- Mar 5, 2005
> hello,

As far as i know, traditional geostatistics as originated in Matheron is

> 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.

distribution-free. The analysis does not require a pre-experimental

probability model for the data, thus it does not rely on any

post-experimental likelihood function. So it does not matter, for kriging,

if the data are skewed or has any shape whatsoever. That is the theory at

least. People may still want to work with symmetrical distributions

because they may not be entirely confortable with the theory?

> Now,

Yes, though the data may only be a little lognormal. If it is exactly

> 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).

lognormal then the parameter of the Box-Cox transformation is 0, but

values like -0.1 or +0.1 can produce more symetrical distributions . This

parameter can be estimated along with spatial correlation function

parameters to let the data decide what precise transformation makes it

look more Gaussian. For this you would need to set up a formal statistical

model for the data instead of following the traditional distribution-free

methodology. Check the info on geoR, a contributred package to R.

> If at least some data are not point samples, but correspond to the

If you don't have raw data but averages then within the likelihood-based

> 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.

approach you may want to think of a marginal likelihood model to carry

over the uncertainty associated with the averaging into the final

analysis. I think this is rather complicated. On the other hand, maybe

there is no such problem within the traditional distribution-free school

because the uncertainty associated to the fitting of the spatial model

ususally is ignored.

[snip the rest for brevity]

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