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GEOSTATS: Re: Normality

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  • Donald Myers
    Just a word of caution about tests for normality, i.e., your question about whether normality is required for geostatistical tools. In geostatistics the data
    Message 1 of 6 , Feb 14, 2000
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      Just a word of caution about tests for normality, i.e., your question about
      whether normality is required for geostatistical tools.

      In geostatistics the data is considered to be a non-random sample from one
      realization of the random function (otherwise the underlying concepts of
      geostatistics would not apply). There are at least two different notions of
      normality to consider:

      1. If one considered the data obtained from all the locations in the region
      of interest, would the distribution of this data be normal? The data is a
      sample from this distribution but is not a random sample (random location
      selection is not quite the same thing as random sampling).

      Note that the region of interest is often defined after the fact, i.e.,
      after data has been collected.

      2. Assuming that the random function is strongly stationary, is the
      univariate distribution normal ? Note that the data is not a sample from
      this distribution.

      The standard tests for normality are based on random sampling FROM the
      distribution, it is difficult to modify the tests to allow for the spatial
      correlation (especially without knowing the "true" variogram/covariance).
      Hence one should be cautious in treating the results of a test of
      hypothesis (for normality) as really definitive.

      Although not often mentioned, we use a form of egodicity in geostatistics
      but the simplest form of this pertains to moments not distributions. For
      example, the weak law of Large numbers implies that the sample proportion
      (for a random sample) will converge (in a certain sense) to the population
      proportion. Similarly, we expect the sample mean to converge to the
      population mean (as the sample size increases) but note that the
      distribution of the sample mean tends to the standard normal not to the
      original distribution.

      We do know that if the random function is multivariate normal that the
      simple kriging estimator is the conditional expectation and hence is THE
      minimum variance, unbiased estimator/predictor. In general however it is
      only the minimum variance, unbiased LINEAR estimator/predictor.

      The bottom line however is probably not a statistical question, do the
      geostatistical tools produce useful results? "Useful" has to be decided by
      the user, not the statistician. Across a wide spectrum of applications the
      answer seems to be yes but in specific instances the answer may be no
      because of a lack of data, difficulty in estimating/modeling the variogram,
      sensitivity of any linear estimator to unusual data values, etc.

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

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

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    • Dragoljub Pokrajac - EECS
      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.
      Message 2 of 6 , Feb 14, 2000
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        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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      • 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 3 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 4 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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            >

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