One should be a little careful about accepting the validity of a test for
the "equality" of two variograms. If one uses an estimator such as the
sample variogram, one only obtains estimates of the values of the variogram
for a finite number of lags (note that dealing with a possible anisotropy
makes it even more complicated). Moreover the reliability of these
estimates varies, in part because the numbers of pairs will vary. If one is
using the variogram for kriging or simulation then one is most interested
in the behavior of the variogram, i.e., the values for short lags and
unfortunately the short lags usually have the smallest numbers of pairs. If
one uses least squares or maximum likelihood then one must first choose a
model (or models in the case of a nested model) and then one of these is
used to estimate the parameters.
There is an old paper by Davis and Borgman in Mathematical Geology (circa
1980) on the distribution of the sample variogram, they give two results:
(1) beginning with an assumption of multivariate normality (which is not
testable) and an assumed model type then they obtain numerical results for
the distribution , (2) they obtain asymptotic results which are
theoretically interesting but probably not much help in practice.
There is also a paper in Mathematical Geology, circa 1990, on the "true"
numbers of pairs. The problem as is well known is that there is an
interdependence between the pairs used to estimate for one lag and those
used to estimate for another. The author has to assume multivariate
normality to derive the results.
It is known that the kriging estimator is relatively robust with respect to
the variogram, i.e., slight changes in the variogram will result in only
slight changes in the kriging weight vector and hence in general only
slight changes in the kriged values. There are at least two different ways
to quantify the "distance" between two variograms, these correspond to a
notion of continuity. A third one corresponds to differentiability, none of
the three implies the others.
In practice one often uses a search neighborhood in kriging hence it is
only of interest whether the variograms match or are at least close for up
to some maximum lag. One will have very little information about the
variogram for longer lags anyway.
In general statistical tests will require some distributional assumptions
and these are hard to obtain for variograms/variogram estimators. It is an
interesting question to ask, i.e., are the variograms for two different
variables or the same variable for two different regions the same but one
that will be hard to test without making very strong assumptions
Finally one might want to consider the question of sample location pattern
design relative to testing the equality of two variograms. I have an old
paper with A.W. Warrick on the design of sampling plans in order to control
the numbers of pairs for each lag. If one assumes isotropy (it is even more
complicated in the case of anisotropy) then the pattern that generates an
equal number of pairs is a spiral, not a very practical result.
Note also that if one assumes normality then the distribution of the
half-squared differences will be Chi-Squared (one can see this effect in
most sample variograms, the VARIO component of GEOEAS will provide
histograms for these distributions). Not a particularly nice distribution
for testing because of the "fat" tails.
Donald E. Myers
Department of Mathematics
University of Arizona
Tucson, AZ 85721
At 05:02 PM 2/14/00 -0800, you wrote:
>Assume that we have two sets of geostatistical data. Is there any
>statistical test to determine whether variograms on those two sets are the
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