## GEOSTATS: Summary of: How to detect a trend ?

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• Summary of: How to detect a trend ? Hello ! Some weeks ago I asked how I could detect a trend in my data and how I could get information if it is significant.
Message 1 of 1 , Apr 25, 2000
Summary of: How to detect a trend ?

Hello !
Some weeks ago I asked how I could detect a trend in my data and how I
could get information if it is significant.
From Don Myers:
> There are actually two questions; is there a trend in the DATA, is the
> mean of the random function non-constant? These are not the same.
> Unfortunately since the data is a sample from only one realization we
have to relate the questions. Some indicators of a trend and hence indicators
of a non-constant mean
>
> 1. Plot the data values against the E-W coordinate, also plot a
> regression line (non-zero slope?)
>
> 2. Plot the data values against the N-S coordinate, also plot a
> regression
> line (non-zero slope?)
>
> 3. Does the sample variogram of the original data show a growth rate
> that
>
> 4. Is there a difference between the sample variogram of the original
> data
> and the sample variogram of the residuals?
>
> 5. As an alternative to 1. & 2. or in addition make a coded plot of the
> data locations (each location color coded by the data value at that
> location.
>
> The above questions mostly relate to problems pertaining to
> variogram/covariance estimation. The sample variogram does not estimate
> the variogram (it estimates half of the expected value of the square of
> a
> first order difference, the two are the same if the mean is constant)
> and in order for a variogram model to be valid it has to grow at a rate
> which is less than quadratic.
>
> Having said all that, when fitting a variogram we usually fit it only to
> a
> part of the sample variogram, i.e., up to a fixed lag. The rapid growth
> rate may not show for short lags and hence we may be able to use the
> sample
> variogram for the original data if we only use the beginning lags. That
> is,
> we may be able to fit a valid model to the sample variogram for short
> lags.
> In order to use this model we should be sure to use a moving
> neighborhood
> for the kriging (Matheron has referred to this practice as assuming
> "local" stationarity)
>
>
> Now turn to the question of the kriging. The universal kriging
> estimator/universal kriging equations allow the incorporation of a
> non-constant mean (represented by a polynomial in the position
> solution. These do not explicitly show in the kriging estimator but do
> appear in the kriging variance.
>
> Journel and Rossi (see a paper in Math Geology) discuss "when do we need
> a
> trend model?" with respect to the difference in the results when using
> universal vs ordinary kriging. Note that the use of universal kriging
> does
> not avoid the problems that might be encountered in estimating/modeling
> the
> variogram if there is a non-constant mean.
>
> N. Cressie has written several times on the use of "Median Polish" as a
> technique for "removing" the trend.
>
> Theoretically there is a clear distinction between the random component
> (with constant or zero mean) and the deterministic, non-constant, mean.
> However when we only have data available the distinction/separation is
> not
> so clear.
>
> I suggest avoiding simple minded "black box" solutions, look at your
> data.
> Look at the plots suggested above. Is one or both of the slopes
> "non-zero"
> only because of a few plotted points at one end or the other? I.e., is
> it
> possibly an artifact of the analysis? Would a non-constant mean make
> sense
> for the particular phenomenon you are studying?
>
> In the case of a linear variogram it is often difficult to distinguish
> between an anisotropy and a non-constant mean.
>
> 1991, Myers,D.E., On Variogram Estimation. in Proceedings of the First
> Inter.
> Conf. Stat. Comp., Cesme, Turkey, 30 Mar.-2 April 1987, Vol
> II,
> American
> Sciences Press, 261-281
>
> 1991, Myers,D.E., Interpolation and Estimation with Spatially Located
> Data,
> Chemometrics and Intelligent Laboratory Systems 11, 209-228
>
> 1989, Myers,D.E.,To Be or Not to Be...Stationary:That is the Question.
> Math.
> Geology, 21, 347-362
>
> 1985, J.Tabor, A.Warrick, D. Pennington and D.E. Myers, Spatial
> Variability of
> Nitrate in Irrigated Cotton II: Soil Nitrate and Correlated
> Variances. Soil Sci.
> Soc. Amer.J., 49, 390-394
>
> 1984, J.Tabor, A. Warrick, D. Pennington and D.E. Myers,
> Spatial
> Variability of
> Nitrate in Irrigated Cotton I:Petioles. Soil Sci. Soc.
> Amer.J.,48, 602-607
>
> Donald E. Myers
> Department of Mathematics
> University of Arizona
Ø Tucson, AZ 85721

From Andrew (ne100fia@...)
> What I would do is subtract the original data from the trend and see if
> anys
> apatial autocorrelation exists in the residuals (you can do this very
> easily, for one, in Surfer (www.goldensoftware.com). If it disappears,
> then
> you know that trend is singificant. If the spatial autocorrelation
> remains
> unchanged (the variogram/correlogram appear similar), then you probably
>
> don't need to worry about the trend.
> Look at Isaaks webpage www.isaaks.com and check out his discussion of
> variogram vs. correlogram for dealing with trend; interesting...
>
>
> Andrew
I found these page of isaaks very worth to read !
From Robert Reynolds:
> If you are testing for trend with low-order polynomials, you can do
> significance
> tests on the coefficients.
He suggested as a reference:
> Statistics and Data Analysis In Geology
> John C. Davis
> Second Edition
> ISBN: 0-471-08079-9
>
> Details on pages 419-425
>
> Davis uses ANOVA for significance of Regression and
ANOVA for
> Significance of
> Increase of polynomial degree.
From Ulrich Leupold:
He told me elemantary aspects of trends and stationary in geostatistics, I
have not known and when to use universal kriging etc.
Thanks to all, who have answered me !
Klemens

--
Klemens Barfus
Department of Geography
University of Wuerzburg
Germany

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