[R-sig-Geo] kriging

Fri Jul 4 09:08:29 CEST 2008

Sorry for dropping in late in this thread, which I have not followed closely. 

Perhaps Paulo Ribeiro can correct me but thinking of geostatistics in terms of statistical models then I think that the book by Diggle & Ribeiro:

http://www.springer.com/geosciences/computer+&+mathematical+applications/book/978-0-387-32907-9

would give some regular methods to do what some call regression kriging (and related methods) based on statistical models. I know this requires some distributional assumptions, but there are straight forward methods for Gaussian data as well as data with more general distribution as in "generalized linear models". Also moderate departure from the assumption of Gaussian data do not have that big effect on inferences in such models. The above reference is supporting the geoR package in R.

Med venlig hilsen / Regards

Frede Aakmann Tøgersen
Forsker / Scientist

 	 AARHUS UNIVERSITET / UNIVERSITY OF AARHUS	
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Fra: r-sig-geo-bounces at stat.math.ethz.ch på vegne af Edzer Pebesma
Sendt: to 03-07-2008 13:33
Til: Hengl, T.
Cc: r-sig-geo at stat.math.ethz.ch; Dave Depew
Emne: Re: [R-sig-Geo] kriging

Hengl, T. wrote:
> I agree with Paulo - gstat can work with any linear model including the transforms of the original predictors e.g.:
>
> Z ~ X + X^2 + Y + Y^2    etc.
>
> The problem is that gstat implements the so-called Kriging-with-external-trend algorithm to make predictions (see section 2.1 of my lecture notes), which is mathematically more elegant, but then it accepts only a family of linear models (and not GLMs, regreesion-trees etc.). I have been promoting the concept of regression-kriging (deterministic and stochastic predictions seperated), but we still did not implement it in any package so far.
>  
And I can see why, as there are quite a few problems still to solve
(afaik) ahead of you. When you cut the problem in two, do the regression
estimation and residual prediction in two separate processes (often
under different assumptions, e.g. wrt spatial correlation) you ignore
the correlation between the two. Finding a prediction variance by
naively adding the variances of the two components e.g. does not yield
zero variance at observation locations, because a non-zero correlation
is ignored. At other locations, this correlation is also non-zero.
Furthermore, if you cut the problem in two for e.g. binomial or Poisson
distributed cases, in this approach you likely end up with negative
predictions or predictions above one for the binomial case.

Does the paper you refer to (by yourself) give solutions to these two
problems?
> You can at any time separate the predictions (e.g. krige only the residuals), but then gstat will not give you the regression-kriging variance, and you can not run geostatistical simulations.
>  
No, of course not, for the reasons mentioned above. The gstat approach
is: if you want to make a mess, please take responsibility for it by
yourself (and don't blame me--through the package). There is a paper I
did it with count data, though, which is

E.J. Pebesma, R.N.M. Duin, P.A. Burrough, 2005. Mapping Sea Bird
Densities over the North Sea: Spatially Aggregated Estimates and
Temporal Changes. Environmetrics 16
<http://www3.interscience.wiley.com/cgi-bin/jissue/110577560>, (6), p
573-587 <http://dx.doi.org/10.1002/env.723>.

and (part of) the analysis is found in

library(gstat)
demo(fulmar)

I'm also confused by this term "regression kriging". Would you claim
that the universal kriging/kriging with (one or more) external drifts
implemented by gstat is not regression kriging? Are you actually working
on a package that does do regression kriging as you define it?
--
Edzer

> see also:
> https://stat.ethz.ch/pipermail/r-sig-geo/2008-February/003174.html
>
>
> All the best,
>
> Tom Hengl
> http://spatial-analyst.net <http://spatial-analyst.net/> 
>
> Hengl, T., 2007. A Practical Guide to Geostatistical Mapping of
> Environmental Variables. EUR 22904 EN Scientific and Technical Research
> series, Office for Official Publications of the European Communities,
> Luxemburg, 143 pp.
> http://bookshop.europa.eu/uri?target=EUB:NOTICE:LBNA22904:EN:HTML
>
>
> -----Original Message-----
> From: r-sig-geo-bounces at stat.math.ethz.ch on behalf of Dave Depew
> Sent: Mon 6/16/2008 10:54 PM
> To: Paulo Justiniano Ribeiro Jr
> Cc: r-sig-geo at stat.math.ethz.ch
> Subject: Re: [R-sig-Geo] kriging
> 
> Ok,
> What about higher order polynomials? I have fitted one using a gam to
> the data which which helps to normalize the residuals, and reduce the
> variance of the residuals.
> Is it simply a matter of plugging in the function into the gstat command
> line? Or is it simpler to krig the residuals and then add the trend back
> to the interpolated residual grid?
>
>
> Paulo Justiniano Ribeiro Jr wrote:
>  
>> Dave,
>>
>> what is necessary for UK is a relation expressed by a linear model, not
>> necessaraly a linear relation between the variables.
>> e.g. you could have a second degree polinomial and still work within the
>> scope of universal kriging.
>>
>>
>> On Mon, 16 Jun 2008, Dave Depew wrote:
>>
>>  
>>    
>>> Hi all,
>>> I have a data set that I would like to krige to interpolate between
>>> transects. There is a non-linear trend between two of the variables...my
>>> impression from reading the gstat help file is that there must be a
>>> linear relationship between the data to use universal kriging?
>>> Second, would a method of non-linear regression followed by modelling of
>>> the residuals with a semivariogram be an appropriate solution?
>>>
>>> Thanks,
>>>
>>> Dave
>>>
>>> _______________________________________________
>>> R-sig-Geo mailing list
>>> R-sig-Geo at stat.math.ethz.ch
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>>>
>>>    
>>>      
>> Paulo Justiniano Ribeiro Jr
>> LEG (Laboratorio de Estatistica e Geoinformacao)
>> Universidade Federal do Parana
>> Caixa Postal 19.081
>> CEP 81.531-990
>> Curitiba, PR  -  Brasil
>> Tel: (+55) 41 3361 3573
>> Fax: (+55) 41 3361 3141
>> e-mail: paulojus AT  ufpr  br
>> http://www.leg.ufpr.br/~paulojus
>>
>>
>>
>>
>>    
>
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--
Edzer Pebesma
Institute for Geoinformatics (IfGI)
University of Münster
http://ifgi.uni-muenster.de/

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