Loading ...
Sorry, an error occurred while loading the content.

[ai-geostats] Re: extreme values

Expand Messages
  • 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
    • 0 Attachment
      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





      ___________________________________________________________ALL-NEW Yahoo! Messenger - all new features - even more fun! http://uk.messenger.yahoo.com
    • 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 2 of 10 , Aug 30, 2004
      • 0 Attachment
        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 3 of 10 , Aug 30, 2004
        • 0 Attachment
          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 4 of 10 , Aug 31, 2004
          • 0 Attachment
            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 5 of 10 , Aug 31, 2004
            • 0 Attachment
              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@...
            Your message has been successfully submitted and would be delivered to recipients shortly.