# 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 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 .
• The value of for is computed as , where . The result is a binned variogram:
• • The resulting binned variogram is typically not a valid variogram. It needs to be conditionally negative definite 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 Automatic specify the function to compute distance Method "NonLattice" specify the shape of the locations for binning SpatialTrendFunction None specify 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" 1/3 data pairs for which the ratio of their distance to the max pair distance is greater are dropped "MinPairs" 30 bins 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)

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: 