[R-sig-Geo] gstat::krige() - regression kriging vs. kriging with external drift

[Tomislav Hengl]

I usually use "Kriging with External Drift (KED)" and "Universal  
kriging" as synonyms. I think "Universal kriging" is the oldest term  
(Matheron, 1963), but then 'universal' can seem a bit vague.

Regression-kriging (RK) is when you krige the residuals and add them  
to the trend (hence predict 'm' and 'e' separately). Edzer (gstat)  
implements mainly KED/UK, but then you need the regression residuals  
to estimate the variogram (I guess getting the residuals before you  
fit variogram is unavoidable). Both RK and KED/UK are equivalent i.e.  
give exactly the same predictions under same input settings  
(http://dx.doi.org/10.1016/j.cageo.2007.05.001), so this is where many  
get confused (two paths to the same destination).

The code you have sent is definitively RK.

The real difference between RK and KED is that with RK you can use  
different modelling techniques including random forest and similar. As  
long as the residuals are normal, you can then add interpolated  
residuals back to the trend.

HTH,

T. Hengl

Quoting Edzer Pebesma <edzer.pebesma at uni-muenster.de>:

> On 14/11/15 17:32, Bede-Fazekas Ákos wrote:
>> Dear List, dear Edzer,
>> is it correct if I use the term "regression kriging" when I run this
>> function?:
>> kriged_value <- gstat::krige(z ~ x + y, [...])@data$var1.pred
>>
>> Or should I call it "kriging with external drift" (or "universal kriging" if
>> x and y are coordinates), and use the term "regression kriging" only in the
>> case of running this?:
>> linear_model <- lm(z ~ x + y, [...])
>> residuals <- linear_model$residuals
>> kriged_residuals <- gstat::krige(residuals ~ 1, [...])@data$var1.pred
>> kriged_value <- linear_model$fitted.values + kriged_residuals
>>
>
> For what it's worth, my opinion:
>
> I would call it universal kriging, in line with the software you're
> using (which I wrote):
>
>> x = krige(zinc~x+y, meuse, meuse.grid, m)
> [using universal kriging]
>
> Most of the (older) literature associates external drift kriging with a
> single external drift variable (e.g. Goovaerts), and universal kriging
> with using coordinates as covariates.
>
> The statistician might call everything best linear unbiased prediction
> under the the general linear model y = X beta + e, with a spatial
> covariance function describing the covariance of e, and in that
> perspective both methods are equivalent.
>
> I'm not the person to tell what regression kriging is and what it is
> not, but it might include the above models as well as those where trend
> fitting and residuals are done in different steps, and potentially under
> different assumptions for e.
>
> --
> Edzer Pebesma
> Institute for Geoinformatics (ifgi),  University of Münster,
> Heisenbergstraße 2, 48149 Münster, Germany; +49 251 83 33081
> Journal of Statistical Software:   http://www.jstatsoft.org/
> Computers & Geosciences:   http://elsevier.com/locate/cageo/
> Spatial Statistics Society http://www.spatialstatistics.info