types of Bayesian kriging, within a general framework, according to how

you specify the prior distributions of your model parameters, i.e. the

parameters describing your mean field as well as your covariance

structure. For example you can chose uniform priors for the parameters

describing your mean field, and assume that the parameters used to

characterise your covariance structure are well-known. This corresponds to

the classical universal or ordinary kriging model, depending on how you

specify your mean field.

From my point of view a highly relevant question regarding sample size

is: How many data points do we need before predictions, computed by

Bayesian kriging, are no longer influenced by our choice of

priors. And, how does this depend on the design ??

On Tue, 31 Aug 2004, Gregoire Dubois wrote:

> Hello everyone,

>

> I'm profiting from the discussion about Bayesian kriging to update my

> knowledge. Are there not various types of Bayesian kriging?

>

> I remember having applied in 1998 methodologies and codes (in C)

> developed in Klagenfurt, by the team of the Juergen Pilz (see

> http://www.math.uni-klu.ac.at/?language=en ). If I remember well, I have

> used functions like

>

> - Subjective Bayesian kriging (SBK) is a scenario that is between Simple

> Kriging (mean is known) and Ordinary kriging (mean unknown). In the case

> of SBK, one has some knowledge about the min and max values taken by the

> mean value of the variable that is analysed. In other words, the values

> of the mean values are constrained. Various scenarios were implemented

> in the code depending on the shape of the probability distribution

> function. For what concerns the kriging variance, the theory predicts a

> lower kriging variance for SBK only if the experimental semivariogram is

> the true one. A case study I did in my PhD was to improve estimations of

> radioactivity in Switzerland, using information provided by measurements

> made in a neighbouring country. Although the statistical distribution of

> these two datasets were very different but with similar mean values,

> this information could be efficiently used to improve to clearly reduce

> estimation errors. On the other hand, I often got a higher kriging

> variance with SBK than with OK.

>

> - Empirical Bayesian kriging (EBK): one has a much better knowledge of

> the pdf of the analysed dataset than in SBK. I did apply it to

> investigate two contaminated regions with similar distributions. Mean

> errors were lower for EBK than for Ordinary kriging. However, I also

> encountered many cases in which I got terrible results with EBK.

>

> Are other versions of Bayesian kriging not those with known

> semivariograms (Cui & Stein?) or those for which some knowledge about a

> number of parameters of the semivariogram is known, etc. Thus, going

> back to my first question, is there not a standard vocabulary that would

> allow readers to distinguish the type of prior knowledge used when one

> is talking about Bayesian kriging?

>

> For what concerns the number of points to be used etc... I don't

> understand the discussion. Should the correct question not be "how far

> does the number of samples used reflect the prior knowledge?".

>

> I hope I did not add too much confusion here :((

>

> Cheers,

>

> Gregoire

>

> PS: useful resources about the above described methods:

>

> Practically, the codes I used were written by Albrecht Gebhard( I think

> they are still available from his web site)and had a number of bugs at

> that time (in 1998-1999). The codes may have been updated since.

>

> For what concerns the mathematical developments, I used papers from

> Klagenfurt (all of them are in German, sorry). I enjoyed reading Pilz &

> Knospe (1997): Eine Anwendung des Bayes Kriging in der

> Lagerstaettentmodellierung. Glueckauf-Forschungshefte, 58(4): 670-677. I

> also recommend the master's thesis of Gerhard Buchacher: Bayes'sche und

> Empirisch Bayes'sche Methoden in der Geostatistik.

>

> More recent codes and papers should be available from Juergen Pilz's and

> Albrecht Gebhardt's homepages (again, see

> http://www.math.uni-klu.ac.at/?language=en )

>

> Hope this helps a bit.

>

> __________________________________________

> Gregoire Dubois (Ph.D.)

> JRC - European Commission

> IES - Emissions and Health Unit

> Radioactivity Environmental Monitoring group

> TP 441, Via Fermi 1

> 21020 Ispra (VA)

> ITALY

>

> Tel. +39 (0)332 78 6360

> Fax. +39 (0)332 78 5466

> Email: gregoire.dubois@...

> WWW: http://www.ai-geostats.org

> WWW: http://rem.jrc.cec.eu.int

>

> "The views expressed are purely those of the writer and may not in any

> circumstances be regarded as stating an official position of the

> European Commission."

>

>

>

>

>

> -----Original Message-----

> From: Soeren Nymand Lophaven [mailto:snl@...]

> Sent: 30 August 2004 22:13

> To: Edzer J. Pebesma

> Cc: Monica Palaseanu-Lovejoy; kai.zosseder@...;

> ai-geostats@...

> Subject: Re: [ai-geostats] extreme values

>

>

>

> Based on my relatively limited knowledge on Bayesian kriging I have a

> few comments to the current discussion:

>

> - Bayesian kriging gives better predictions than the classical approach

> if you have relatively few data points and at the same time is able to

> come up with good prior distributions for your model parameters.

>

> - The two approaches gives similar predictions if you have many data

> points.

>

> - The Bayesian approach always results in higher prediction variances,

> i.e. the classical kriging approach under estimates the prediction

> variances, because it is assumed that the parameters are known, which in

> practice they are not.

>

> - I chapter 2 in the reference below there is a figure showing

> predictions computed by the two approaches. Predictions were computed

> from a subset of the Swiss rainfall dataset (SIC97) consisting of 100

> data values. It is seen that the predictions are very close to being

> exactly equal. This means that if you are interested in prediction and

> have more than 100 data values it does not matter which approach you

> use. If you for some reason are interested in prediction variance, e.g.

> for comparing the efficiency of different designs, then Bayesian kriging

> gives you the best answer.

>

> Best regards / Venlig hilsen

>

> SÃ¸ren Lophaven

> ************************************************************************

> ******

> Master of Science in Engineering | Ph.D. student

> Informatics and Mathematical Modelling | Building 321, Room 011

> Technical University of Denmark | 2800 kgs. Lyngby, Denmark

> E-mail: snl@... | http://www.imm.dtu.dk/~snl

> Telephone: +45 45253419 |

> ************************************************************************

> ******

>

> On Mon, 30 Aug 2004, Edzer J. Pebesma wrote:

>

> >

> >

> > Monica Palaseanu-Lovejoy wrote:

> > ....

> >

> > >If you are still interested in predicting values, a better solution,

> > >in

> > >my experience, is to use a bayesian kriging method. Such

> > >methods are implemented in the package R (which is free) with the

> > >geoR routine (http://cran.r-project.org/)({ HYPERLINK

> "http://cran.r-project.org/" }. Using this method i

> > >always had smaller error standard deviations, and the precision and

> > >accuracy are better than the "normal" kriging method.

> > >

> > Thanks for sharing your experiences with us, Monica. I wondered if you

> > published

> > your results somewhere, because there is, AFAIK, little published

> > material on

> > comparisons of the "traditional" and the "model based" geostatistical

> > approaches.

> >

> > You mention smaller error standard deviations -- I assume that you

> > refer to cross validation error standard deviations, and not kriging

> > prediction standard errors? How did you calculate precision and

> > accuracy? In addition to specifying

> > a variogram model, you also need to specify prior distribution on all

> > variogram

> > parameters in the model-based approach, how did you choose these?

> >

> > One paper that does the comparison is Moyeed and Papritz, Math Geol

> > 34(4), 365-386 but they found little improvement in using model-based

> > as opposed to regular kriging; in their comparison case they used a

> > large (n>2500) data set

> > though.

> >

> > Anyone else who wants to shed light on this issue? Is there e.g. a

> > minimum sample size above which both approaches become hard to

> > distinguish?

> > --

> > Edzer

> >

> >

> >

>

>

>

>