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1940[ai-geostats] question about kriging with skewed distribution

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  • Ing. Marek Brabec PhD
    Mar 4, 2005
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      hello,
      I have a question about what is/should be typically done when kriging is
      used for spatial interpolation of a process X(z) where z gives spatial
      location (e.g. z=(x,y) with cartesian coordinates x,y) and X(z) has a
      skewed continuous distribution with nonnegative support. For instance
      lognormal.
      Now,
      if all data are in the form of point samples, X(z)'s can obviously be
      transformed by taking logs to Y(z)=log(X(z)) which are exactly (with
      lognormal X's) or approximately Gaussian, so that kriging can be done
      comfortably (and the result backtransformed with easy correction for the
      fact that E f(X) is generally not equal to f(E X), based on the formula
      for lognormal expected value or Taylor expansion).
      If at least some data are not point samples, but correspond to the
      regional averages, then problem occurs due to the facts that: i) sum
      of lognormals is not lognormal, ii) the log of the sum (or average)
      of lognormals is not normal.
      Obviously, one can do:
      i) the kriging on logs anyway with some hand-waving (effectively
      replacing sums by products based on delta method),
      ii) or one can (quite inefficiently) work with original data without log
      transformation and argue that at least method of moments estimators
      are invoked (with proper weighting),
      iii)or one can use some kind of Monte Carlo computationally-intensive
      approach to compute likelihood (or posterior) based on sums of
      lognormals.
      At this point, I am not interested in either of the three. My question
      is whether people used some other parametric family (it cannot be
      lognormal) of marginal distributions with positive support, positive
      skew, that is closed under convolution (or under taking weighted
      averages, to be more general) - so that the regional averages and point
      values will have distribution of the same type, differing only in
      parameters (just like in normal case and real support case). One
      possibility would be gamma, what about others?
      Thanks in advance for any suggestions.
      Best Regards
      Ing. Marek Brabec, PhD
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