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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