[R-sig-Geo] question about regression kriging

[Tomislav Hengl]

The two components of the regression-kriging model are not independent, hence you are doing a wrong
thing if you are just summing them. You should use instead the universal kriging variance that is
derived in gstat. The complete derivation of the Universal kriging variance is available in Cressie
(1993; p.154), or even better Papritz and Stein (1999; p.94). See also pages 7-8 of our technical
note:

Hengl T., Heuvelink G.B.M. and Stein A., 2003. Comparison of kriging with external drift and
regression-kriging. Technical report, International Institute for Geo-information Science and Earth
Observation (ITC), Enschede, pp. 18.
http://www.itc.nl/library/Papers_2003/misca/hengl_comparison.pdf

Edzer is right, you can not back-transform prediction variance of the transformed variable (logits).
However, you can standardize/normalize the UK variance by diving it with global variance (see e.g.
http://dx.doi.org/10.1016/j.geoderma.2003.08.018), so that you can evaluate the success of
prediction in relative terms (see also http://spatial-analyst.net/visualization.php).


Tom Hengl
http://spatial-analyst.net 


-----Original Message-----
From: r-sig-geo-bounces at stat.math.ethz.ch [mailto:r-sig-geo-bounces at stat.math.ethz.ch] On Behalf Of
Edzer Pebesma
Sent: dinsdag 8 april 2008 20:50
To: David Maxwell (Cefas)
Cc: r-sig-geo at stat.math.ethz.ch
Subject: Re: [R-sig-Geo] question about regression kriging

David Maxwell (Cefas) wrote:
> Hi,
>
> Tom and Thierry, Thank you for your advice, the lecture notes are very useful. We will try geoRglm
but for now regression kriging using the working residuals gives sensible answers even though there
are some issues with using working residuals, i.e. not Normally distributed, occasional very large
values and inv.logit(prediction type="link" + working residual) doesn't quite give the observed
values.
>
> Our final question about this is how to estimate standard errors for the regression kriging
predictions of the binary variable?
>
> On the logit scale we are using
>  rk prediction (s0) = glm prediction (s0) + kriged residual prediction (s0) 
> for location s0
>
> Is assuming independence of the two components adequate?
>  var rk(s0) ~= var glm prediction (s0) + var kriged residual prediction (s0) 
>   
In principle, no. The extreme case is prediction at observation 
locations, where the correlation is -1 so that the final prediction 
variance becomes zero. I never got to looking how large the correlation 
is otherwise, but that shouldn't be hard to do in the linear case, as 
you can get the first and second separately, and also the combined using 
universal kriging.

Another question: how do you transform this variance back to the 
observation scale?
--
Edzer

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