Hello.
> I'm obviously missing something in my litterature review. I did a new
> MC simulation, with a much smaller number of observation points
> (namely 3, to fit a straight line!!!). It turns out that the formula
> you are advocating for is the best estimate of the standard deviation
> of the parameters. Could you please explain why this fomula differs
> from formulas (34) and (35) in
> http://mathworld.wolfram.com/LeastSquaresFitting.html?
Independently of the explanation to be provided by Dimitri, I think that
there are code design arguments in favour of deprecating (and later,
deleting) the "guessParametersErrors" method, as follows.
In the context of the "optimization.general" package, one assumes that a
Jacobian matrix is available. From there, the code in "AbstractLeastSquares"
computes the covariance matrix, from which one can readily extract the
"sigma".
This can be done without computing the chi-square! [While, as you have
probably noticed, the "guessParametersErrors" will not behave nicely if you
don't call "updateResidualsAndCost()" beforehand.]
For the class to be self-consistent, the story can end here: Any additional
utilities can lead to wrong expectations from different types of users (as
we've demonstrated here).
Indeed, confidence intervals refer to additional variables (as Dimitri
wrote: "By how much can a parameter change before the normalized chi2
changes by ?"). Being able to answer those questions also
involves the correlations between the parameters (cf. the plot I've attached
to MATH-784), whereas "guessParametersErrors" does not take them into
account.
> I hope I'm not bothering you too much. I really would like to
> understand, so that we could write an accurate javadoc and possibly
> rename the method appropriately.
For clarity's sake (design-wise), I propose to remove the
"guessParametersErrors" method, and add a "getSigma" (as syntactic sugar).
If you want to dig further into the confidence interval issues in order to
provide the related functionality (similar to, but not limited to, the
current "guessParametersErrors"), I propose that that code be located in the
"stat" package (where, by the way, some of the utilities might already
exist!).
What do you think?
Best regards,
Gilles
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