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.

I got several answers:

From Don Myers:> There are actually two questions; is there a trend in the DATA, is the

have to relate the questions. Some indicators of a trend and hence indicators

> mean of the random function non-constant? These are not the same.

> Unfortunately since the data is a sample from only one realization we

of a non-constant mean>

Ø Tucson, AZ 85721

> 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

> is quadratic or higher?

>

> 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

> coordinates). This leads to additional Lagrange multipliers in the

> 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

From Andrew (ne100fia@...)> What I would do is subtract the original data from the trend and see if

I found these page of isaaks very worth to read !

> 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

From Robert Reynolds:> If you are testing for trend with low-order polynomials, you can do

He suggested as a reference:

> significance

> tests on the coefficients.

> Statistics and Data Analysis In Geology

ANOVA for

> John C. Davis

> Second Edition

> ISBN: 0-471-08079-9

>

> Details on pages 419-425

>

> Davis uses ANOVA for significance of Regression and

> Significance of

From Ulrich Leupold:

> Increase of polynomial degree.

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

Sent through GMX FreeMail - http://www.gmx.net

--

*To post a message to the list, send it to ai-geostats@....

*As a general service to list users, please remember to post a summary

of any useful responses to your questions.

*To unsubscribe, send email to majordomo@... with no subject and

"unsubscribe ai-geostats" in the message body.

DO NOT SEND Subscribe/Unsubscribe requests to the list!