Comparing predictors

From: Lionel B (me_at_privacy.net)
Date: 06/24/04 

Date: Thu, 24 Jun 2004 17:08:35 +0100

Greetings,

Suppose I have (jointly distributed, real-valued) random variables Y,
X_1, ..., X_n and a real function f(x_1, ..., x_n) which is used to
define the "predictor" Y' = f(X_1, ..., X_n) for Y. It is then standard
to measure the quality of Y' by the mean square error E((Y'-Y)^2).

Now I have the following situation: I have two such predictors; the
first, derived from f(x_1, ..., x_n), say, is actually quadratic in the
x_i (it is, in fact a simple least squares fit to a 2nd order
polynomial). The other, derived from g(x_1, ..., x_n), say, is the
output of a (trained) neural network.

I now have the suspicion that the neural network predictor Y'' = g(X_1,
..., X_n) is actually doing "more-or-less the same thing" as the
quadratic fit predictor Y' = f(X_1, ..., X_n). But how do I test this
suspicion?

So far the only way sensible way I can think of for comparing predictors
is to measure their correlation. Indeed, in my case the correlation
corr(Y',Y'') comes out at a significantly high (approx.) 0.8. But
correlation alone doesn't somehow quite seem to confirm that the neural
network really is doing more-or-less the same as the quadratic fit - it
misses out on the joint distribution of the respective predictors with
the independents X_1, ..., X_n.

If this sounds confused, it is ... my real question is probably: what
question should I be asking here? 

-- 
Lionel B