[R-sig-Geo] question about regression kriging

[Edzer Pebesma]

Tom,

I'm afraid things are harder than you sketch. In glm's, the parameter 
estimation is done using iteratively reweighted least squares, where the 
weights depend on a variance function that links the variance of 
observations to the mean. So, observations (residuals) are assumed to be 
unstationary, in principle, and because of the mean-dependency this 
changes over the iterations. The equations and references you mention 
afaik all assume a known, and fixed variogram, and one-step solutions, 
no iteration.

Also, you falsly accuse me of claiming one cannot back-transform 
prediction variances. I did not claim this (I have seen suggestions on 
how to do this), I just asked how David would do this.
--
Edzer

Tomislav Hengl wrote:
> 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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