList}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/BinnedVariogramList.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_binnedvariogramlist, organization={Wolfram Research}, title={BinnedVariogramList}, year={2021}, url={https://reference.wolfram.com/language/ref/BinnedVariogramList.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_binnedvariogramlist, organization={Wolfram Research}, title={BinnedVariogramList}, year={2021}, url={https://reference.wolfram.com/language/ref/BinnedVariogramList.html}, note=[Accessed: 26-March-2025 ]}