Free combination of models from basic models¶

A fitting function can be freely combined from different basic models, such as peak forms or correction factors and background. In combining the model objects, they can be given names, so that every parameter gets a unique name, which can then be referenced in constraints by other parameters.

For example, in this model:

model = Background(bkgd=1) + Gauss('peak1', pos=97.5, ampl=100, fwhm=0.5) + \
    Gauss('peak2', pos=105, ampl=10, fwhm='peak1_fwhm')

the two Gauss peaks have different names “peak1” and “peak2”, and the model parameters are “peak1_pos”, “peak2_pos”, “peak1_ampl” and so forth. The Background model has no name of its own, so its “bkgd” parameter will also be called “bkgd” in the final model.

Parameter constraints and limits¶

Parameters can have limits (if the fitting backend supports this), and can be set to expressions involving other parameters and metadata from the data files.

For example, in this model:

model = Background(bkgd=1) + Gauss('peak1', pos=97.5, ampl=fixed(100), fwhm=0.5) + \
    Gauss('peak2', pos=105, ampl=limited(0, 20, 10), fwhm=expr('2*peak1_fwhm'))

the parameter “peak1_ampl” is fixed to a value of 100, the parameter “peak2_ampl” has a limited range, and the parameter “peak2_fwhm” is constrained to twice the value of “peak1_fwhm”.

Global fits¶

ufit can perform a so-called “global fit”, i.e. a fit where multiple datasets are used as the data points. In the model, some parameters can be treated as “global” (defined using overall()), while others are specific to each dataset.

This allows one to e.g. fit a series of spectra with a common background and incoherent line width.

An example:

from ufit.lab import *
set_datatemplate('path/to/data%04d.dat')

# read several data files
datas = [read_data(n, 'EN', 'CNTS') for n in range(100, 110)]

# create a model of a simple Gaussian peak with given initial guess
model = Background(bkgd=overall(1)) + \
    Gauss('peak', pos=97.5, ampl=100, fwhm=overall(0.5))

# fit the model, then print and plot all the results
results = model.global_fit(datas)
for result in results:
    result.printout()
    result.plot()
show()

In this example, a Gaussian peak is fitted to all 10 datasets. The background level and peak width are defined as global parameters, while the peak position and amplitude will be allowed to differ for each dataset.

Model components¶

ufit thinks of a model as consisting of several “components”. For example, when fitting several peaks to a spectrum, each peak is one of those components. When plotting a model fit, ufit can extract components from the model and plot them separately, together with the combined model.

Backends¶

ufit uses other libraries to do the fitting. For this purpose it has several backends:

• lmfit – uses the lmfit package written by Matthew Newville.

  lmfit builds upon the optimize functions available in SciPy, but adds the important capability to apply constraints and limits to parameters.

  By default, lmfit uses Levenberg-Marquardt least-squares, but other methods can be selected by a “method” keyword to the models’ fit() method when using this backend.

• minuit – uses the MINUIT package from CERN.

  Minuit has been the standard package for minimizing general N-dimensional functions in high-energy physics since its introduction in 1972. It is the minimization engine used behind-the-scenes in most high-energy physics curve fitting applications.

  The Python bindings necessary to use this backend can be retrieved from Google code. Note that the PyMinuit2 package currently seems to be buggy. This backend supports parameter limits.

• scipy – uses the scipy.optimize.leastsq function without any further wrapping. This backend doesn’t support parameter limits.

A backend is selected automatically on import, and a different one can be selected using set_backend().