It may be mis-leading to talk about confidence intervals for variograms

since they are functions but are only estimated at a small number of

points. One might talk about confidence intervals for the variogram values

at each of several lags (one confidence interval for each such lag).

More generally one should talk about the robustness of kriging with respect

to the variogram, i.e., if the variogram model is changed slightly (or one

might say is slightly mis-modeled) how much do the results change? Note

that the kriged results depend on both the weights obtained from the

kriging equations (which do not directly depend on the data) and also on

the data, changing the variogram (model types, parameters, etc) only

affects the weights.

There are at least three ways to quantify "closeness" for variograms. Let

g(r), h(r) be two isotropic variograms (isotropy is not essential but

makes it easier to discuss at first)

(1) d(g,h) = sup | g(r) - h(r)|, 0 < r < search radius

This would be analogous to a "confidence interval" with width constant for

all lags, this definition is symmetric in h,g.

(2) d(g,h) = sup | g(r)/h(r) - 1|, 0 < r < search radius

This definition is NOT symmetric in g, h. This is essentially the distance

introduced by Armstrong and Diamond (Math Geology). In this case the width

of the interval is smaller for small lags.

The first two correspond to a notion of continuity, one might also consider

differentiability. SUPPOSE that the space of valid variograms was finite

dimensional (which is not true) and that every variogram could be written

as a nested model of g1, g2,...., gp. Then we might say that g is close

to h "in the direction gj" if

g(r) = h(r) + c* gj

for c a small positive number. This would allow us to define a Frechet

derivative and then use the same idea on the kriging estimator.

Unfortunately these three definitions are not equivalent nor are they

ordered in terms of implications.

Haiyan Cui in her dissertation (University of Arizona, 1994) considered all

three of these and obtained bounds on the change in the weight vector in

terms of changes in the variogram.

Mike Stein (U. Chicago) also has some results on showing the relationship

of the kriging variance using the "wrong" variogram vs the kriging variance

using the "correct" one.

Generally speaking it is known that the kriging estimator is relatively

robust with respect to the variogram, i.e small changes in the variogram

result in small changes in the kriging results.

Bruce Davis in his dissertation (U. Wyoming) obtained some results on

confidence intervals for variogram estimates using m-normality.

With respect to cross-validation there are at least six statistics that

might be evaluated (mean error, mean square error, mean square normalized

error, correlation between observed and estimated values, correlation

between error and estimated, correlation between normalized error and

estimated, fraction of normalized errors exceeding 2.5 in magnitude). These

are not equally sensitive to changes in the variogram and moreover they can

be quite sensitive to changes in the search neighborhood (radius, min and

max number of points used).

Donald Myers

Department of Mathematics

University of Arizona

myers@...

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

--

*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!