AI-GEOSTATS: Summary: indicator kriging with a trend
- Dear Colleagues,
Two responses to my post to the list concerning indicator kriging in the
presence of a trend were posted to the list by Isobel Clark and Pierre
Goovaerts, and will not be repeated here. I got a third response from
Donald Myers (pasted at the bottom of this message), complementing on
I tested two approaches:
In the first approach, I used Universal kriging with my binary data and
assumed a linear trend, because most of the signs of non-stationarity
disappeared from the semivariance estimators after linear trend removal.
In the second approach (suggested by Donald), I performed a logistic
regression of my binary data as a function of x and y coordinates, used
the residuals to estimate and model the semivariance, used ordinary
kriging of the logistic regression residuals, and added back the trend
surface predicted by the logistic model.
Both methods generated values outside the range 0..1, but these points
were located outside boundaries delimited by the sampling points, so, they
were easily masked in the prediction map. Both methods provided similarly
good results, although the second method provided slightly finer contours.
In any case, they both provided much more realistic predictions than when
the trend was ignored. Those interested in seing additionnal material
regarding this (pictures, used semivariograms etc...), please feel free to
Thank you for your help,
A couple of additional observations.
As you have noted and as Pierre has suggested, for real valued data (as
opposed to 0-1 data) there are both "theoretical" and "practical" ways
to deal with a non-stationarity. One way, already mentioned, is to fit a
Trend Surface to the data, compute the residuals and then estimate/model
the variogram using the residuals. You could then krig the residuals and
add back the Trend Surface. This and the use of a small search
neighborhood are "practical" ways to handle the non-stationarity. Note
also that some authors have suggested the use of "Median Polish", see
for example some papers by N. Cressie, in place of the Trend Surface.
Universal Kriging is the "theoretical" way to deal with the
non-stationarity but the problem is how to estimate and model the
variogram (or generalized covariance) See an old paper by Pierre
Delfiner in the proceedings of the NATO conference of 1975 (Advanced
Geostatistics in the Mining Industry, D. Reidel, 1976). Allso see the
book co-authored by Chiles and Delfiner.
If you are tryin to estimate a variogram you need "residuals", Matheron
has shown (see his 1971 Summer School Notes) that kriging is the optimal
way to estimate the drift (non-constant mean), unfortunatley you need
the variogram first so you have a circular problem. Hence the interest
in "practical" alternatives.
Now however, your problem is slightly different. For the usual forms of
kriging, second order or intrinsic stationarity is the right kind. This
means that one is only interested in trasnlation invariance of the first
and second order moments. In the case of Indicator Kriging, however one
really needs a slightly stronger former of stationarity, namely,
translation invariance of the marginal distribution function and of the
bi-variate distribution functions.
Since there is nothing in the derivation of thekriging equations that
ensures that the kriged values will be of the same "kind" as the data
(in your case the data are 0's, 1's) you have to worry about
interpretation. For Indicator kriging, the values are usually
interpreted as cumulative probabilities. This suggests that perhaps
instead of an ordinary Trend Surface you may want to use something
closer to a logistic regression. I don't think I have seen this done but
it is reasonable.
Since you are apparently coding your data as simply, the tree is
infested or not infested, you didn't really do an indicator transform
(you don't have multiple cuttoffs).
I think you will find a couple of somewhat relevant papers in the
proceedings of the GEOENV conferences (the most recent one was just held
Donald E. Myers
Dr. Marius Gilbert
Collaborateur Scientifique FNRS
Laboratoire de biologie animale et cellulaire
Universite Libre de Bruxelles CP 160/12
50, av F.D. Roosevelt 1050, Bruxelles BELGIUM
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