Fitting Models to Data¶
This module provides wrappers, called Fitters, around some Numpy and Scipy
fitting functions. All Fitters can be called as functions. They take an
instance of FittableModel
as input and modify its
parameters
attribute. The idea is to make this extensible and allow
users to easily add other fitters.
Linear fitting is done using Numpy’s numpy.linalg.lstsq
function. There are
currently two non-linear fitters which use scipy.optimize.leastsq
and
scipy.optimize.fmin_slsqp
.
The rules for passing input to fitters are:
- Non-linear fitters currently work only with single models (not model sets).
- The linear fitter can fit a single input to multiple model sets creating
multiple fitted models. This may require specifying the
model_set_axis
argument just as used when evaluating models; this may be required for the fitter to know how to broadcast the input data.
Fitting examples¶
Fitting a polynomial model to multiple data sets simultaneously:
>>> from astropy.modeling import models, fitting >>> import numpy as np >>> p1 = models.Polynomial1D(3) >>> p1.c0 = 1 >>> p1.c1 = 2 >>> print(p1) Model: Polynomial1D Inputs: ('x',) Outputs: ('y',) Model set size: 1 Degree: 3 Parameters: c0 c1 c2 c3 --- --- --- --- 1.0 2.0 0.0 0.0 >>> x = np.arange(10) >>> y = p1(x) >>> yy = np.array([y, y]) >>> p2 = models.Polynomial1D(3, n_models=2) >>> pfit = fitting.LinearLSQFitter() >>> new_model = pfit(p2, x, yy) >>> print(new_model) Model: Polynomial1D Inputs: 1 Outputs: 1 Model set size: 2 Degree: 3 Parameters: c0 c1 c2 c3 --- --- ------------------ ----------------- 1.0 2.0 -5.86673908219e-16 3.61636197841e-17 1.0 2.0 -5.86673908219e-16 3.61636197841e-17
Fitters support constrained fitting.
All fitters support fixed (frozen) parameters through the
fixed
argument to models or setting thefixed
attribute directly on a parameter.For linear fitters, freezing a polynomial coefficient means that a polynomial without that term will be fitted to the data. For example, fixing
c0
in a polynomial model will fit a polynomial with the zero-th order term missing. However, the fixed value of the coefficient is used when evaluating the model:>>> 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: 1 Outputs: 1 Model set size: 2 Degree: 2 Parameters: c0 c1 c2 --- ------------- ------------- 1.0 2.38641216243 2.96827885742 1.0 2.38641216243 2.96827885742
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']