Fitting with constraints

Fitters support constrained fitting.

• All fitters support fixed (frozen) parameters through the fixed argument to models or setting the fixed attribute directly on a parameter.

  For linear fitters, freezing a polynomial coefficient means that the corresponding term will be subtracted from the data before fitting a polynomial without that term to the result. For example, fixing c0 in a polynomial model will fit a polynomial with the zero-th order term missing to the data minus that constant. However, the fixed coefficient value is restored when evaluating the model, to fit the original data values:

  >>> import numpy as np
  >>> from astropy.modeling import models, fitting
  >>> x = np.arange(1, 10, .1)
  >>> p1 = models.Polynomial1D(2, c0=[1, 1], c1=[2, 2], c2=[3, 3],
  ...                          n_models=2)
  >>> p1  
  <Polynomial1D(2, c0=[1., 1.], c1=[2., 2.], c2=[3., 3.], n_models=2)>
  >>> y = p1(x, model_set_axis=False)
  >>> p1.c0.fixed = True
  >>> pfit = fitting.LinearLSQFitter()
  >>> new_model = pfit(p1, x, y)
  >>> print(new_model)  
  Model: Polynomial1D
  Inputs: ('x',)
  Outputs: ('y',)
  Model set size: 2
  Degree: 2
  Parameters:
       c0  c1  c2
      --- --- ---
      1.0 2.0 3.0
      1.0 2.0 3.0
  

  The syntax to fix the same parameter c0 using an argument to the model instead of p1.c0.fixed = True would be:

  >>> p1 = models.Polynomial1D(2, c0=[1, 1], c1=[2, 2], c2=[3, 3],
  ...                          n_models=2, fixed={'c0': True})
  

• A parameter can be tied (linked to another parameter). This can be done in two ways:

  >>> def tiedfunc(g1):
  ...    mean = 3 * g1.stddev
  ...    return mean
  >>> g1 = models.Gaussian1D(amplitude=10., mean=3, stddev=.5,
  ...                        tied={'mean': tiedfunc})
  

  or:

  >>> g1 = models.Gaussian1D(amplitude=10., mean=3, stddev=.5)
  >>> g1.mean.tied = tiedfunc
  

Bounded fitting is supported through the bounds arguments to models or by setting min and max attributes on a parameter. Bounds for the LevMarLSQFitter are always exactly satisfied–if the value of the parameter is outside the fitting interval, it will be reset to the value at the bounds. The SLSQPLSQFitter handles bounds internally.

• Different fitters support different types of constraints:

  >>> fitting.LinearLSQFitter.supported_constraints
  ['fixed']
  >>> fitting.LevMarLSQFitter.supported_constraints
  ['fixed', 'tied', 'bounds']
  >>> fitting.SLSQPLSQFitter.supported_constraints
  ['bounds', 'eqcons', 'ineqcons', 'fixed', 'tied']
  

Note that there are two “constraints” (prior and posterior) that are not currently used by any of the built-in fitters. They are provided to allow possible user code that might implement Bayesian fitters (e.g., https://gist.github.com/rkiman/5c5e6f80b455851084d112af2f8ed04f).