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Re: [ai-geostats] extreme values

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  • Soeren Nymand Lophaven
    Based on my relatively limited knowledge on Bayesian kriging I have a few comments to the current discussion: - Bayesian kriging gives better predictions than
    Message 1 of 10 , Aug 30, 2004
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      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
      >
      >
      >
    • 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 2 of 10 , Aug 31, 2004
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        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
        >
        >
        >
      • 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 3 of 10 , Aug 31, 2004
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          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@...
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