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AI-GEOSTATS: Answers on question about simulation

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  • Joe Geo
    Dear ai-geostats A question about simulation I understand in the simulation methods SGS that each node the data a kriging is performed than an value is drawn
    Message 1 of 1 , Jun 9, 2003
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      Dear ai-geostats

      A question about simulation

      I understand in the simulation methods SGS that each node the data a kriging
      is performed than an value is drawn from a normal distribution with the
      local mean (from data and previously simulated nodes captured in a search
      neigbourhood) and variance defined by the kriging variance.

      My understanding of kriging variance is that this variance is really an
      index of data configuration independent of the data values. This leads to
      my question.

      How a set of multiple simulations capture information about the variability
      of conditioning data values when the kriging variance is only a data
      location index. Specifically, it is possible to have the same conditioning
      data configuration but the variability of the values attached to the data
      can be quite different. How is this recognised in the simulation process.



      Hi Joe,

      After the backtransform, the distribution of simulated values
      at each node is not Gaussian anymore and its variance can be
      used as a local index of uncertainty, which accounts for both
      the range of surrounding values and their closeness in
      terms of data configuration.


      Pierre Goovaerts


      Dr. Pierre Goovaerts
      President of PGeostat, LLC
      Chief Scientist with Biomedware Inc.
      710 Ridgemont Lane
      Ann Arbor, Michigan, 48103-1535, U.S.A.

      E-mail: goovaert@...
      Phone: (734) 668-9900
      Fax: (734) 668-7788


      Answer Donald E. Myers

      A couple of additional observations about SGS. This algorithm is based on
      properties of the multivariate normal distribution. Let Z0, Z1,....., Zn be
      jointly distributed with all having finite variances then the best estimator
      of Z0 given Z1,...., Zn is the conditional expectation of Z0 given Z1,...,
      Zn. "Best" in this case means unbiased and with minimal estimation variance.
      IF in addition the joint distribution is multivariate normal then the
      conditional expectation has a particularly simple form, i.e. it is the same
      as the simple kriging estimator. MOREOVER the conditional distribution of Z0
      given the Z1,..., Zn is normal AND the mean of this distribution is the
      conditional mean and its variance is the estimation variance. All of the
      above is true without any reference to spatial problems (except in drawing
      the analogy to simple kriging).

      One critical practical problem then that arises is how does one know that
      the assumption of multivariate normality is satisfied? In practice one does
      not, it is simply an assumption. Note that doing a statistical test on the
      data does not answer the question, the data is not the right kind to test
      for multivariate distributional properties.

      In the SGS algorithm we sort of turn things around. The simple kriging
      equations are derived without any distributional assumptions and we know how
      to compute the kriging variance and the kriged estimate without the
      distributional assumptions. IF the multivariate normality assumption were
      true then the simple kriging value would be the conditional mean, i.e., the
      mean of the conditional distribution and the kriging variance would be the
      variance of the conditional distribution. As you have noted the kriging
      variance is computed without using the data values and is completely
      determined by (1) the covariance model, (2) the pattern of the data
      locations. Under the multivariate normality assumption however this is the
      right quantity. There is no way to get away from that important assumption.
      A histogram and other descriptive statistics are useful in deciding whether
      that assumption is "reasonable" but you can't absolutely answer the

      Donald E. Myers

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