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Math Help - Nonlinear regression correlation coefficient

  1. #1
    Forum Admin topsquark's Avatar
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    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.
    Last edited by topsquark; July 10th 2013 at 05:33 PM.
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  2. #2
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    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.
    Thanks from Shakarri and topsquark
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    Forum Admin topsquark's Avatar
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    Re: Nonlinear regression correlation coefficient

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

    -Dan
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