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Re: GEOSTATS: Test of equal variograms

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  • Donald Myers
    One should be a little careful about accepting the validity of a test for the equality of two variograms. If one uses an estimator such as the sample
    Message 1 of 6 , Feb 15 9:52 AM
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      One should be a little careful about accepting the validity of a test for
      the "equality" of two variograms. If one uses an estimator such as the
      sample variogram, one only obtains estimates of the values of the variogram
      for a finite number of lags (note that dealing with a possible anisotropy
      makes it even more complicated). Moreover the reliability of these
      estimates varies, in part because the numbers of pairs will vary. If one is
      using the variogram for kriging or simulation then one is most interested
      in the behavior of the variogram, i.e., the values for short lags and
      unfortunately the short lags usually have the smallest numbers of pairs. If
      one uses least squares or maximum likelihood then one must first choose a
      model (or models in the case of a nested model) and then one of these is
      used to estimate the parameters.

      There is an old paper by Davis and Borgman in Mathematical Geology (circa
      1980) on the distribution of the sample variogram, they give two results:
      (1) beginning with an assumption of multivariate normality (which is not
      testable) and an assumed model type then they obtain numerical results for
      the distribution , (2) they obtain asymptotic results which are
      theoretically interesting but probably not much help in practice.

      There is also a paper in Mathematical Geology, circa 1990, on the "true"
      numbers of pairs. The problem as is well known is that there is an
      interdependence between the pairs used to estimate for one lag and those
      used to estimate for another. The author has to assume multivariate
      normality to derive the results.

      It is known that the kriging estimator is relatively robust with respect to
      the variogram, i.e., slight changes in the variogram will result in only
      slight changes in the kriging weight vector and hence in general only
      slight changes in the kriged values. There are at least two different ways
      to quantify the "distance" between two variograms, these correspond to a
      notion of continuity. A third one corresponds to differentiability, none of
      the three implies the others.

      In practice one often uses a search neighborhood in kriging hence it is
      only of interest whether the variograms match or are at least close for up
      to some maximum lag. One will have very little information about the
      variogram for longer lags anyway.

      In general statistical tests will require some distributional assumptions
      and these are hard to obtain for variograms/variogram estimators. It is an
      interesting question to ask, i.e., are the variograms for two different
      variables or the same variable for two different regions the same but one
      that will be hard to test without making very strong assumptions
      (non-testable assumptions).

      Finally one might want to consider the question of sample location pattern
      design relative to testing the equality of two variograms. I have an old
      paper with A.W. Warrick on the design of sampling plans in order to control
      the numbers of pairs for each lag. If one assumes isotropy (it is even more
      complicated in the case of anisotropy) then the pattern that generates an
      equal number of pairs is a spiral, not a very practical result.

      Note also that if one assumes normality then the distribution of the
      half-squared differences will be Chi-Squared (one can see this effect in
      most sample variograms, the VARIO component of GEOEAS will provide
      histograms for these distributions). Not a particularly nice distribution
      for testing because of the "fat" tails.

      Donald E. Myers
      Department of Mathematics
      University of Arizona
      Tucson, AZ 85721

      http://www.u.arizona.edu/~donaldm

      At 05:02 PM 2/14/00 -0800, you wrote:
      >Assume that we have two sets of geostatistical data. Is there any
      >statistical test to determine whether variograms on those two sets are the
      >same?
      >
      >Thanks,
      >
      >A. Lazarevic
      >
      >
      >
      >
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