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.
  • The LinearLSQFitter currently works only with simple (not compound) models.
  • The current fitters work only with models that have a single output (including bivariate functions such as Chebyshev2D but not compound models that map x, y -> x', y').

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

  • Iterative fitting with sigma clipping:

import numpy as np
from astropy.stats import sigma_clip
from astropy.modeling import models, fitting
import scipy.stats as stats
from matplotlib import pyplot as plt

# Generate fake data with outliers
np.random.seed(0)
x = np.linspace(-5., 5., 200)
y = 3 * np.exp(-0.5 * (x - 1.3)**2 / 0.8**2)
c = stats.bernoulli.rvs(0.35, size=x.shape)
y += (np.random.normal(0., 0.2, x.shape) +
      c*np.random.normal(3.0, 5.0, x.shape))
g_init = models.Gaussian1D(amplitude=1., mean=0, stddev=1.)

# initialize fitters
fit = fitting.LevMarLSQFitter()
or_fit = fitting.FittingWithOutlierRemoval(fit, sigma_clip,
                                           niter=3, sigma=3.0)

# get fitted model and filtered data
or_fitted_model, mask = or_fit(g_init, x, y)
filtered_data = np.ma.masked_array(y, mask=mask)
fitted_model = fit(g_init, x, y)

# plot data and fitted models
plt.figure(figsize=(8,5))
plt.plot(x, y, 'gx', label="original data")
plt.plot(x, filtered_data, 'r+', label="filtered data")
plt.plot(x, fitted_model(x), 'g-',
         label="model fitted w/ original data")
plt.plot(x, or_fitted_model(x), 'r--',
         label="model fitted w/ filtered data")
plt.legend(loc=2, numpoints=1)

()

../_images/fitting-1.png
  • Fitting with weights from data uncertainties
import numpy as np
from astropy.stats import sigma_clip
from astropy.modeling import models, fitting
import scipy.stats as stats
from matplotlib import pyplot as plt

# Generate fake data with outliers
np.random.seed(0)
x = np.linspace(-5., 5., 200)
y = 3 * np.exp(-0.5 * (x - 1.3)**2 / 0.8**2)
c = stats.bernoulli.rvs(0.35, size=x.shape)
y += (np.random.normal(0., 0.2, x.shape) +
      c*np.random.normal(3.0, 5.0, x.shape))
y_uncs = np.sqrt(np.square(np.full(x.shape, 0.2))
                 + c*np.square(np.full(x.shape,5.0)))
g_init = models.Gaussian1D(amplitude=1., mean=0, stddev=1.)

# initialize fitters
fit = fitting.LevMarLSQFitter()

# fit the data w/o weights
fitted_model = fit(g_init, x, y)

# fit the data using the uncertainties as weights
fitted_model_weights = fit(g_init, x, y, weights=1.0/y_uncs)

# plot data and fitted models
plt.figure(figsize=(8,5))
plt.errorbar(x, y, yerr=y_uncs, fmt='kx', label="data")
plt.plot(x, fitted_model(x), 'g-', linewidth=4.0,
         label="model fitted w/o weights")
plt.plot(x, fitted_model_weights(x), 'r--', linewidth=4.0,
         label="model fitted w/ weights")
plt.legend(loc=2, numpoints=1)

()

../_images/fitting-2.png

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:

    >>> 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

  • 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).

Plugin Fitters

Fitters defined outside of astropy’s core can be inserted into the astropy.modeling.fitting namespace through the use of entry points. Entry points are references to importable objects. A tutorial on defining entry points can be found in setuptools’ documentation. Plugin fitters are required to extend from the Fitter base class. For the fitter to be discovered and inserted into astropy.modeling.fitting the entry points must be inserted into the astropy.modeling entry point group

setup(
      # ...
      entry_points = {'astropy.modeling': 'PluginFitterName = fitter_module:PlugFitterClass'}
)

This would allow users to import the PlugFitterName through astropy.modeling.fitting by

from astropy.modeling.fitting import PlugFitterName

One project which uses this functionality is Saba, which insert its SherpaFitter class and thus allows astropy users to use Sherpa’s fitting routine.