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,

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