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

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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 1 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
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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 2 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 3 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 4 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 5 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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