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All about Variogram

Nugget effect

Mikyoung Jun (Texas A&M) stat647 lecture 4 September 6, 2012 1 / 17

What is the variogram?

Let us consider a stationary (or isotropic) random field Z with acovariance function K .

Note Var{Z (s1) − Z (s2)} = 2K (0) − 2K (s1 − s2) = 2γ(s1 − s2)

Why do you think we are interested in this quantity?

2γ(·) is called a variogram and γ(·) is called a semivariogram

If Z is isotropic, then γ(h) = γ0(|h|)

Most of the variogram models available so far are isotropic

Just like covariance function, variogram model should be negativedefinite to be valid. Why?


Variogram models

Some isotropic variogram models (let h = |s1 − s2|):

(a) Spherical model (valid in Rd , d = 1, 2, 3):

γ(h) =

0, h = 0

c0 + cs{3

2(h/as) −

1

2(h/as)

3}, 0 < h ≤ as

c0 + cs , h ≥ as

(b) Wave (hole-effect) model (valid in Rd , d = 1, 2, 3):

γ(h) =

{

0, h = 0

c0 + cw{1 − aw sin(h/aw )/h}, h > 0

(c) How can you write down an exponential variogram model?


Some example pictures

0 200 400 600 800 1000

01

23

45

(a) Spherical model

distance

vario

gram

0 200 400 600 800 1000

01

23

4

(b) Hole−effect model


General shape of variogram

Nugget effect gives the variation of the process at a finer scale than thesmallest distance measured


Nugget effect

Formally we define nugget effect as a nonzero constant c0 such thatc0 = lim

h→0

γ(h)

Example: spherical model

γ(h) =

0, h = 0

c0 + cs{3

2(h/as) −

1

2(h/as)

3}, 0 < h ≤ as

c0 + cs , h ≥ as


How do we plot variogram?

Suppose you have the data points Z (s1), · · · ,Z (sn) and assumeisotropy for the data

First plot squared differences of the data for every possible pairsagainst the distances between the two locations

This plot is called a variogram cloud

Then you bin the distances and in each bean, you take average of thesquared differences

Sometimes you may look at the quantiles instead of the mean


Example

Variogram of TOMS ozone data:

0 500 1500 2500

050

0010

000

2000

0

distance

vario

gram

0 500 1500 2500

020

4060

8012

0

sqrt

(VG

)


Example

What are the blue lines?

Do you see a nugget effect?

Do you think the data is isotropic?

Notice the y-scale difference between the two figures


Variogram plots using R

In fields package, you can use vgram

In geoR package, you may use variog

See help pages for the above commands and related commands forexamples and other usages


How to use variogram pictures

First check if the process is isotropic

If it is, you can guess what variogram model should be appropriate,what are the parameter values (roughly), and whether there should bea nugget effect in the model or not

If not, you can plot variogram against the differences of somecoordinates

To check if the process is geometrically anisotropic or not, you canuse directional variograms

You can plot variograms for some parts of the data to checknonstationarity


Directional variogram

For some non-isotropic data, it is the case that the variograms showsignificant differences in different directions

Suppose Z is geometrically anisotropic in that Z (V x) is isotropic,

V =

(

4 00 1

)(

cos (π/4) sin (π/4)−sin(π/4) cos (π/4)

)

=

(

2.83 2.83−0.707 0.707

)

,

and x ∈ R2

Then we can imagine variograms for NE-SW direction should begreater than variograms for NW-SE direction at same distance lag


Directional variogram

A directional variogram of the process Z based on 900 observationson [0, 1] × [0, 1] (exponential covariance model)

Use “geoR” package in R (function “variog4”)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.4

0.8

1.2

distance

sem

ivar

ianc

e

0°45°90°135°


Variogram for nonstationary data

If you suspect certain nonstationarity in the data set, you may tryvariograms for subsets of the data

For global data set, if there is a dependence of covariance structure onlatitude, try variograms for each latitude levelsFor a dataset that covers land and sea, you may try variograms overthe land and the sea separatelyIf you look at air pollution data, you may try variograms for urban andrural areas separately. You may also try variograms over majorhighways etc separately

Don’t be fooled by some patterns that may well be due to randomvariation in the data

Also don’t be fooled by a variogram that may hide many differentpatterns


Variogram of TOMS ozone data

Variogram of TOMS ozone data for May 1, 1990

0 500 1500 2500

050

0010

000

2000

0

distance

vario

gram

0 500 1500 2500

020

4060

8012

0

sqrt

(VG

)


Variogram of TOMS ozone data

0 500 1500 2500

010

0020

0030

0040

00

latitude 60.5

0 500 1500 2500

010

0020

0030

0040

00

latitude −0.5

0 500 1500 2500

010

0020

0030

0040

00

latitude −40.5


Issues with variogram display

If the data are not isotropic...

if the process is stationary, we could display variograms against thespatial lagif the process is not stationary, it is not clear what is the best choice forthe x-axisfor TOMS data, we might display variograms against longitude lag ateach latitude levels

If we have space-time data, then the variogram should be displayed as3 dimensional (or higher dimensional) plot even if the process isisotropic in space


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