Re: Covariance with bounds in leasqr

[Pablo]

On 03/22/2012 03:41 AM, Olaf Till wrote:
> On Wed, Mar 21, 2012 at 06:31:51PM -0300, Pablo wrote:
> > Hi everyone,
> >    I've been intrigued by a detail in leasqr (from Octave-Forge) for
> > a while. In the function's documentation it's stated that "If
> > constraints (or bounds) are set, returned guesses of corp, covp, and
> > Z are generally invalid, even if no constraints are active for the
> > final parameters".
> >
> >    I wonder why is that, particularly for covp (it's usually the most
> > useful for me). Unfortunately, I don't seem to have access to the
> > reference given (Bard) so I can't check if that's mentioned there.
> > Also, a few web searches didn't give me any clue.
> >    I'm familiar with the way the covariance of the parameters is
> > calculated for a linear fit, and as far as I can see it's
> > essentially the same for leasqr.
> >
> >    Could anyone give me some pointers as to why that statement is
> > true? Any references or links to help me understand it would also be
> > highly appreciated.
> >
> >    Best Regards,
> >      Arnoques
>
> This is not specific to leasqr. Roughly speaking, the covariance
> matrix of estimated parameters is a guess on how estimated parameters
> change due to random in the input data. But if they change, they might
> hit (and so are influenced by) a constraint, even if the constraint is
> not active for the current estimation of parameters. This prevents an
> easy estimation of the covariance matrix of parameters from (possibly
> estimated) (co)variance of input data.
>
> Olaf

Ok, so the covariance estimation is only affected by the bounds if the 
probability distribution for the parameters is not negligible at the bounds.

Just to check if I got that right, if I set [0,1] as the bounds for the 
parameter, and the fit gives me 0.3 with a variance of 0.1, the variance 
is fine. If, on the other hand, the fit gives me a variance of 1, this 
value is meaningless.

Thank you,
  Arnoques