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

Pierre's message:

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

contact me.

Thank you for your help,

Marius

============================================

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

in Barcelona).

Donald E. Myers

http://www.u.arizona.edu/~donaldm

--------------------------------------------------

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

http://lubies.ulb.ac.be

---------------------------------------------------

--

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

* As a general service to the users, please remember to post a summary of any useful responses to your questions.

* To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

* Support to the list is provided at http://www.ai-geostats.org