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Re: [ai-geostats] extreme values

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  • Monica Palaseanu-Lovejoy
    Hi, I am dealing with PAHs contamination data in soils as well. In my experience, depending where this contamination is (i mean if it is an old industrial
    Message 1 of 10 , Aug 30, 2004
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      Hi,

      I am dealing with PAHs contamination data in soils as well. In my
      experience, depending where this contamination is (i mean if it is
      an old industrial site, a dump site, or something else) you may
      have actually more than one population, so your data you want to
      krige is a mixture of populations. There are statistical tools through
      which you can check if this is the case. But usually if you suspect
      at least 2 different pollution processes, for sure you will have a
      mixture of populations sampled.

      In this case, an indicator kriging or probability kriging or disjunctive
      indicator kriging might be more appropriate than actually predicting
      values. For environmental purposes, most of the times we are
      interested to see the probability with which a contaminant may be
      above (or not) the environmental threshold.

      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.

      I hope this helps a little, good luck,

      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@...
    • Edzer J. Pebesma
      Monica Palaseanu-Lovejoy wrote: ... Thanks for sharing your experiences with us, Monica. I wondered if you published your results somewhere, because there is,
      Message 2 of 10 , Aug 30, 2004
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        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
      • Glover, Tim
        Just a quick point on PAHs - are you aware that there is a general background concentration of PAHs everywhere? These come from air-deposition of PAHs from
        Message 3 of 10 , Aug 30, 2004
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          Just a quick point on PAHs - are you aware that there is a general
          background concentration of PAHs everywhere? These come from
          air-deposition of PAHs from many sources, including natural forest
          fires, auto exhaust, incinerators, jet engines, etc. There are also
          PAHs in asphalt. Any site that has PAH contamination WILL be at least
          bi-modal - the "background" and any contamination.

          Tim Glover
          Senior Environmental Scientist - Geochemistry
          Geoenvironmental Department
          MACTEC Engineering and Consulting, Inc.
          Kennesaw, Georgia, USA
          Office 770-421-3310
          Fax 770-421-3486
          Email ntglover@...
          Web www.mactec.com

          -----Original Message-----
          From: Monica Palaseanu-Lovejoy
          [mailto:monica.palaseanu-lovejoy@...]
          Sent: Monday, August 30, 2004 7:36 AM
          To: kai.zosseder@...; ai-geostats@...
          Subject: Re: [ai-geostats] extreme values

          Hi,

          I am dealing with PAHs contamination data in soils as well. In my
          experience, depending where this contamination is (i mean if it is
          an old industrial site, a dump site, or something else) you may
          have actually more than one population, so your data you want to
          krige is a mixture of populations. There are statistical tools through
          which you can check if this is the case. But usually if you suspect
          at least 2 different pollution processes, for sure you will have a
          mixture of populations sampled.

          In this case, an indicator kriging or probability kriging or disjunctive

          indicator kriging might be more appropriate than actually predicting
          values. For environmental purposes, most of the times we are
          interested to see the probability with which a contaminant may be
          above (or not) the environmental threshold.

          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.

          I hope this helps a little, good luck,

          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@...
        • 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 4 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 5 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 6 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 7 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 8 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 9 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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