[SciPy-Dev] On the leastsq/curve_fit method
Paul Kuin
npkuin@gmail....
Mon Sep 26 09:31:31 CDT 2011
I am just a simple user, but perhaps this gives some idea of what us users
experience:
I got so fed up with leastsq not having good documentation, not being able to
set limits to the parameters to be fit, and not handling errors in
input measurements
in a transparent way, that I was very happy to replace it with the
pkfit routine
based on Craig Markwards IDL code. I am happy now, but I waisted a lot of time
because of these leastsq issues.
Anyway - I am happy now.
Paul Kuin
On Mon, Sep 26, 2011 at 2:57 PM, Gianfranco Durin <g.durin@inrim.it> wrote:
> ----- Original Message -----
>> On Thu, Sep 22, 2011 at 1:23 PM, Gianfranco Durin <g.durin@inrim.it>
>> wrote:
>> > Dear all,
>> > I wanted to briefly show you the results of an analysis I did on
>> > the performances of the optimize.leastsq method for data fitting.
>> > I presented these results at the last Python in Physics workshop.
>> > You can download the pdf here:
>> > http://emma.inrim.it:8080/gdurin/talks.
>> >
>> > 1. The main concern is about the use of cov_x to estimate the error
>> > bar of the fitting parameters. In the docs, it is set that "this
>> > matrix must be multiplied by the residual standard deviation to
>> > get the covariance of the parameter estimates -- see curve_fits.""
>> >
>> > Unfortunately, this is not correct, or better it is only partially
>> > correct. It is correct if there are no error bars of the input
>> > data (the sigma of curve_fit is None). But if provided, they are
>> > used as "as weights in least-squares problem" (curve_fit doc), and
>> > cov_x gives directly the covariance of the parameter estimates
>> > (i.e. the diagonal terms are the errors in the parameters). See
>> > for instance here:
>> > http://www.gnu.org/s/gsl/manual/html_node/Computing-the-covariance-matrix-of-best-fit-parameters.html.
>> >
>> > This means that not only the doc needs fixing, but also the
>> > curve_fit code, those estimation of the parameters' error is
>> > INDEPENDENT of the values of the data errors in the case they are
>> > constant, which is clearly wrong.
>> > I have never provided a patch, but the fix should be quite simple,
>> > just please give me indication on how to do that.
>>
>> Since this depends on what you define as weight or sigma, both are
>> correct but use different definitions.
>
> Of course, but this is not the point. Let me explain.
>
> In our routines, the cov_x is calculated as (J^T * J) ^-1, where J is the jacobian. The Jacobian, unless provided directly, is calculated using the definition of the residuals, which in curve_fit method are "_general_function", and "_weighted_general_function". The latter function explicitly uses the weights, so the cov_x is more precisely (J^T W J) ^-1, where W is the matrix of the weights, and J is just the matrix of the first derivatives. Thus in this case, the diagonal elements of cov_x give the variance of the parameters. No need to multiply by the residual standard deviation.
> In case of W == 1, i.e. no errors in the data are provided, as reported here http://en.wikipedia.org/wiki/Linear_least_squares_%28mathematics%29#Weighted_linear_least_squares
>
> one uses the variance of the observations, i.e. uses the
>
> s_sq = (func(popt, *args)**2).sum()/(len(ydata)-len(p0))
>
> as an estimate of the variance, as done in curve_fit.
>
> BUT, we cannot multiply the cov_x obtained with the _weighted_general_function by s_sq. As I told, we already took it into account in the definition of the residuals. Thus:
>
> if (len(ydata) > len(p0)) and pcov is not None:
> s_sq = (func(popt, *args)**2).sum()/(len(ydata)-len(p0))
> pcov = pcov * s_sq
> else:
> pcov = inf
>
>
> where func can be both "_general_function" and "_weighted_general_function", is not correct.
>
>>
>> Since we just had this discussion, I'm not arguing again. I just want
>> to have clear definitions that the "average" user can use by default.
>> I don't really care which it is if the majority of users are
>> engineers
>> who can tell what their errors variances are before doing any
>> estimation.
>>
>
> Oh, interesting. Where did you have this discussion? On this list? I could not find it...
>
> Here the problem is not to decide an 'average' behaviour, but to correctly calculate the parameters' error when the user does or does not provide the errors in the data.
>
> Hope this helps
> Gianfranco
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