
If $\w = \v + \e$ with $\v \cdot \e =0$ then
$\v \cdot \w = \v \cdot \v $ so
$\R ^2 = \cos ^ \theta 
= {( v \cdot \w )^2 \over  (\v \cdot \v ) ( \w \cdot \w) }
=
= { v \cdot \v  \over  \w \cdot \w }$.
Also by $\v \cdot \e =0$,
$v \cdot \v  + \e \cdot \e  = \w \cdot \w $
so
$\R ^2 = 
{ w \cdot \w - \e \cdot \e  \over  \w \cdot \w }
= 
1 - { \e \cdot \e  \over  \w \cdot \w } $.
which shows how these two common formulas for this ratio of inner
products in terms of sums of squares, 
${\hbox{SSR} \over \hbox{SST} } = 1 - {\hbox{SSE} \over \hbox{SST} } $
depends on the orthogonality of the fit model $\v$ and the residual $\e$.
Here SSR is the ``sum of squares of the regression"
(not the residuals!),
SST is the ``total sum of squares"
and
SSE is the ``sum of squares of the errors" (or residuals).

\bigskip
Also, in many fits,
both model and measurement are typically normalized to
have mean zero (i.e. when the fit is polynomial 
$\v \to \v - ( \v \cdot \one )$
- this doesn't make sense in an abstract inner product space.)
Inner products are not invariant under
translation of origin!
Hence the alternative term for SST, the sum of squares
about the mean.

\bigskip
Since our model is not the orthogonal fit of the data, we
will simply report  $\R ^2$ computed directly from
inner products.




