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GEOSTATS: Linear Model of Coregionalization -- Devil's Advocate

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  • Todd Mowrer
    Let me precede my comment with a clear statement that I would never ignore the linear model of coregionalization (LMC) in my analyses. However, there are
    Message 1 of 6 , Jun 4, 1999
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      Let me precede my comment with a clear statement that I would never
      ignore the linear model of coregionalization (LMC) in my analyses.
      However, there are citable instances in the non-theoretical
      subject-matter literature of people apparently ignoring it, and it is
      conceivable that one might be asked to peer review such an article.
      Now for argument's sake, from a "devil's advocate" position, my
      question is: What is the consequence of negative cokriging variances?
      Does one no longer have a best linear unbiased estimator? It would seem
      to me that one would still obtain a minimum variance (from the
      semi-variogram) weighted combination of measured values to estimate
      unknown locations. If you don't use cokriging variances for anything,
      e.g., co-conditional simulation, (or even look at them, much less
      mention them ), why care?
      Now if you can't invert a matrix, you'll know it. And if you get a
      "spikey" response surface for your primary response variable, that's
      obviously not good for predictive purposes. But if you get a
      "reasonable" response surface (particularly one with a lower
      cross-validation score than from a conservative coregionalized model),
      then it would seem one has gotten off free, perhaps by stumbling into a
      set of (cross) variograms that work (conform to the LMC). Thank you for
      your comments to help me understand the LMC better.
      Best regards...
      Todd Mowrer, Research Scientist
      Rocky Mountain Research Station
      USDA Forest Service
      Fort Collins, Colorado 80526 USA
      tmowrer/rmrs@...
      tmowrer@...
      tel: +970 498-1255
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    • L Scott Baggett
      Greetings, At the risk of deviating from the LMC discussion and extending to a more general problem, I would like to comment on Todd Mowrer s note and include
      Message 2 of 6 , Jun 5, 1999
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        Greetings,

        At the risk of deviating from the LMC discussion and extending to a more
        general problem, I would like to comment on Todd Mowrer's note and include
        the relevance of negative prediction variances in general.

        By definition, the loss function for the BLUP used to compute the kriging
        weights involves the true variance/covariance structure of the random
        field which itself is by definition positive definite. Otherwise it would
        not be a true covariance. It's fairly straight-forward to add an nxn bias
        matrix and an nx1 bias vector to the covariance matrix and vector used to
        directly compute the weights. This shows that if the covariance function
        is misspecified then the predictions will be biased as well by a term
        involving the bias matrix and vector, true covariance matrix and vector,
        and observation vector as well. This may happen for example when a
        covariance function is estimated without properly defining the
        constraints.

        Therefore, what I suggest is that negative prediction variances are more
        to be interpreted as symptoms of a greater problem, e.g. that being proper
        definition of the true parameter space with respect to the covariance
        function where the covariance function itself is the parameter.

        I agree the literature is riddled with examples of non-admissible
        variograms and covariograms. Zonal anisotropic models are a good example.
        But I would submit that simply not having negative prediction variances
        does not necessarily get you home free. To say the least, having negative
        prediction variances should be cause for concern. The covariance model
        itself has to be proven admissible (and of course properly estimated).


        L. Scott Baggett
        Rice University
        Department of Statistics, MS138
        6100 Main Street
        Houston, TX 77005-1892


        On Fri, 4 Jun 1999, Todd Mowrer wrote:

        > Let me precede my comment with a clear statement that I would never
        > ignore the linear model of coregionalization (LMC) in my analyses.
        > However, there are citable instances in the non-theoretical
        > subject-matter literature of people apparently ignoring it, and it is
        > conceivable that one might be asked to peer review such an article.
        > Now for argument's sake, from a "devil's advocate" position, my
        > question is: What is the consequence of negative cokriging variances?
        > Does one no longer have a best linear unbiased estimator? It would seem
        > to me that one would still obtain a minimum variance (from the
        > semi-variogram) weighted combination of measured values to estimate
        > unknown locations. If you don't use cokriging variances for anything,
        > e.g., co-conditional simulation, (or even look at them, much less
        > mention them ), why care?
        > Now if you can't invert a matrix, you'll know it. And if you get a
        > "spikey" response surface for your primary response variable, that's
        > obviously not good for predictive purposes. But if you get a
        > "reasonable" response surface (particularly one with a lower
        > cross-validation score than from a conservative coregionalized model),
        > then it would seem one has gotten off free, perhaps by stumbling into a
        > set of (cross) variograms that work (conform to the LMC). Thank you for
        > your comments to help me understand the LMC better.
        > Best regards...
        > Todd Mowrer, Research Scientist
        > Rocky Mountain Research Station
        > USDA Forest Service
        > Fort Collins, Colorado 80526 USA
        > tmowrer/rmrs@...
        > tmowrer@...
        > tel: +970 498-1255
        > --
        > *To post a message to the list, send it to ai-geostats@....
        > *As a general service to list users, please remember to post a summary
        > of any useful responses to your questions.
        > *To unsubscribe, send email to majordomo@... with no subject and
        > "unsubscribe ai-geostats" in the message body.
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        >

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      • Denis ALLARD
        ... Hello Todd, it seems to me that if you seek a minimum variance estimator, an important constraint is that your variance MUST be non negative ! In this
        Message 3 of 6 , Jun 7, 1999
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          On Fri, 4 Jun 1999, Todd Mowrer wrote:

          > question is: What is the consequence of negative cokriging variances?
          > Does one no longer have a best linear unbiased estimator? It would seem
          > to me that one would still obtain a minimum variance (from the
          > semi-variogram) weighted combination of measured values to estimate
          > unknown locations. If you don't use cokriging variances for anything,
          > e.g., co-conditional simulation, (or even look at them, much less
          > mention them ), why care?


          Hello Todd,

          it seems to me that if you seek a minimum variance estimator, an important
          constraint is that your variance MUST be non negative ! In this case, a
          null variance would mean exact interpolation, as it is the case when
          kriging at a data point (kriging is an exact interpolator). If you allow
          negative variance, there is no minimum anymore, or more exactly the
          theoretical minimum variance is - infinity !!!!

          Now, if you don't care about kriging variances, and don't even look at
          them, wht bother with a statistical, probability based, approach ?
          Why don't you use any interpolation package ?

          The main point of geostatistics, vs mere interpolation is that it provides
          a measure of the estimation uncertainty (the kriging variance) along
          with the interpolation itself, at the cost of a model -- the variogram(s).

          For me, mismodeling the variogram and ignoring the kriging variance is
          simply missing the point.



          Denis Allard


          .------------------------Denis ALLARD--------------------------------.
          | Unite de Biometrie allard@... |
          | INRA Domaine St Paul, Site Agroparc tel: (33) 4 90 31 62 30 |
          | 84914 AVIGNON cedex 9, FRANCE fax: 62 52 |
          `--------- http://www.avignon.inra.fr/biometrie/welcome.html --------'

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        • Octavi Fors Aldrich
          Hello everybody, I m novice using GSLIB2, and after taking a quicklook in the User s Guide, I ve missed one capability in gamv.f program. When calculating
          Message 4 of 6 , Jun 9, 1999
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            Hello everybody,

            I'm novice using GSLIB2, and after taking a quicklook in the User's Guide,
            I've missed one capability in gamv.f program. When calculating variogram
            gamma(h) (whatever option you choose) it could be interesting to have h
            as a non-equally distributed distances array. Unfortunately, it
            seems that gamv.f outputs h as an equally-grided one. This is not
            important when number of points per lag is high, but not the same when
            it's low.

            I was wondering if anybody has "patched" this issue by adding some
            code lines to gamv.f. Any experiences?

            Thanks in advance,

            Octavi.

            =================================================================

            Octavi Fors Aldrich

            Astronomy Department
            Physics Faculty
            Avgda. Diagonal 647
            08028 Barcelona
            SPAIN

            Telf: 34-934021122
            Fax: 34-934021133
            e-mail: octavi@...

            =================================================================




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          • Ulrich Leopold
            Hi all, I have some question concerning the modeling. I have calculated different variogram estimators. The traditional semivariogram reveals spatial
            Message 5 of 6 , Jun 10, 1999
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              Hi all,

              I have some question concerning the modeling.

              I have calculated different variogram estimators.
              The traditional semivariogram reveals spatial continuity. But the
              shape, the ranges and anisotropies are better estimated by the more
              robust pairwise relative semivariogram.

              My questions are:

              (1) Could I model the experimental pairwise relative semivariogram
              (or other more robust variograms) or does it affect the
              kriging estimator and it's estimation variance considering the
              accuracy? And if yes do I have to standardize the sill to one?

              (2) Is there a difference between modeling the traditional
              semivariogram with it's original sill value and with a sill
              standardized to one.

              (3) What is really necessary to yield the correct estimates? Only the
              ratios of nugget and sill structures and their corresponding ranges
              or the real values provided by the traditional semivariogram with a
              non standardized sill?

              Goovaerts 1997 ("Geostatistics for natural resources estimation")
              warns against using the more robust estimators as substitutes for
              the traditional semivariogram. But for example Srivastava and Parker
              1989 ("Robust measures of spatial continuity") did model several
              robust estimators besides the traditional semivariogram.

              If a robust measure provides a better spatial continuity I would say
              that I can use it for modeling. Only the estimation variance will be
              affected and should not be taken as an absolute value but used in
              relative terms to compare the variances.


              Thanks in advance

              Ulrich

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

              Ulrich Leopold

              Department of Soil Science
              The University of Trier

              E-mail: leop6101@...
              phone: 0049-(0)651-140764
              address: Engelstr.104, 54292 Trier, Germany
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            • Mark Evans
              Hi. I m looking for a simple plug-in or extension for ARC VIEW in order to calculate some semivariograms for my forest landscape. ANyone know of any scripts
              Message 6 of 6 , Jul 8, 1999
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                Hi.

                I'm looking for a simple plug-in or extension for ARC VIEW in order
                to calculate some semivariograms for my forest landscape.

                ANyone know of any scripts for download etc ?

                Thanks,

                Mark
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