[R-sig-Geo] question about regression kriging

[David Maxwell (Cefas)]

Hi,
Both Edzer's example (extreme case when prediction is at observation locations) and Tom's tech report (pg8 "note that the Eq.(19) looks very much like (17), except it will give slightly lower values") suggest that assuming independence of the two variances will give values that are too large, if so this is useful to know.

For the follow-up question: how to present the prediction uncertainty? I would follow the usual approach for a binary glm, calculate a confidence interval on the logit scale, then back-transform the limits to the 0,1 scale. If space to present mapped outputs is limited I plan to calculate the width of the confidence interval on the 0,1 scale and map this. 

Thanks again, this list is an excellent catalyst for learning
David
david.maxwell at cefas.co.uk

-----Original Message-----
From: Edzer Pebesma [mailto:edzer.pebesma at uni-muenster.de]
Sent: 09 April 2008 09:49
To: Tomislav Hengl
Cc: David Maxwell (Cefas); r-sig-geo at stat.math.ethz.ch
Subject: Re: [R-sig-Geo] question about regression kriging


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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