[R-sig-Geo] Question about back-transformation(geoR)

[Paulo Justiniano Ribeiro Jr]

Samuel

No, it is not exp(prediction on the normal scale)
by
exp(porediction in the normal scale + 0.5*variance or the prediction in 
the normal scale)

geoiR folows lcosely what is describe in Diggle and Ribeiro (2007)

The Section 3.8 (transformed Gaussian models) it describes the 
basis for the implemented computations :
---------------------------------------------------------------------
The log-transformation is perhaps the most widely used in
practice, and explicit expressions can be derived for its
mean and covariance structure. Suppose that $T(x)= \exp\{S(x)\}$,
where $S(x)$ is a stationary Gaussian process with mean
$\mu$, variance $\sigma^2$ and correlation function
$\rho(u)$. The moment generating function of $S(x)$ is
\begin{equation}
M(a) = {\rm E}[\exp\{aS(x)\}]
      =\exp\{a \mu + \frac{1}{2} a^2 \sigma^2\}.
\end{equation}
It follows from~(\ref{eqn03:mgf}), setting $a=1$, that $T(x)$ has
expectation
\begin{equation}
\mu_T = \exp\left(\mu + \frac{1}{2} \sigma^2\right).
\end{equation}
Similarly, setting $a=2$ in (\ref{eqn03:mgf}) gives ${\rm E}[T(x)^2]$,
and hence the variance of $T(x)$ as
\begin{equation}
\sigma_T^2 = \exp(2 \mu + \sigma^2)\{\exp(\sigma^2)-1\}.
\label{eqn03:lognormalvar}
\end{equation}
Finally,
for any two locations $x$ and $x^\prime$,
$T(x)T(x^\prime) = \exp\{S(x)+S(x^\prime)\}$, and
$S(x)+S(x^\prime)$ is Gaussian with mean $m=2 \mu$ and
variance $v=2\sigma^2\{1+\rho(||x-x^\prime||)\}$. It follows
that
${\rm E}[T(x)T(x^\prime)] = \exp(m+v/2)$, and straightforward
algebra gives the
correlation function of $T(x)$ as
\begin{equation}
\rho_T(u) = [\exp\{\sigma^2 \rho(u)\}- 1]/
           [\exp\{\sigma^2\} - 1].


Paulo Justiniano Ribeiro Jr
LEG (Laboratorio de Estatistica e Geoinformacao)
Universidade Federal do Parana
Caixa Postal 19.081
CEP 81.531-990
Curitiba, PR  -  Brasil
Tel: (+55) 41 3361 3573
VOIP: (+55) (41) (3361 3600) 1053 1066
Fax: (+55) 41 3361 3141
e-mail: paulojus AT  ufpr  br
http://www.leg.ufpr.br/~paulojus

On Fri, 8 Oct 2010, Samuel Turgeon wrote:

> Hi,
>
> I'm working with the function krige.conv of the library geoR. I'm wondering
> how the predicted result are back-transformed when the lambda parameter is
> set to 0 (log-transformation)? Is it a simply transformation like
> exp(result$prediction)? I saw some code example on the web where the exp()
> function was used after the kriging of log-transformed data. Those examples
> were using gstat and the krige function. I have limited knowledge in
> geostatistic, but I think that we cannot use exp() to do the
> backtransformation of the predicted data.... that's why I would like to know
> how the result are back-transformed in the function krige.conv.
>
>
> Thanks
>
> 	[[alternative HTML version deleted]]
>
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