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RE: [ai-geostats] Bayesian kriging(s)?

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  • Gregoire Dubois
    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
    Message 1 of 10 , Aug 31, 2004
      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 :((



      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)

      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@...;
      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

      - 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
    • Monica Palaseanu-Lovejoy
      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
      Message 2 of 10 , Aug 31, 2004

        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 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@...
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