[SciPy-User] Question about errors (uncertainties) in non-linear least squares fitting

Kevin Gullikson kevin.gullikson@gmail....
Tue Aug 7 16:37:09 CDT 2012


Pawell,

I do some fitting with leastq and fix parameters by just giving them as
part of the 'args' keyword. I would assume you could do the same.
On Aug 7, 2012 4:32 PM, "Paweł Kwaśniewski" <pawel.kw@gmail.com> wrote:

> Dear Jonathan,
>
> Thank you for the quick answer and pointing me to the updated code and the
> literature. I'll certainly have a look into it. Just for the record, I
> wanted to use constrained fitting because the unconstrained fit sometimes
> gave me unphysical parameters.
>
> It's interesting what you wrote about how the constraint influences the
> parameter error estimate. Just to give it a shot I tried an unconstrained
> fit on one data set. It did converge, but still - I got enormous chi^2
> (reduced, and yes - I'm using the experimental errors to calculate it) with
> ridiculously small parameter errors... I guess I need to understand better
> what's going on.
>
> By the way, the constrained fitting function is great, but I'd like to add
> another feature - the possibility to fix a parameter. This would largely
> facilitate testing of the fitting function, or can even be useful in real
> applications, where one of the parameters is known from a previous fit of a
> slightly different model or a different measurement or whatever... Anyway -
> maybe you have any ideas how to implement that? I already tried setting the
> upper and lower bound to the same value - it doesn't work. The method you
> use to apply the bounds throws division by zero error. **
>
> Cheers,
>
> Paweł
>
>
> 2012/8/7 Jonathan Helmus <jjhelmus@gmail.com>
>
>>  Pawel,
>>
>>     First off you may want to use a more up to date version of
>> leastsqbound which can be found at
>> https://github.com/jjhelmus/leastsqbound-scipy
>>
>>     Second, when you perform a constrained optimization using internal
>> parameters like leastsqbound does,
>> if one of more of the parameters is close to a bound, the values in the
>> covariance matrix can take on meaningless values.  Section 1.3 of the The
>> Minuit User's Guide [1] gives a good overview of this, especially look at
>> the discussion on page 5.  For best results an unconstrained optimization
>> should be performed, often times you can rewrite your model in such a way
>> that the constraints are automatically imposed (this is what is done
>> internally in leastsqbound, but transforming back to the original model can
>> introduce large errors if a parameter is close to the bounds).
>>
>>     Third, since you have measurement uncertainties make sure you include
>> them in the chi^2 calculation.  I find the discussion by P.H. Richter [2]
>> to be quite good.
>>
>> Cheers,
>>
>>     - Jonathan Helmus
>>
>>
>>
>>
>> [1] http://seal.cern.ch/documents/minuit/mnusersguide.pdf
>> [2] Estimating Errors in Least-Squares Fitting, P.H. Richter TDA Progress
>> Report 42-122
>>     http://tmo.jpl.nasa.gov/progress_report/42-122/122E.pdf
>>
>>
>> On 08/07/2012 09:16 AM, Paweł Kwaśniewski wrote:
>>
>> Hi,
>>
>> I'm fitting some data using a wrapper around the scipy.optimize.leastsq
>> method which can be found under
>> http://code.google.com/p/nmrglue/source/browse/trunk/nmrglue/analysis/leastsqbound.pyBasically it allows for putting bounds on the fitted parameters, which is
>> very important for me.
>>
>> I'm using the covariance matrix, returned by leastsq() function to
>> estimate the errors of the fitted parameters. The fitting is done using
>> real measurement uncertainties (which are ridiculously small, by the way),
>> so I would expect the resulting parameter error to be reasonable. What
>> don't understand, is that I'm getting extremely small errors on the fitted
>> parameters (I calculate the errors as perr = sqrt(diag(fitres[1])), where
>> fitres[1] is the covariance matrix returned by leastsq() function). For
>> example, a parameter which has a fitted value of ~100 gets an error of
>> ~1e-6. At the same time, when I calculate the reduced chi squared of the
>> fit I'm getting an extremely large number (of the order of 1e8). I can
>> understand the large chi^2 value - the data variance is extremely small and
>> the model curve is not perfect, so even slight deviations of the fitted
>> model from the data will blow up chi^2 into space. But how can the fitted
>> parameter variance be so small, while at the same time the fit is garbage
>> according to chi^2?
>>
>> I guess this requires a better understanding of how the covariance matrix
>> is calculated. Some suggestions anyone?
>>
>> Cheers,
>>
>> Paweł
>>
>>
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