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RE: GEOSTATS: Test of equal variograms

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  • Daniel Bebber
    I asked a similar question a while back, and was sent the following reference by Andrew Lister Kabrick J. M., Clayton M. K. & McSweeney K. 1997. Spatial
    Message 1 of 6 , Feb 15, 2000
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      I asked a similar question a while back, and was sent the following
      reference by Andrew Lister

      Kabrick J. M., Clayton M. K. & McSweeney K. 1997. Spatial patterns of carbon
      texture on
      drumlins in northeastern Wisconsin. Soil Sci. Soc. Am. J. 61(2):541-548

      This contains a method for estimating the errors on a semivariogram value,
      and testing for differences between them. If anyone has any comments on this
      methodology I would be interested to hear them.

      Dan
      _____________________________________
      Mr. Daniel P. Bebber
      Department of Plant Sciences
      University of Oxford
      South Parks Road
      Oxford OX1 3RB
      UK
      Tel. 01865 275000 Fax. 01865 275074

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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 2 of 6 , Feb 15, 2000
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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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