# 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)
```

- 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)
```

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.