Argument of a linear model

Hey

I have a little problem here:


I have an experimental space, lets say [-1,+1]^2, and I fit a second  
order
model above it. Regarding the whole experimental space the regression
function maps within [-3,+4], which means nothing else than
f^-1([-3,+4])=[-1,+1]
Now for example the question is: What is f^-1([-1,+2])=?

Is there any inverse function available in order to get the argument  
of a
regression function f?

In a general sense such a problem is ill-defined mathematically  
because the inverse of a second order quadratic (my assumption  
regarding what you meant by "second order") will not necessarily be a  
function in the mathematical sense of being one-to-one. You could  
first plot and then see if it makes sense to fit a surface to the  
point set generated by:

   sapply( seq(-1, 1, by=0.01) , f)

(Which could then be limited in your more restricted domain.) You  
could then predict those elements in seq(-1, 1, 0.01) which had  
corresponding images in f(seq(.)).

It would be more productive if you assembled a test case in R code.

David Winsemius, MD
Alameda, CA, USA