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GEOSTATS: SUMMARY: Trend Analysis

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  • William C. Thayer
    Thanks to everyone that replied to my request for information on what methods to use to to dermine if a trend exists in a set of data. The replies are shown
    Message 1 of 1 , Nov 8, 1998
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      Thanks to everyone that replied to my request for information on what
      methods to use to to dermine if a trend exists in a set of data. The
      replies are shown below.

      1. From Professeur Paul Chiasson, PhD:

      For a polynomial trend of order k defined as follows :
      m(x)=a_0 x^0 + a_1 x_1 + ... +a_k x^k
      You can compute an experimental generalized covariance of order k, i.e.
      K(h) (Intrinsic Random Function of order k theory) by using a method
      developed by Chiasson and SouliƩ (1997). Then you investigate if :
      K(h)/h^(2k+2) vanishes to 0, when h tends to infinity
      Or you can also compute the variogram (unknown constant expectation),
      linvariogram (linear trend) or quadvariogram described by Cressie
      (1987, 1993). Then you investigate if :
      g(h)/(h^2) tends to 0 when h tends to infinity
      The lowest order k where these expressions vanishes to zero is the order
      of your polynomial trend. See the references for more details. Both
      approaches are equivalent and are based on a property demonstrated by
      Matheron (1973).
      References :
      Chiasson, P. and M. SouliƩ. 1997. Nonparametric Estimation of
      Generalized Covariances by Modeling On-Line Data. Mathematical Geology,
      29(1) :153-172.
      Cressie, N.A. 1993.Statistics for Spatial Data. J. Wiley, New York
      Cressie, N.A. 1987. A nonparametric view of generalized covariances
      for kriging. Math. Geology, 19(5) :425-449.

      2. From John Kern:

      The approach which I have been using is three steps which are repeatable
      and
      defensable if only approximately statistically rigorous,
      1) Start with Ordinary least squares regression to test for
      regression
      effects in ignorance of spatial correlation knowing that the estimates
      of
      the regression coefficients are unbiased but not minimum variance and
      hence
      the statistical inferences are not quite right.
      2) Second test the residuals for for spatial correlation using
      MOran's I.
      3) If Spatial correlations exist in the residuals, then model the
      correlation function or variogram.
      4) Finally, refit the regression using General Least Squares to
      estimate
      the regression coefficients and get tests which incorporate the
      correlations.
      Another alternative is to estimate the corelation structure using
      Restricted
      Maximum Likelihood (Available in SAS) Which estimates the correlations
      and
      regression simultaneously. The draw back is that the variogram
      estimation
      step is left in a black box without the (I believe) preferred step in
      which
      sample variograms are smoothed before fitting a model.

      3. From Ashley Francis:

      This is essentially a question about non-stationary estimation
      problems. The book "Quantitative Hydrogeology" by Ghislain de Marsily
      (Academic Press, 1986) has an excellent section (Chapter 11) on the
      procedures for non-stationary modelling and kriging. In fact the book
      has excellent sections on geostats all the way through - thoroughly
      recommended.

      Thanks again for the help.

      Bill
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