## GEOSTATS: SUMMARY: Trend Analysis

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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
Message 1 of 1 , Nov 8, 1998
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 SouliĆ© (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. SouliĆ©. 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,
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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