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Re: AI-GEOSTATS: average semi-variogram

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  • Gerald Boogaart
    Dear Koen, ... Thats my view of the forest: When you directly measure the mean value of blocks, with the blocks all of equal shape B, that is nothing else than
    Message 1 of 9 , Mar 2, 2004
      Dear Koen,

      > I did measure the point values in a grid in a plot/block. I don't have to
      > estimate them, I just need a spacially depended estimate of the whole
      > plot/block variance.
      >
      > The problem is that I would like to automate the procedure of calculating
      > that within block variance because variogram models need to be fit
      > visually most of the time, wich is time consuming. When repeating
      > calculations on randomized data this isn't an option. So if gamma(V,V)
      > really is model property rather then a data property I have a problem...
      >
      > If you can see the trees for the forest, because I don't.
      > Koen.

      Thats my view of the forest:

      When you directly measure the mean value of blocks, with the blocks all of
      equal shape B, that is nothing else than measuring points x in another random
      process Y , which is related to original process Z by the simple relation
      Y(x) = \int_{B} Z(x+w) dw / Area(B). (1)
      But anyway that is a random process in its own right and can be handled by
      ordinary pointwise geostatistics. That is in the same way true, when we have
      a block approximated by datapoints:

      Y(x) = 1/|B| \sum_{w in B} Z(x+w) (2)

      That the variograms of Z and Y are related in the way

      gamma_Y(x,x') = 1/|B|^2 \int \int gamma_Z(x+w,x'+w') dw dw'

      or resp.

      gamma_Y(x,x') = 1/|B|^2 \sum_{w in B} \sum_{w' in B} gamma_Z(x+w,x'+w')

      stays a biproduct, which would more or less (or more less) allow to infer
      gamma_Z from gamma_Y. However you don't seem to need relation. You can go
      along with gamma_Y only.

      The first problem is that you measure according to equation (2) but you want
      to estimate according to equation (1). Anyway (2) should well approximate
      (1).

      The second problem is: The variogram of Y should be very smooth in the
      origing, but it is most sure not Gaussian.

      Gerald

      -------------------------------------------------
      Prof. Dr. K. Gerald v.d. Boogaart
      Professor als Juniorprofessor für Statistik
      http://www.math-inf.uni-greifswald.de/statistik/

      office: Franz-Mehring-Str. 48, 1.Etage rechts
      e-mail: Gerald.Boogaart@...
      phone: 00+49 (0)3834/86-4621
      fax: 00+49 (0)89-1488-293932 (Faxmail)
      fax: 00+49 (0)3834/86-4615 (Institut)

      paper-mail:
      Ernst-Moritz-Arndt-Universität Greifswald
      Institut für Mathematik und Informatik
      Jahnstr. 15a
      17487 Greifswald
      Germany
      --------------------------------------------------


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    • Syed Abdul Rahman Shibli
      If you have gridded observed data at close and regular spacing for the block, then why bother with a Gamma(V,V)? Just calculate the variance for the block
      Message 2 of 9 , Mar 2, 2004
        If you have gridded observed data at close and regular spacing for the block, then why bother with a Gamma(V,V)? Just calculate the variance for the block directly.

        On the one hand you need "a spatially dependent estimate of the whole plot/block variance" but on the other hand you want to dispense with modelling any correlation (whether variogram, or correlogram, etc) because it's "time consuming." Not sure how to tackle this one, something has to give. How about implementing an inverse distance weighting scheme to derive the within block variance instead of working with a variogram measure?

        Cheers,

        Syed

        >> Note that gamma(V,V) is a variogram model property, not a data
        >> property: in the figure you mention, the big dots are data points,
        >> the small dots discretize the block but do not indicate observed
        >> data.
        >
        >I know that you can discretize block estimates with a pretty good accuracy
        >with a limited amount of points. But, I was convinced that I could use
        >grided observed data and gamma(V,V) as an estimate for the variance within
        >my grided block.
        >
        >To clarify,
        >
        >I did measure the point values in a grid in a plot/block. I don't have to
        >estimate them, I just need a spacially depended estimate of the whole
        >plot/block variance.
        >
        >The problem is that I would like to automate the procedure of calculating
        >that within block variance because variogram models need to be fit
        >visually most of the time, wich is time consuming. When repeating
        >calculations on randomized data this isn't an option. So if gamma(V,V)
        >really is model property rather then a data property I have a problem...
        >
        >If you can see the trees for the forest, because I don't.
        >Koen.
        >
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        >
        >

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      • Koen Hufkens
        ... This is indeed an option and a simple one. The original sampling design was roughly tested with a RMSE/MSE analysis. Or, taking the statistics of all the
        Message 3 of 9 , Mar 2, 2004
          > If you have gridded observed data at close and regular spacing for the
          > block, then why bother with a Gamma(V,V)? Just calculate the variance
          > for the block directly.

          This is indeed an option and a simple one. The original sampling design
          was roughly tested with a RMSE/MSE analysis. Or, taking the statistics of
          all the grid data as a reference and then sampling smaller and smaller
          portions of this data to get an idea of the block variance and how many
          samples could suffice.

          > On the one hand you need "a spatially dependent estimate of the whole
          > plot/block variance" but on the other hand you want to dispense with
          > modelling any correlation (whether variogram, or correlogram, etc)
          > because it's "time consuming."

          Time consuming isn't exactly bad but I think that if I have to repeat
          fitting curves a lot of times because of the randomizations etc. Result, I
          would be fitting curves more then I would like and probably is useful.

          > Not sure how to tackle this one, something has to give.

          Indeed, that's what I'm finding out right now. Implementing a technique in
          my work that's as efficient as possible without losing to much accuracy.

          > How about implementing an inverse distance weighting scheme to derive
          > the within block variance instead of working with a variogram measure?

          I'll look into that...
          Thanks,
          Koen.

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        • Koen Hufkens
          ... Hard one without the mathematical writing, I looked it up and you refer to following calculations I presume (end of the page):
          Message 4 of 9 , Mar 2, 2004
            > ...
            > gamma_Y(x,x') = 1/|B|^2 \sum_{w in B} \sum_{w' in B} gamma_Z(x+w,x'+w')
            >
            > stays a biproduct, which would more or less (or more less) allow to infer
            > gamma_Z from gamma_Y. However you don't seem to need relation. You can go
            > along with gamma_Y only.

            Hard one without the mathematical writing, I looked it up and you refer to
            following calculations I presume (end of the page):

            http://uk.geocities.com/drisobelclark/PG1979/Chapter_3/Part2.htm

            > The first problem is that you measure according to equation (2) but you
            > want
            > to estimate according to equation (1). Anyway (2) should well approximate
            > (1).

            It could do the trick, but then again... as someone asked before is it
            worth the effort, considering the pretty dense sampling grid in the
            plot/block maybe this isn't the best way to approach the problem!?

            > The second problem is: The variogram of Y should be very smooth in the
            > origing, but it is most sure not Gaussian.

            Correct on that one.
            Or, I've got data that is highly skewed (LAI, leaf area indexes) and other
            data that isn't (that much, reflectance/albedo measurements). So getting a
            variogram in the first place has horror qualities (transforming,
            backtransforming and that mess) even if I do the actual fitting on all the
            data from multiple plots/blocks.

            Thanks for all the input,
            Koen.

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