- Hello Kai

> 1. I get a quite good fitting with an

Your kriging standard deviation is a direct

> omnidirectional spherical variogram but the kriging

> standard deviation is relativly high and the results

> of the cross validation aren´t very good. How can I

> interpret that? Is it possible that extreme values

> in my data set can be responsible for that ?

consequence of the semi-variogram model which you

fitted. This, of course, is a direct reflection of the

variance of your data. If your data follows a skewed

distribution (or, at least, not very Normal) then the

variance is affected by other factors than simple

variability -- such as, extreme values in the 'tails'.

You can probably get a much better semi-variogram by

transforming your data in some way. Most software

packages have a mechanism for this. This assumes that

your extreme values are in the tail and not anomalies

of some kind.

> 2. I ´ve read that it is useful to use standardized

Not if you want to do cross validation. See my paper

> variograms for minimize the influence of extreme

> values. Can I use the variogram parameters of the

> standaridized variogram as an input for the kriging

> system like the paramteres of a 'normal'

> semivariogram ?

'Does Geostatistics Work', 1979. Download from

http://uk.geocities.com/drisobelclark/resume or Noel

Cressie's paper which I cited last week.

> 3. I get a very good fitting with another data set

Only if the power is approaching 2 or greater.

> by a Power model. Can I interpret that as only a

> trend function ?

Isobel Clark

http://geoecosse.bizland.com/whatsnew.htm

___________________________________________________________ALL-NEW Yahoo! Messenger - all new features - even more fun! http://uk.messenger.yahoo.com - Hi,

Yes i know. For these reasons i have suggested to look if the data

does not come from 2 different populations. Also, usually the

background is not above the environmental threshold, so indicator

kriging or probability kriging are more appropriate, in my opinion,

than doing predictions using a set of data coming from a mixture of

populations.

Thanks for stressing out that usually there is a "background" for

PAHs we should take into consideration.

Monica

Monica Palaseanu-Lovejoy

University of Manchester

School of Geography

Mansfield Cooper Bld. 3.21

Oxford Road

Manchester M13 9PL

England, UK

Tel: +44 (0) 275 8689

Email: monica.palaseanu-lovejoy@... - 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

Søren 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

>

>

> - 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

Søren 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

>

>

> - Hi,

Well, the bayesian kriging methods you are describing are

somewhat different than what i am using. I am using R and geoR

by Ribeiro and Diggle (2001).

Web pages for R:{ HYPERLINK "http://cran.r-project.org/" }http://cran.r-project.org/

Web page for geoR: www.est.ufpr.br/geoR

Usually with Bayesian kriging you will have higher variance just

because the uncertainty is incorporated in all (some) parameters,

while for the geostatistical kriging (or the "other kriging") there is no

uncertainty assumed for the semi-variogram model. So, in a way

kriging is a particular case of bayesian kriging as it is described by

Ribeiro and Diggle.

Uncertainty can be assumed for nugget, variance, mean and range,

or only for one parameter, or a combination of parameters. Usually

everything is depending on how well one is understanding the data,

or at least so i think. Citing from Ribeiro the inference is done by

Monte Carlo simulations, and samples are taken from the posterior

and predictive distributions and used for inference and predictions.

One of his algorithms looks like that:

1. Choose a range of values for phi (range parameter in

geostatistical kriging) which is sensible for the given data, and

assign a discrete uniform prior for phi on a set of values spanning

the chosen range;

2. compute the posterior probabilities on this discrete support set,

defining a discrete posterior distribution with probability mass

function pr(phi | y);

3. sample a value of phi from this discrete distribution pr(phi | y);

4. attach the sampled value phi to the distribution [beta, sigma

square |y, phi] and sample from this distribution (beta = mean

param., sigma square = variance, phi = range)

5. repeat steps 3 and 4 as many times as required / desired. the

resulting sample of the triplets (beta, sigma square, phi) is a

sample from the joint posterior distribution.

In my experience, if the data set is highly skewed and the spatial

autocorrelation is weak, bayesian kriging does a better job than

geostatistical kriging, even if the data is transformed to approach

normality. From literature (see the paper mentioned by Edzer

Pebesma - Moyeed and Papritz, Math Geol 34(4), 365-386) it

seems that for very large sets of data (n > 2500) the advantage

Bayesian kriging has over geostatistical kriging is minimal, while

with the data sets i am using (random locations, weak spatial

autocorrelation, areas of spatial heterogeneity, n in between 200 to

350 points), Bayesian kriging seems to be superior.

I hope this helps a little,

Monica

Monica Palaseanu-Lovejoy

University of Manchester

School of Geography

Mansfield Cooper Bld. 3.21

Oxford Road

Manchester M13 9PL

England, UK

Tel: +44 (0) 275 8689

Email: monica.palaseanu-lovejoy@...