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GEOSTATS: Summary of: How to detect a trend ?

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  • Klemens Barfus
    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.
      I got several answers:
      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
      > 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
      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 Barfus
      Department of Geography
      University of Wuerzburg

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