## 1940[ai-geostats] question about kriging with skewed distribution

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• Mar 4, 2005
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hello,
I have a question about what is/should be typically done when kriging is
used for spatial interpolation of a process X(z) where z gives spatial
location (e.g. z=(x,y) with cartesian coordinates x,y) and X(z) has a
skewed continuous distribution with nonnegative support. For instance
lognormal.
Now,
if all data are in the form of point samples, X(z)'s can obviously be
transformed by taking logs to Y(z)=log(X(z)) which are exactly (with
lognormal X's) or approximately Gaussian, so that kriging can be done
comfortably (and the result backtransformed with easy correction for the
fact that E f(X) is generally not equal to f(E X), based on the formula
for lognormal expected value or Taylor expansion).
If at least some data are not point samples, but correspond to the
regional averages, then problem occurs due to the facts that: i) sum
of lognormals is not lognormal, ii) the log of the sum (or average)
of lognormals is not normal.
Obviously, one can do:
i) the kriging on logs anyway with some hand-waving (effectively
replacing sums by products based on delta method),
ii) or one can (quite inefficiently) work with original data without log
transformation and argue that at least method of moments estimators
are invoked (with proper weighting),
iii)or one can use some kind of Monte Carlo computationally-intensive
approach to compute likelihood (or posterior) based on sums of
lognormals.
At this point, I am not interested in either of the three. My question
is whether people used some other parametric family (it cannot be
lognormal) of marginal distributions with positive support, positive
skew, that is closed under convolution (or under taking weighted
averages, to be more general) - so that the regional averages and point
values will have distribution of the same type, differing only in
parameters (just like in normal case and real support case). One
possibility would be gamma, what about others?
Thanks in advance for any suggestions.
Best Regards
Ing. Marek Brabec, PhD
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