[R-sig-Geo] Variogram Paradox

[Edzer Pebesma]

On 04/08/2013 08:12 PM, Saman Monfared wrote:
> Dear All,
> We know that the estimation of covariance parameters is an important
> problem for spatial processes because the variogram shows the spatial
> variation. In many cases to select the best variogram model some
> parametric models considered and some criterions such as mean
> prediction error, mean square error, correlation between the observed
> and predicted values and correlation between the predicted and the
> residual values in cross validation method uses to select the best
> variogram model.
> 
> Below codes get an example whit two variogram models which are have
> very different parameters (sill, range and nugget) but values of
> mentioned criterions are approximately equal for them.
> Why?
> What is the role of variogram?
> What is the role of empirical variogram when a variogram function
> which is so far away than it can has approximately equaled cross
> validation results.
> 
> library(gstat)
> data(meuse)
> coordinates(meuse)<-~x+y
> v<-variogram(log(zinc)~1,meuse)
> v.f<-fit.variogram(v,vgm(.205,"Mat",700,0.008,kappa=1))
> plot(v,v.f)
> v.ff<-fit.variogram(v,vgm(.205,"Mat",700,0.008,kappa=1)
> ,fit.sills =F, fit.ranges =F)
> plot(v,v.ff)
> k1<-krige.cv(log(zinc)~1,meuse,v.f)
> k2<-krige.cv(log(zinc)~1,meuse,v.ff)
> mean(k1$residual)
> mean(k2$residual)
> mean(k1$residual^2)
> mean(k2$residual^2)
> cor(k1$var1.pred,k1$observed)
> cor(k2$var1.pred,k2$observed)
> cor(k1$var1.pred,k1$residual)
> cor(k2$var1.pred,k2$residual)
> 
> Best,
> Saman.

Because the ratio of the two models,

v1 = variogramLine(v.f, 500)
v2 = variogramLine(v.ff, 500)
plot(v1[,1], v1[,2]/v2[,2])

is fairly constant.

If a variogram model gets multiplied by a positive constant, the kriging
predictions will remain identical:

v.fff = v.ff
v.ff$psill = v.ff$psill * 1234 # or any pos. number
k3 <- krige.cv(log(zinc)~1,meuse,v.fff)
summary(k3$var1.pred-k2$var1.pred)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
      0       0       0       0       0       0

-- 
Edzer Pebesma
Institute for Geoinformatics (ifgi), University of Münster
Weseler Straße 253, 48151 Münster, Germany. Phone: +49 251
8333081, Fax: +49 251 8339763  http://ifgi.uni-muenster.de
http://www.52north.org/geostatistics      e.pebesma at wwu.de