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

[josef.pktd at gmail.com]

On Thu, Jun 10, 2010 at 2:58 PM, Sebastian Haase <seb.haase at gmail.com> wrote:
> 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 !?)

fmin_slsqp  is the most flexible for constraints, but so far I have
use the constraint maximizers only for toy examples, and don't know
how robust they are.

Imposing constraints by reparamaterization or penalization is in my
experience not much of a problem, except getting a slightly interior
solution instead of exact boundary value.
The multinomial logit parameterization for A1, A2, A3 is pretty common
in econometrics, I'm not sure what's the most common for
non-negativity constraints.

If you have analytical gradients, then one of the other optimizers
might be better than leastsq.

(These are just my impression from my use cases.)

Josef


> Thanks,
> Sebastian
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