[R-sig-Geo] kriging

Fri Jul 4 13:57:25 CEST 2008

Dear Frede

I would not say that the methods discussed in the MBG book should be
refered as "regression kriging".

As far as I understand, the term "regression kriging", despite of all
variations, refer to proposals/algorithms for which predictions are
obtained combining two stages:
1. get predictions by some method (linear model, GLM, GAM, trees, etc)
   without accounting for the spatial correlations by a stochastic
   mechanism, without assuming a random field
2. either use or combine the above with some sort of kriging, i.e.,
   spatial predictions assuming and using the spatial covariance structure
This brings issue on how to assess prediction variances and so on.

The algorithms in geoR and geoRglm **does not** follow
this kind of two stage route.
**Given the assumed model**, the inference and prediction
follows from it and therefore likelihood based methods
(including Bayesian) can be adopted.

This does come with some price ---
As Edzer already pointed, for very large number of data points
this can be prohibitive for dealing with large covariance matrices
and this is a limitation of geoR/geoRglm.
There are possible workarounds still within the same paradigma
but not (yet?) implemented in such packages

best
P.J.

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

On Fri, 4 Jul 2008, Frede Aakmann Tøgersen wrote:

> 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
>
>
>
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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
> >>
> >>
> >>
> >>
> >>
> >
> > _______________________________________________
> > R-sig-Geo mailing list
> > R-sig-Geo at stat.math.ethz.ch
> > https://stat.ethz.ch/mailman/listinfo/r-sig-geo
> >
> >
> >
> >       [[alternative HTML version deleted]]
> >
> > _______________________________________________
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> >
>
> --
> Edzer Pebesma
> Institute for Geoinformatics (IfGI)
> University of Münster
> http://ifgi.uni-muenster.de/
>
>
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