WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

EstimatedVariogramModel
✖
EstimatedVariogramModel

[Experimental]

✖
EstimatedVariogramModel[{loc1val1,loc2val2,…}]

estimates the best variogram function from values vali given at locations loci.

✖
EstimatedVariogramModel[{loc1,loc2,…}{val1,val2,…}]

generates the same result.

✖

estimates the best parameters of the variogram function specified by "model".

✖
EstimatedVariogramModel[…,{"model",params}]

estimates the non-numeric parameters in params.

Details and Options

  • Variogram model is also known as variogram and semivariogram.
  • EstimatedVariogramModel fits a model to the spatial field data and returns a VariogramModel.
  • VariogramModel is typically used as a local model of spatial dependence when predicting values of a spatial field, as in SpatialEstimate.
  • 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 gamma(TemplateBox[{{{p, _, 1}, -, {p, _, 2}}}, Norm]).
  • Typical examples of spatial field data that is stationary and isotropic together with the corresponding variogram.
  • With automatic settings, this is all that is needed, but if you want to provide detailed control over the variogram model there are further aspects to understand. The two main issues are the range and level of smoothing.
  • The range for the variogram indicates how far the dependence on nearby points extends. A larger range variogram will correspond to a slower varying field.
  • The range of the variogram controls the distance at which points influence the range of prediction of values in SpatialEstimate.
  • The smoothness of the prediction is affected by the so-called spatial noise variance of the variogram, which is the value at the origin. It corresponds to adding a white noise model, e.g. measurement errors or true discontinuities in data such as a gold nugget.
  • The size of the spatial noise variance controls the level of smoothing of values in SpatialEstimate. In particular, with a nonzero noise variance, the resulting surface will not interpolate the given values but instead approximate them.
  • The locations loci can have the forms:
  • {p1,…,pd}geometrical locations
    GeoPosition[…],GeoPositionENU[…],…geographical locations
  • The values vali can have the forms:
  • ciscalar value
    Quantity[ci,"unit"]scalar quantity
  • The models need to satisfy consistency conditions to be a valid spatial variogram. It must be a non-negative function and satisfy the conditionally negative definite condition 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. The valid model families can be grouped into tables with similar features, and these are enumerated in VariogramModel.
  • The following options can be given:
  • SpatialNoiseLevel Automaticspecify the noise variance in the model
    SpatialTrendFunction Automaticspecify the global trend model

Examples

open allclose all

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

Estimate a variogram model for concentrations of zinc in soil:

In[8]:=8
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-tzfncx
In[10]:=10
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-vn8l03
Out[10]=10

Estimate exponential variogram model:

In[32]:=32
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-ppdaf0
Out[32]=32

Extract the sill and range information:

In[33]:=33
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-m69cmn
Out[33]=33

Value at a given distance:

In[34]:=34
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-wkwhat
Out[34]=34

Plot variogram function:

In[35]:=35
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-ps0ouk
Out[35]=35

Estimate a white noise variogram model for random data:

In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-xttwcl
In[2]:=2
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-fw7x7
Out[2]=2

Estimate white noise variogram model:

In[3]:=3
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-w2dpeu
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-xq9wod
Out[4]=4

Scope  (2)Survey of the scope of standard use cases

Estimate variogram model for geographical data:

In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-5256rf
In[2]:=2
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-6y0v91
In[3]:=3
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-f7f4pn
Out[3]=3

Fit a few models with slow initial variation:

In[4]:=4
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-6xlag2
In[8]:=8
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-kn5g8c
Out[8]=8

Visualize the fit of variogram models to the binned variogram of the data:

In[7]:=7
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-0q6xl
Out[7]=7

Estimate a variogram model from a binned variogram:

In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-kzrafo

Compute the binned variogram:

In[2]:=2
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-42710s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-xzu9jw
Out[3]=3

Estimate a variogram model:

In[4]:=4
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-4qmjoc
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-xixk99
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-kpzf5y

Options  (3)Common values & functionality for each option

SpatialNoiseLevel  (1)

Specify noise level using SpatialNoiseLevel:

In[92]:=92
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-o7k31g
In[95]:=95
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-ewguzi
Out[95]=95

Estimate variogram model assuming zero noise level:

In[96]:=96
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-z07f8
Out[96]=96
In[97]:=97
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-eew2c
Out[97]=97

Estimate variogram model assuming positive noise level:

In[98]:=98
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-gl2p8a
Out[98]=98
In[99]:=99
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-ziijr
Out[99]=99

Compare model parameters:

In[100]:=100
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-fyrtt7
Out[100]=100

SpatialTrendFunction  (2)

Specify trend function using SpatialTrendFunction:

In[49]:=49
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-t5zjxn
In[52]:=52
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-vyyzkh
Out[52]=52

Estimate variogram model assuming constant trend:

In[53]:=53
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-xobu7h
Out[53]=53
In[54]:=54
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-o694pc
Out[54]=54

Estimate variogram model assuming linear trend:

In[58]:=58
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-t6ndvn
Out[58]=58
In[59]:=59
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-9rnqat
Out[59]=59

Compare model parameters:

In[60]:=60
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-18i5z7
Out[60]=60

Look at data with a linear trend:

In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-d0o8d0

Without detrending, the variogram is well fit by an unbounded model:

In[2]:=2
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-gafs7w
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-f3qbul
Out[3]=3

Removing the large scale trend gives us a better idea of the micro scale variation:

In[4]:=4
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-d9f89c
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-ez7bwy
Out[5]=5

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

Estimate a variogram model for SpatialPointData:

In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-g8lm4f
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-82adb3
Out[2]=2

Compute the BinnedVariogramList to get an idea of which model family to choose:

In[3]:=3
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-59yrn0
Out[3]=3
In[6]:=6
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-ymtjrv
Out[6]=6

Fit a few asymptotic variogram models:

In[12]:=12
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-drcqhv
In[13]:=13
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-zyd2h3
Out[13]=13

Visualize the fit of the variogram models to the binned variogram of the data:

In[14]:=14
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-y1sb6i
Out[14]=14

Out of the four models, Matern fits best and could be used for spatial estimation:

In[15]:=15
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-pwy96y
Out[15]=15
In[17]:=17
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-rkbc8j
Out[17]=17

Use fit residuals to choose the best variogram model:

In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-s4l1w4
In[2]:=2
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-3mn2w0
Out[2]=2

Fit a few variogram models:

In[3]:=3
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-r25bbs
In[4]:=4
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-ll2n2s

Compute the weighted sum of residuals from the fit:

In[5]:=5
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-fe0xkt
Out[5]=5

Visualize the fit of variogram models to the binned variogram of the data:

In[6]:=6
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-irjnyn
Out[6]=6

Choose the model with the smallest weighted sum of residuals and use it to compute spatial estimate:

In[7]:=7
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-xo76hy
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-qj7w9t
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-144ve3
Out[9]=9

Possible Issues  (1)Common pitfalls and unexpected behavior

Some variogram models are badly suited for a given data:

In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-okolqr
In[1]:=1
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-foojel
Out[2]=2

Switch to a different model:

In[3]:=3
✖
https://wolfram.com/xid/0is8ehdmho3gvjhjscdlp-n67twa
Out[3]=3
Wolfram Research (2021), EstimatedVariogramModel, Wolfram Language function, https://reference.wolfram.com/language/ref/EstimatedVariogramModel.html.
✖
Wolfram Research (2021), EstimatedVariogramModel, Wolfram Language function, https://reference.wolfram.com/language/ref/EstimatedVariogramModel.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_estimatedvariogrammodel, author="Wolfram Research", title="{EstimatedVariogramModel}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/EstimatedVariogramModel.html}", note=[Accessed: 26-March-2025 ]}

✖
@misc{reference.wolfram_2025_estimatedvariogrammodel, author="Wolfram Research", title="{EstimatedVariogramModel}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/EstimatedVariogramModel.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_estimatedvariogrammodel, organization={Wolfram Research}, title={EstimatedVariogramModel}, year={2021}, url={https://reference.wolfram.com/language/ref/EstimatedVariogramModel.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_estimatedvariogrammodel, organization={Wolfram Research}, title={EstimatedVariogramModel}, year={2021}, url={https://reference.wolfram.com/language/ref/EstimatedVariogramModel.html}, note=[Accessed: 26-March-2025 ]}