[SciPy-User] Global Curve Fitting of 2 functions to 2 sets of data-curves

[Sebastian Haase]

On Thu, Jun 10, 2010 at 8:27 PM,  <josef.pktd at gmail.com> wrote:
> On Thu, Jun 10, 2010 at 4:05 AM, Sebastian Haase <seb.haase at gmail.com> wrote:
>> Hi,
>>
>> so far I have been using scipy.optimize.leastsq to satisfy all my
>> curve fitting needs.
>> But now I am thinking about "global fitting" - i.e. fitting multiple
>> dataset with shared parameters
>> (e.g. ref here:
>> http://www.originlab.com/index.aspx?go=Products/Origin/DataAnalysis/CurveFitting/GlobalFitting)
>>
>> I have looked here (http://www.scipy.org/Cookbook/FittingData) and here
>> (http://docs.scipy.org/doc/scipy/reference/optimize.html)
>>
>> Can someone provide an example  ? Which of the routines of
>> scipy.optimize are "easiest" to use ?
>>
>> Finally, I'm thinking about a "much more" complicated fitting task:
>> fitting two sets of datasets with two types of functions.
>> In total I have 10 datasets to be fit with a function f1, and 10 more
>> to be fit with function f2. Each function depends on 6 parameters
>> A1,A2,A3, r1,r2,r3.
>> A1,A2,A3 should be identical ("shared") between all 20 sets, while
>> r1,r2,r3 should be shared between the i-th set of type f1 and the i-th
>> set of f2.
>> Last but not least it would be nice if one could specify constrains
>> such that r1,r2,r3 >0 and A1+A2+A3 == 1 and 0<=Ai<=1.
>>
>> ;-)  Is this too much ?
>>
>> Thanks for any help or hints,
>
> Assuming your noise or error terms are uncorrelated, I would still use
> optimize.leastsq or optimize.curve_fit using a function that stacks
> all the errors in one 1-d array. If there are differences in the noise
> variance, then weights/sigma per function as in curve_fit can be used.
>
> common parameter restrictions across functions can be encoded by using
> the same parameter in several (sub-)functions.
>
> In this case, I would impose the constraints through reparameterization, e.g
> r1 = exp(r1_), ...
> A1 = exp(A1_)/(exp(A1_) + exp(A2_) + 1)
> A1 = exp(A2_)/(exp(A1_) + exp(A2_) + 1)
> A1 = 1/(exp(A1_) + exp(A2_) + 1)
>
> (maybe it's more tricky to get the standard deviation of the original
> parameter estimate)
>
> or as an alternative, calculate the total weighted sum of squared
> errors and use one of the constraint fmin in optimize.
>
> Josef

Thanks for the reply,
I will have to think about implementing my constrains by redefining
vars using those kinds of tricks with exp -- are you sure they don't
mess up convergence ? I'm just thinking of the optimization steps
being so different depending on the current parameter value during the
iteration (i.e. the derivative of exp is very non-linear)

What are those other functions in
http://docs.scipy.org/doc/scipy/reference/optimize.html for ?
(Once, long time ago, I did use fmin_cobyla ... but don't remember why
I choose it. Maybe something like one-sided constrains !?)

Thanks,
Sebastian