[R-sig-Geo] Analytical covariance expression of backtransformed Box-Cox data

[Ole F. Christensen]

Dear Anders

As far as I know, an exact expression is not available in any nice form.
However you can create an approximation using a first order Taylor 
expansion of f around y_0=E[Y_i], assuming a constant mean,

f(y) \approx f(y_0)+ (y-y_0) f`(y_0)

and then use the approximation to calculate the covariance.

It may not be accurate, and you can improve on it by using a higher 
order Taylor approximation instead.

Do you actually need the covaraince cov(X_i,X_j) ? or do you only want 
mean and varinnce ?

Ole


Anders Malmberg wrote:

> Dear Readers,
>
> I have a question not related to R itself. It is of a theoretical 
> nature. I hope this
> is ok.
>
> I am facing the problem where some spatial data follows the Gaussian 
> distribution
> after a Box-Cox transformation. I am using geoR and I understand that 
> geoR (in the latest version)
> for lambda \neq 0 or 0.5 calculates the mean and variance through 
> simulation
> (using backtransform.moments()).
>
> If X is original data and Y is Box-Cox transformed data for which I have
> estimates of the parameters in exponential covariance function, then
> it should be that (mean=0),
>
> cov(X_i,X_j)=E(X_i X_j) = E(f(Y_i) f(Y_j))
>
> where "f" is the inverse of the Box-Cox transform. But this turns out 
> to be quite messy.
> Is it possible to find an analytical expression for any lambda?
>
> Thanks in advance,
>
> Anders
>
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-- 
Ole F. Christensen
BiRC - Bioinformatics Research Center
University of Aarhus