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

RE: [ai-geostats] extreme values

Expand Messages
  • 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 1 of 10 , Aug 30, 2004
    • 0 Attachment
      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 2 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, 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 3 of 10 , Aug 30, 2004
        • 0 Attachment
          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 4 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 5 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 6 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 7 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.