BinnedVariogramList

BinnedVariogramList[{loc1val1,loc2val2,}]

computes a variogram using binned values.

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

generates the same result.

BinnedVariogramList[,spec]

allows binning spec to be specified as given in HistogramList.

Details

  • 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:
  • DistanceFunctionAutomaticspecify the function to compute distance
    Method"NonLattice"specify the shape of the locations for binning
    SpatialTrendFunctionNonespecify 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

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Basic Examples  (2)

Compute BinnedVariogramList from data:

Compute binned variogram:

Compute BinnedVariogramList from geo data:

Specify values via annotation key:

Scope  (3)

Basic Uses  (3)

Compute the binned variogram for random locations:

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

Use HistogramList specs for binning:

Plot the data:

Compute binned variogram with Automatic bin specification:

Use fixed number of bins:

Use bins of given width:

Use a named binning method (Scott is default):

Plot all binned variograms:

Compute BinnedVariogramList for geographical data:


Binned variogram:

Visualize:

Options  (7)

DistanceFunction  (1)

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

EuclideanDistance is the default distance function for Cartesian coordinates:

Use the p-norm directly:

Method  (5)

Method  (1)

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

BinCenter  (1)

Compute BinnedVariogramList for various "BinCenter" specifications:

Specify "BinCenter" settings:

Visualize:

MaxDistanceRatio  (1)

Compute BinnedVariogramList for various "MaxDistanceRatio" specifications:

Specify "MaxDistanceRatio" settings:

Visualize:

MinPairs  (1)

Compute BinnedVariogramList for various "MinPairs" specifications:

Specify "MinPairs" settings:

Visualize:

ScaleEstimator  (1)

Compute BinnedVariogramList for various "ScaleEstimator" specifications:

Specify "ScaleEstimator" settings:

Visualize:

SpatialTrendFunction  (1)

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

Specify trend settings using SpatialTrendFunction and compute binned variogram:

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

Applications  (2)

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

Compute binned variogram for values specified by the annotation key:

Fit a few models with slow initial variation:

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

Compute binned variogram with no trend specification:

The plot shows that data has a trend:

Compute binned variogram with linear trend:

Compare the linearly detrended binned variogram with the default:

Possible Issues  (1)

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

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.

CMS

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

BibTeX

@misc{reference.wolfram_2022_binnedvariogramlist, author="Wolfram Research", title="{BinnedVariogramList}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/BinnedVariogramList.html}", note=[Accessed: 03-December-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_binnedvariogramlist, organization={Wolfram Research}, title={BinnedVariogramList}, year={2021}, url={https://reference.wolfram.com/language/ref/BinnedVariogramList.html}, note=[Accessed: 03-December-2022 ]}