1. Nonlinear regression correlation coefficient

A question I've been pondering recently is the derivation of the correlation coefficient for nonlinear regression. I'm more interested in the LA approach so I thought I'd ask here.

I just looked at this paper and noted the formula for r. (Second panel down on the left.) Is it really this simple a formula? If so what is the geometric meaning of it? I've never seen that form when talking about linear regression (just the form with all the sums and no geometric explanation.)

-Dan

PS For those who don't want to look up the page
$r = \sqrt{1 - \frac{ \sum_{i = 1}^n ( y_i - yf_i )^2 }{ \sum_{i = 1}^n (y_i - \bar{y} )^2 }}$
where yf_i are the "fitted" values.

2. Re: Nonlinear regression correlation coefficient

Hey topsquark.

The geometric meaning can be associated in terms of an inner product with the Cauchy-Schwartz inequality:

Cauchy?Schwarz inequality - Wikipedia, the free encyclopedia

If you look at a lot of the results regarding variance operators and positive definite-ness, you will see a relationship between metrics/norms and the variance operator.

This correlation aspect has a relationship with an inner product (normalized) just like Cauchy-Schwrdinartz except we are talking about variation and relationship between random variables regarding the relationship of variation between variables and not about a geometric relationship between the orientation of two vectors (which is what a normalized inner product looks at).

So yes in short, think about the correlation in terms of an angle.

3. Re: Nonlinear regression correlation coefficient

(sighs) So simple once it's explained! Thanks mucho.

-Dan