Re: Fitting with leasqr and uncertainties

[Olaf Till]

On Mon, Jan 20, 2014 at 01:15:00AM -0800, st.michal88 wrote:
> Hello everybody,
> 
>  i am pretty new with Octave and I need some advice. I am using leasqr
> function to fit my experimental data with exponential function. The problem
> is that uncertainties of parameters are very important to me. I diged up
> internet with this question and didn't find the answer. What I want to get
> at the end is a set of parameters in form: parameter +- uncertainty. Is it
> possible with this function? I found out also about nonlin_curvefit but it
> doesn't give me the answer eighter. Thank you in advance for your help.

You can only get confidence regions for all parameters together. The
choice of the shape of this region is arbitrary, a popular choice is
an ellipsoid. The length of the axes of such an ellipsoid would
probably more or less correspond to the length of the interval of
"uncertainty" you desire for single parameters.

There are different methods to determine such an ellipsoid. Usually
some assumptions on the model functions are taken and the resulting
determination of the ellipsoid is not perfect. It is outside the scope
of an optimization function to decide whether a certain assumption
probably yields a sufficiently accurate determination of the
ellipsoid; this decision must be left to the user.

The ellipsoid can usually be determined by, among others, the
covariance matrix of the parameters. The latter can be returned both
by residmin_stat/curvefit_stat and by leasqr. leasqr can even return a
ready made matrix 'Z' so that, assuming near linearity of the model
equations at the result parameters, dp.'*Z*dp follows an F
distribution (call leasqr with 'global verbose = true;' for details),
where 'dp' is the vector of differences from the result
parameters. This determines an ellipsoid for a given degree of
confidence. leasqr does not compute the length of the ellipsoid axes,
but this is easily done.

Olaf

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