Re: gstat and backtransformation

["Edzer J. Pebesma" <e.pebesma@geog.uu.nl>]

Markus Neteler wrote:
> 
> Hi Edzer,
> 
> while being close to have our upcoming GRASS book finalized,
> we are searching for weak points in the text. A question
> came up how to treat back-transformations of log-transformed
> data and the related standard errors:
> 
> > p. 353-359: I don't think that natural logarithms and backtransforming is
> > so simple, certainly it always creates discussion. In particulat
> > backtransforming predictions and their standard errors is controversial,
> > because +/- 1se isn't symmetric any more, is it? If you use natural logs
> > in gstat, then do so here too, but I'd get advice on how to handle the
> > backtransforms for the standard errors. (what does Edzer do with that?)
> 
> Probably you have described this in the gstat manual, but I didn't find
> it yet (back-transformation). I must say, that I am not a long term
> geostats expert (more focused on remote sensing etc).
> 
> Thanks for any hint,
> 
>   Markus Neteler

Markus, this questions pops up every now and then in the geostats scene.
And for good reasons. There are numerous old and recent papers that deal
with it, mainly found in Mathematical Geology.

It's a bit of a nasty business. There is a thing called
"lognormal kriging", which transforms the simple kriging mean
on the log-scale Y* and simple kriging variance on the log-scale s2,
back to estimated expected value on the original scale by:

  Z* = exp(Y* + 0.5 s^2)
  
See Journal and Huybregts, 1989, p. 572. The trouble with this thing
is that it takes huge values when s^2 is large. There's a lot of
literature on this issue whether it still is a good, or a bad thing.
With backtransforming kriging variances, things get even worse it
seems.

Another approach is to stick with medians: exp(Y*) is an estimate of
the median of Z*, and the confidence interval [exp(Y*-2s),exp(Y*+2s)]
is an approximate 95% confidence interval for the median value of Z.

I took this latter approach in my PhD work, see

E.J. Pebesma and J.W. de Kwaadsteniet, 1997, Mapping Groundwater Quality
in the Netherlands. Journal of Hydrology 200, pp. 364-386.

gstat doesn't deal with either -- it only lets the user choose to
log-transform the data for convenience. Back-transforming is the users
responsibility.

On a side (but closely related) issue, see the discussion following
http://www.gstat.org/gstat-info/msg00058.html

Good luck with finishing up your book,
--
Edzer