GEOSTATS: Confidence intervals for variograms
- Some thoughts
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).
Department of Mathematics
University of Arizona
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