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[ai-geostats] Re: extreme values

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  • Isobel Clark
    Hello Kai ... Your kriging standard deviation is a direct consequence of the semi-variogram model which you fitted. This, of course, is a direct reflection of
    Message 1 of 10 , Aug 30, 2004
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      Hello Kai

      > 1. I get a quite good fitting with an
      > 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 ?
      Your kriging standard deviation is a direct
      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
      > 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 ?
      Not if you want to do cross validation. See my paper
      '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
      > by a Power model. Can I interpret that as only a
      > trend function ?
      Only if the power is approaching 2 or greater.

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





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    • Monica Palaseanu-Lovejoy
      Hi, This is more or less the subject of my PhD thesis i hope to submit this January 2005 ;-)) I am working with small sample size (around 300 values).
      Message 2 of 10 , Aug 30, 2004
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        Hi,

        This is more or less the subject of my PhD thesis i hope to submit
        this January 2005 ;-))

        I am working with small sample size (around 300 values).
        Specifying prior distribution may be tricky and i have to recognize
        that for me it is still a "try and error". Besides, when i want to
        predict values / or probabilities at different locations- i am doing
        that on a grid of 10 by 10 metres which gives me about 9000 cells.
        This stretches the limit of my computer at maximum.

        I represented the precision and accuracy graphically by plotting
        together the bayesian density and krige density curves for a certain
        measured value - for example. For kriging i have always a bell
        shape curve since the assumption is Gaussian. For the bayesian
        method the curve may resemble a bell shaped curve by i never got
        a true Gaussian shape until now. Usually the bayesian density
        curve is more "narrower" yielding a smaller prediction interval for a
        95% confidence. The validation error standard deviations are
        usually smaller for the bayesian method than for kriging. For the
        grid predictions, always the bayesian method yields smaller error
        standard deviations, does not matter how "good" the kriging was.

        I hope to be able to publish some of my results next year.
        Meanwhile i will test this bayesian method on the SIC2004 data as
        well.

        Thanks for the encouragements,

        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@...
      • Monica Palaseanu-Lovejoy
        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
        Message 3 of 10 , Aug 30, 2004
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          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@...
        • 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 4 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 5 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 6 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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