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AI-GEOSTATS: How reliable are your kriging variances?

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  • Chris Howden
    G day all, I reckon we need to quantify the reliability of the kriging variance map. Because sometimes its going to be an accurate map, and other times its
    Message 1 of 5 , Feb 18, 2003
      G'day all,

      I reckon we need to quantify the reliability of the kriging variance
      map. Because sometimes its going to be an accurate map, and other times
      its going to be way off the mark.

      Imagine the situation when there are two maps with similar kriging
      variances. However when we look at the semivariagram fit one of them
      closely follow the line of fit while the other has a much larger
      scatter. This means that one of the maps is actually much more accurate
      then the other.

      But as maps are currently presented we would never know!!

      Could this be a big problem? I think it could. Particularly when the
      estimation is quite bad, meaning that the variances have been
      underestimated and should likely be much larger.

      One solution could be to make the kriging variances proportional to the
      model fit. Maybe the error between the kriging variance (as estimated
      using the semivariagram) and the estimation variance (using real data
      points) could be used to do this?

      Does anyone know if this has been discussed before? Has it ever been
      considered. Or am I totally off the trail and should activate my GIS
      beacon?




      For those that are interested I'll explain how I got to the above
      conclusion:

      Kriging can be summarised by the following:
      Var(est) = f(weights and semivariance between all points that have a
      positive weight), and we obtain the Var(krig) by minimising Var(est)
      with respect to the weights. This is how we get the weights.

      But in order to do this we need to know what the semivariance between
      the points is. However if we're estimating a point we don't have then we
      can't calculate the semi-variance, so we can't find the appropriate
      weights. However, if we have a model for the semi-variance then we can
      predict what the semi-variance should be using this model and we can
      then calculate the appropriate weights. Which is why we require a
      semivariagram model. So the semivariagram fit is vital in generating not
      only the estimates, but their reliability also.

      What this all boils down to is that the most important thing when
      kriging is the ASSUMPTION that the points used to generate the
      semi-variagram are capable of representing the semivariance for all
      points. As well as the ASSUMPTION that the correct model has been fit,
      and that its a good fit.

      If either of these assumption fails then the kriging variance is
      incorrect.

      More to the point if the model is a poor fit then the kriging variance
      is less likely to be accurate.

      This brought me to my question. Should we have some statitistic that
      quantifies how reliable our kriging variances are?





      Christopher G Howden
      Statistical Ecologist
      Department of Land and Water Conservation
      (Work) 02 9895 7130
      (Fax) 02 9895 7867
      (Mob) 0410 689 945


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    • Soeren Nymand Lophaven
      Dear Chris Bayesian kriging is what you should use, if you want to include estimation uncertainty into the kriging variances. Some useful references are: Le,
      Message 2 of 5 , Feb 19, 2003
        Dear Chris

        Bayesian kriging is what you should use, if you want to include estimation
        uncertainty into the kriging variances. Some useful references are:

        Le, N.D. and Zidek, J.V. (1992).
        Interpolation with uncertain covariances: a Bayesian alternative to
        Kriging.
        Journal of Multivariate Analysis, 43, p. 351-74.

        Handcock, M.S. and Stein, M.L. (1993).
        A Bayesian analysis of kriging.
        Technometrics, 35, p. 403-10.

        Kitanidis, P.K. (1986).
        Parameter uncertainty in estimation of spatial functions: Bayesian
        analysis.
        Water Resources Research, 22, p. 499-507.

        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 Wed, 19 Feb 2003, Chris Howden wrote:

        > G'day all,
        >
        > I reckon we need to quantify the reliability of the kriging variance
        > map. Because sometimes its going to be an accurate map, and other times
        > its going to be way off the mark.
        >
        > Imagine the situation when there are two maps with similar kriging
        > variances. However when we look at the semivariagram fit one of them
        > closely follow the line of fit while the other has a much larger
        > scatter. This means that one of the maps is actually much more accurate
        > then the other.
        >
        > But as maps are currently presented we would never know!!
        >
        > Could this be a big problem? I think it could. Particularly when the
        > estimation is quite bad, meaning that the variances have been
        > underestimated and should likely be much larger.
        >
        > One solution could be to make the kriging variances proportional to the
        > model fit. Maybe the error between the kriging variance (as estimated
        > using the semivariagram) and the estimation variance (using real data
        > points) could be used to do this?
        >
        > Does anyone know if this has been discussed before? Has it ever been
        > considered. Or am I totally off the trail and should activate my GIS
        > beacon?
        >
        >
        >
        >
        > For those that are interested I'll explain how I got to the above
        > conclusion:
        >
        > Kriging can be summarised by the following:
        > Var(est) = f(weights and semivariance between all points that have a
        > positive weight), and we obtain the Var(krig) by minimising Var(est)
        > with respect to the weights. This is how we get the weights.
        >
        > But in order to do this we need to know what the semivariance between
        > the points is. However if we're estimating a point we don't have then we
        > can't calculate the semi-variance, so we can't find the appropriate
        > weights. However, if we have a model for the semi-variance then we can
        > predict what the semi-variance should be using this model and we can
        > then calculate the appropriate weights. Which is why we require a
        > semivariagram model. So the semivariagram fit is vital in generating not
        > only the estimates, but their reliability also.
        >
        > What this all boils down to is that the most important thing when
        > kriging is the ASSUMPTION that the points used to generate the
        > semi-variagram are capable of representing the semivariance for all
        > points. As well as the ASSUMPTION that the correct model has been fit,
        > and that its a good fit.
        >
        > If either of these assumption fails then the kriging variance is
        > incorrect.
        >
        > More to the point if the model is a poor fit then the kriging variance
        > is less likely to be accurate.
        >
        > This brought me to my question. Should we have some statitistic that
        > quantifies how reliable our kriging variances are?
        >
        >
        >
        >
        >
        > Christopher G Howden
        > Statistical Ecologist
        > Department of Land and Water Conservation
        > (Work) 02 9895 7130
        > (Fax) 02 9895 7867
        > (Mob) 0410 689 945
        >
        >
        > --
        > * To post a message to the list, send it to ai-geostats@...
        > * As a general service to the users, please remember to post a summary of any useful responses to your questions.
        > * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
        > * Support to the list is provided at http://www.ai-geostats.org
        >


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      • Pierre Goovaerts
        Hi Christopher, I believe you forgot a key assumption, homoscedasticity. In most situations this assumption is not realistic and we would like the kriging
        Message 3 of 5 , Feb 19, 2003
          Hi Christopher,

          I believe you forgot a key assumption, homoscedasticity.
          In most situations this assumption is not realistic
          and we would like the kriging variance to somehow
          depend on the local variability of data.
          Rescaling globally the kriging variance to
          account for uncertainty about variogram model
          won't solve this problem.
          Your map might be "globally" more accurate
          but locally it will still fail to indicate
          where prediction errors might be larger.

          Regarding statistics to account for reliability of kriging variance,
          the key question is what do you want to do with that variance.
          If it's used to derive local probability distributions
          under the multiGaussian model, you can assess
          precision and accuracy of uncertainty models
          using cross-validation. I addressed this issue
          in the following paper:
          Goovaerts, P. 2001.
          Geostatistical modelling of uncertainty in soil science.
          Geoderma, 103: 3-26.
          and would be glad to send you a PDF copy of the paper if needed.

          Regards,

          Pierre
          <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

          Dr. Pierre Goovaerts
          President of PGeostat, LLC
          Chief Scientist with Biomedware Inc.
          710 Ridgemont Lane
          Ann Arbor, Michigan, 48103-1535, U.S.A.

          E-mail: goovaert@...
          Phone: (734) 668-9900
          Fax: (734) 668-7788
          http://alumni.engin.umich.edu/~goovaert/

          <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

          On Wed, 19 Feb 2003, Chris Howden wrote:

          > G'day all,
          >
          > I reckon we need to quantify the reliability of the kriging variance
          > map. Because sometimes its going to be an accurate map, and other times
          > its going to be way off the mark.
          >
          > Imagine the situation when there are two maps with similar kriging
          > variances. However when we look at the semivariagram fit one of them
          > closely follow the line of fit while the other has a much larger
          > scatter. This means that one of the maps is actually much more accurate
          > then the other.
          >
          > But as maps are currently presented we would never know!!
          >
          > Could this be a big problem? I think it could. Particularly when the
          > estimation is quite bad, meaning that the variances have been
          > underestimated and should likely be much larger.
          >
          > One solution could be to make the kriging variances proportional to the
          > model fit. Maybe the error between the kriging variance (as estimated
          > using the semivariagram) and the estimation variance (using real data
          > points) could be used to do this?
          >
          > Does anyone know if this has been discussed before? Has it ever been
          > considered. Or am I totally off the trail and should activate my GIS
          > beacon?
          >
          >
          >
          >
          > For those that are interested I'll explain how I got to the above
          > conclusion:
          >
          > Kriging can be summarised by the following:
          > Var(est) = f(weights and semivariance between all points that have a
          > positive weight), and we obtain the Var(krig) by minimising Var(est)
          > with respect to the weights. This is how we get the weights.
          >
          > But in order to do this we need to know what the semivariance between
          > the points is. However if we're estimating a point we don't have then we
          > can't calculate the semi-variance, so we can't find the appropriate
          > weights. However, if we have a model for the semi-variance then we can
          > predict what the semi-variance should be using this model and we can
          > then calculate the appropriate weights. Which is why we require a
          > semivariagram model. So the semivariagram fit is vital in generating not
          > only the estimates, but their reliability also.
          >
          > What this all boils down to is that the most important thing when
          > kriging is the ASSUMPTION that the points used to generate the
          > semi-variagram are capable of representing the semivariance for all
          > points. As well as the ASSUMPTION that the correct model has been fit,
          > and that its a good fit.
          >
          > If either of these assumption fails then the kriging variance is
          > incorrect.
          >
          > More to the point if the model is a poor fit then the kriging variance
          > is less likely to be accurate.
          >
          > This brought me to my question. Should we have some statitistic that
          > quantifies how reliable our kriging variances are?
          >
          >
          >
          >
          >
          > Christopher G Howden
          > Statistical Ecologist
          > Department of Land and Water Conservation
          > (Work) 02 9895 7130
          > (Fax) 02 9895 7867
          > (Mob) 0410 689 945
          >
          >
          > --
          > * To post a message to the list, send it to ai-geostats@...
          > * As a general service to the users, please remember to post a summary of any useful responses to your questions.
          > * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
          > * Support to the list is provided at http://www.ai-geostats.org
          >



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        • Ruben Roa
          ... Because sometimes its going to be an accurate map, and other times its going to be way off the mark. ... variances. However when we look at the
          Message 4 of 5 , Feb 19, 2003
            >G'day all,
            >
            >I reckon we need to quantify the reliability of the kriging variance map.
            Because sometimes its going to be an accurate map, and other times its
            going to be way off the mark.
            >
            >Imagine the situation when there are two maps with similar kriging
            variances. However when we look at the semivariagram fit one of them
            closely follow the line of fit while the other has a much larger scatter.
            This means that one of the maps is actually much more accurate then the
            other.

            Statistically, your question is related to the fact that the covariance
            matrix from the fit of the variogram model is usually disregarded in
            subsequent analyses. Some time ago i asked about this discarding of the
            variogram parameters covariance matrix and geostatisticians replied that
            this source of uncertainty was usually less important than other problems
            and should not be of much importance.
            Rubén
            http://webmail.udec.cl

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          • Ruben Roa
            ... they are completely and perfectly reliable IOW, the kriging variances are the objective functions to be optimized not the parameters to be estimated (the
            Message 5 of 5 , Feb 20, 2003
              >
              >A couple of observations on your problem
              >
              >1. The kriging variances are computed, not estimated, hence in that sense
              they are completely and perfectly "reliable"

              IOW, the kriging variances are the objective functions to be optimized not
              the parameters to be estimated (the latter ones are the values of the
              spatial variable at non observed locations). It seems to me they are
              perfectly "reliable" (or perhaps 'exact') in linear kriging but not so in
              non linear kriging, in which they should be approximate.

              >2. However, the kriging variances are computed using (1) the variogram or
              covariance model, (2) the coordinates of the data locations used to
              generate the kriged value, (2) the coordinates of the location being
              kriged, (3) the degree of the polynomial used to represent the drift (if
              Universal Kriging is used). (4) Decisions made about the search
              neighborhood. Therein lies the problem, while it may be that the
              coordinates are known (nearly perfectly), the variogram or covariance must
              be estimated from the data and that is not a "perfect" process.

              Under standard geostatistical procedures the variogram or covariance is
              taken as known during kriging. I once reviewed a ms. sent to a statistical
              journal in which the authors assessed by simulation the degree to which
              this assumption affects OK (IIRW). They found that for fairly low sample
              sizes, the effect of ignoring the covariance matrix of parameters in the
              model variogram was very minor.

              >While changing the search neighborhood may not change the estimates much,
              it can sometimes have a big effect on the kriging variances, an easy way to
              see this is to do a little experimenting in cross-validation.
              >
              >3. Note that if you multiply the variogram by a positive number (either
              bigger or smaller than one) you will NOT change the kriging weights and
              hence not change the kriged values but you will change the kriging
              variance.

              That is why the use of standardized experimental variograms to fit the
              model variogram do not change the kriging weights as compared with the
              standard experimental variogram.

              Your other points (snipped) are very well taken.
              Note that i did not posit the original question as may appear by the way
              Donald's message showed up on the list.

              Rubén
              http://webmail.udec.cl

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