Fitting theory to data

At the moment, only the standard least-squares fitting is implemented, and it uses the iminuit Python interface for the Minuit2 C++ library.

Gepard user creates a MinuitFitter object, where first argument is a dataset (collection of DataPoint objects), and second argument is a Theory object. Then fit() method of this MinuitFitter objects starts the fit (by calling migrad optimizer of Minuit). Before this, user should release some parameters of the Theory because all parameters are by default fixed at the moment of the creation of Theory object.

So, the minimal example of fitting is:

>>> import gepard as g
>>> pts = g.dset[36]     # 12 H1 DVCS measurements
>>> class FitTest(g.gpd.PWNormGPD, g.cff.MellinBarnesCFF, g.dvcs.BMK):
...     pass
>>> th = FitTest()
>>> f = g.MinuitFitter(pts, th)
>>> f.release_parameters('ns', 'ms2', 'secs')

Final values of chi-square and of parameters are available as

>>> f.minuit.fval
>>> f.print_parameters()
ns    =    1.46 +- 0.45
ms2   =    0.93 +- 0.07
secs  =   -0.32 +- 0.01

After successful fit of theory object th, user can access parameter uncertainties as th.parameters_errors dictionary, and full covariance matrix (inverse of the chi-square Hessian matrix) as th.covariance dictionary. Correlation matrix is also available as th.correlation.

Covariance matrix is then internally used to propagate parameter uncertainties to prediction of observables, like this:

>>> th.predict(pts[0], uncertainty=True)
(13.25, 1.62)

where parameter-dependent form factors (such as CFFs) can also be “predicted”, i. e., calculated together with their uncertainty:

>>> pt = g.DataPoint(xB=0.01, t=-0.2, Q2=10)
>>> th.predict(pt, observable='ImH', uncertainty=True)
(273.2, 25.8)


These uncertainties are “naive”. They ignore unknown part of uncertainty due to the rigidity of the model. It is likely that true uncertainty is always significantly larger.

Fitting using is simple but limited. For better control over fitting procedure and determination of parameter uncertainties, user should use many functionalities of the iminuit package, and call directly its functions like f.minuit.migrad, f.minuit.minos etc. For details one should consult documentation of this package. It is important to keep in mind that although parameter values are automatically kept in sync between iminuit and Gepard, parameter uncertainties and covariances are not. So, in order to correctly use uncertainties in Gepard, one should first syncronize covariances using special function f.covsync. For example, simple fitting with is equivalent to first doing f.minuit.migrad() and then f.covsync().