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AI-GEOSTATS: Summary: indicator kriging with a trend

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  • Marius Gilbert
    Dear Colleagues, Two responses to my post to the list concerning indicator kriging in the presence of a trend were posted to the list by Isobel Clark and
    Message 1 of 1 , Dec 18, 2002
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      Dear Colleagues,

      Two responses to my post to the list concerning indicator kriging in the
      presence of a trend were posted to the list by Isobel Clark and Pierre
      Goovaerts, and will not be repeated here. I got a third response from
      Donald Myers (pasted at the bottom of this message), complementing on
      Pierre's message:

      I tested two approaches:

      In the first approach, I used Universal kriging with my binary data and
      assumed a linear trend, because most of the signs of non-stationarity
      disappeared from the semivariance estimators after linear trend removal.

      In the second approach (suggested by Donald), I performed a logistic
      regression of my binary data as a function of x and y coordinates, used
      the residuals to estimate and model the semivariance, used ordinary
      kriging of the logistic regression residuals, and added back the trend
      surface predicted by the logistic model.

      Both methods generated values outside the range 0..1, but these points
      were located outside boundaries delimited by the sampling points, so, they
      were easily masked in the prediction map. Both methods provided similarly
      good results, although the second method provided slightly finer contours.
      In any case, they both provided much more realistic predictions than when
      the trend was ignored. Those interested in seing additionnal material
      regarding this (pictures, used semivariograms etc...), please feel free to
      contact me.

      Thank you for your help,



      A couple of additional observations.

      As you have noted and as Pierre has suggested, for real valued data (as
      opposed to 0-1 data) there are both "theoretical" and "practical" ways
      to deal with a non-stationarity. One way, already mentioned, is to fit a
      Trend Surface to the data, compute the residuals and then estimate/model
      the variogram using the residuals. You could then krig the residuals and
      add back the Trend Surface. This and the use of a small search
      neighborhood are "practical" ways to handle the non-stationarity. Note
      also that some authors have suggested the use of "Median Polish", see
      for example some papers by N. Cressie, in place of the Trend Surface.

      Universal Kriging is the "theoretical" way to deal with the
      non-stationarity but the problem is how to estimate and model the
      variogram (or generalized covariance) See an old paper by Pierre
      Delfiner in the proceedings of the NATO conference of 1975 (Advanced
      Geostatistics in the Mining Industry, D. Reidel, 1976). Allso see the
      book co-authored by Chiles and Delfiner.

      If you are tryin to estimate a variogram you need "residuals", Matheron
      has shown (see his 1971 Summer School Notes) that kriging is the optimal
      way to estimate the drift (non-constant mean), unfortunatley you need
      the variogram first so you have a circular problem. Hence the interest
      in "practical" alternatives.

      Now however, your problem is slightly different. For the usual forms of
      kriging, second order or intrinsic stationarity is the right kind. This
      means that one is only interested in trasnlation invariance of the first
      and second order moments. In the case of Indicator Kriging, however one
      really needs a slightly stronger former of stationarity, namely,
      translation invariance of the marginal distribution function and of the
      bi-variate distribution functions.

      Since there is nothing in the derivation of thekriging equations that
      ensures that the kriged values will be of the same "kind" as the data
      (in your case the data are 0's, 1's) you have to worry about
      interpretation. For Indicator kriging, the values are usually
      interpreted as cumulative probabilities. This suggests that perhaps
      instead of an ordinary Trend Surface you may want to use something
      closer to a logistic regression. I don't think I have seen this done but
      it is reasonable.

      Since you are apparently coding your data as simply, the tree is
      infested or not infested, you didn't really do an indicator transform
      (you don't have multiple cuttoffs).

      I think you will find a couple of somewhat relevant papers in the
      proceedings of the GEOENV conferences (the most recent one was just held
      in Barcelona).

      Donald E. Myers

      Dr. Marius Gilbert
      Collaborateur Scientifique FNRS
      Laboratoire de biologie animale et cellulaire
      Universite Libre de Bruxelles CP 160/12
      50, av F.D. Roosevelt 1050, Bruxelles BELGIUM

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