Variograms

Modeling & interpolatingspatial dependencies

Ginger Allington - Analyses in R - 17 March 2010

Spatial patterns• Most ecological field studies are inherently

spatial, but this factor is not incorporated intoanalysis (e.g. anova)

• Standard statistics can miss important trendsin the data

• Spatial methods use the underlying spatialvariations to create better estimates ofdifferences among treatments or sites

(Scheiner & Gurevitch 2001)

Similar mean, but the patterns of spatial dependency are different

(Images from R. Barnes, Golden Software)

Variograms of the two datasets

semivariograms• A method to to characterize spatial variance /

quantify spatial dependencies in the data

• Semivariograms present the mean variance found incomparisons of samples of increasing lag intervals(distance).

• The semi-variogram is a function that relates semi-variance (or dissimilarity) of data points to thedistance that separates them.

Interpreting variogramsexpected curvefor randomdistribution

expected curvewhen samples showauto- correlationover a certain range(Ao)

The distance between the originand the sill is known as therange, and this represents thegeneral distance over which thesamples are autocorrelated.

We can only modellag distances to thesmallest distancebetween pairs ofsamples, andvariance that existsat an even smallerscale is representedby the nugget.

Interpreting variograms (con’t)

• When data are randomly distributed we can expect that there will belittle difference in the variance (γ)at any distance comparison.

• owever, when there is a pattern present in the distribution, we canexpect that variance will increase with comparisons of close,autocorrelated samples, but will level off to form a sill when samplesbecome independent.

The nugget:sill ratioindicates what percentof the overall varianceis found at a distancesmaller than thesmallest lag interval,and gives a sense ofhow much varianceyou have successfullyaccounted for in themodel.

Variogram components• Nugget variance: a non-zero value for γ when lag

distance (h) = 0. Produced by various sources ofunexplained error (e.g. measurement error).

• Sill: for large values of h the variogram levels out,indicating that there no longer is any correlationbetween data points. The sill should be equal to thevariance of the data set.

• Range: is the value of h where the sill occurs (or95% of the value of the sill).

• In general, 30 or more pairs per point are neededto generate a reasonable sample variogram.

• The most important part of a variogram is its shapenear the origin, as the closest points are givenmore weight in the interpolation process.

© Arthur J. Lembo, Jr.Cornell University

Variogram models

© Arthur J. Lembo, Jr.Cornell University

Spatial patterns acrosstreatments, or sites

Anisotropy

• Higher spatialautocorrelation in onedirection than in others

• Variation in continuity indifferent directions

• Can be accounted forby calculatingvariograms in differentdirections.

Interpolation via kriging