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GEOSTATS: Confidence intervals for variograms

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  • Donald Myers
    Some thoughts It may be mis-leading to talk about confidence intervals for variograms since they are functions but are only estimated at a small number of
    Message 1 of 1 , Oct 1, 1998
      Some thoughts

      It may be mis-leading to talk about confidence intervals for variograms
      since they are functions but are only estimated at a small number of
      points. One might talk about confidence intervals for the variogram values
      at each of several lags (one confidence interval for each such lag).

      More generally one should talk about the robustness of kriging with respect
      to the variogram, i.e., if the variogram model is changed slightly (or one
      might say is slightly mis-modeled) how much do the results change? Note
      that the kriged results depend on both the weights obtained from the
      kriging equations (which do not directly depend on the data) and also on
      the data, changing the variogram (model types, parameters, etc) only
      affects the weights.

      There are at least three ways to quantify "closeness" for variograms. Let
      g(r), h(r) be two isotropic variograms (isotropy is not essential but
      makes it easier to discuss at first)

      (1) d(g,h) = sup | g(r) - h(r)|, 0 < r < search radius

      This would be analogous to a "confidence interval" with width constant for
      all lags, this definition is symmetric in h,g.

      (2) d(g,h) = sup | g(r)/h(r) - 1|, 0 < r < search radius

      This definition is NOT symmetric in g, h. This is essentially the distance
      introduced by Armstrong and Diamond (Math Geology). In this case the width
      of the interval is smaller for small lags.

      The first two correspond to a notion of continuity, one might also consider
      differentiability. SUPPOSE that the space of valid variograms was finite
      dimensional (which is not true) and that every variogram could be written
      as a nested model of g1, g2,...., gp. Then we might say that g is close
      to h "in the direction gj" if

      g(r) = h(r) + c* gj

      for c a small positive number. This would allow us to define a Frechet
      derivative and then use the same idea on the kriging estimator.

      Unfortunately these three definitions are not equivalent nor are they
      ordered in terms of implications.

      Haiyan Cui in her dissertation (University of Arizona, 1994) considered all
      three of these and obtained bounds on the change in the weight vector in
      terms of changes in the variogram.

      Mike Stein (U. Chicago) also has some results on showing the relationship
      of the kriging variance using the "wrong" variogram vs the kriging variance
      using the "correct" one.

      Generally speaking it is known that the kriging estimator is relatively
      robust with respect to the variogram, i.e small changes in the variogram
      result in small changes in the kriging results.

      Bruce Davis in his dissertation (U. Wyoming) obtained some results on
      confidence intervals for variogram estimates using m-normality.

      With respect to cross-validation there are at least six statistics that
      might be evaluated (mean error, mean square error, mean square normalized
      error, correlation between observed and estimated values, correlation
      between error and estimated, correlation between normalized error and
      estimated, fraction of normalized errors exceeding 2.5 in magnitude). These
      are not equally sensitive to changes in the variogram and moreover they can
      be quite sensitive to changes in the search neighborhood (radius, min and
      max number of points used).

      Donald Myers
      Department of Mathematics
      University of Arizona



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