Models and Fitting (astropy.modeling)

Introduction

astropy.modeling provides a framework for representing models and performing model evaluation and fitting. It currently supports 1-D and 2-D models and fitting with parameter constraints.

It is designed to be easily extensible and flexible. Models do not reference fitting algorithms explicitly and new fitting algorithms may be added without changing the existing models (though not all models can be used with all fitting algorithms due to constraints such as model linearity).

The goal is to eventually provide a rich toolset of models and fitters such that most users will not need to define new model classes, nor special purpose fitting routines (while making it reasonably easy to do when necessary).

Note

astropy.modeling is currently a work-in-progress, and thus it is likely there will still be API changes in later versions of Astropy. Backwards compatibility support between versions will still be maintained as much as possible, but new features and enhancements are coming in future versions. If you have specific ideas for how it might be improved, feel free to let us know on the astropy-dev mailing list or at http://feedback.astropy.org

Getting started

The examples here use the predefined models and assume the following modules have been imported:

>>> import numpy as np
>>> from astropy.modeling import models, fitting

Using Models

The astropy.modeling package defines a number of models that are collected under a single namespace as astropy.modeling.models. Models behave like parametrized functions:

>>> from astropy.modeling import models
>>> g = models.Gaussian1D(amplitude=1.2, mean=0.9, stddev=0.5)
>>> print(g)
Model: Gaussian1D
Inputs: ('x',)
Outputs: ('y',)
Model set size: 1
Parameters:
    amplitude mean stddev
    --------- ---- ------
          1.2  0.9    0.5

Model parameters can be accessed as attributes:

>>> g.amplitude
Parameter('amplitude', value=1.2)
>>> g.mean
Parameter('mean', value=0.9)
>>> g.stddev
Parameter('stddev', value=0.5)

and can also be updated via those attributes:

>>> g.amplitude = 0.8
>>> g.amplitude
Parameter('amplitude', value=0.8)

Models can be evaluated by calling them as functions:

>>> g(0.1)
0.22242984036255528
>>> g(np.linspace(0.5, 1.5, 7))
array([ 0.58091923,  0.71746405,  0.7929204 ,  0.78415894,  0.69394278,
        0.54952605,  0.3894018 ])

As the above example demonstrates, in general most models evaluate array-like inputs according to the standard Numpy broadcasting rules for arrays.

Models can therefore already be useful to evaluate common functions, independently of the fitting features of the package.

Simple 1-D model fitting

In this section, we look at a simple example of fitting a Gaussian to a simulated dataset. We use the Gaussian1D and Trapezoid1D models and the LevMarLSQFitter fitter to fit the data:

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

# Generate fake data
np.random.seed(0)
x = np.linspace(-5., 5., 200)
y = 3 * np.exp(-0.5 * (x - 1.3)**2 / 0.8**2)
y += np.random.normal(0., 0.2, x.shape)

# Fit the data using a box model
t_init = models.Trapezoid1D(amplitude=1., x_0=0., width=1., slope=0.5)
fit_t = fitting.LevMarLSQFitter()
t = fit_t(t_init, x, y)

# Fit the data using a Gaussian
g_init = models.Gaussian1D(amplitude=1., mean=0, stddev=1.)
fit_g = fitting.LevMarLSQFitter()
g = fit_g(g_init, x, y)

# Plot the data with the best-fit model
plt.figure(figsize=(8,5))
plt.plot(x, y, 'ko')
plt.plot(x, t(x), label='Trapezoid')
plt.plot(x, g(x), label='Gaussian')
plt.xlabel('Position')
plt.ylabel('Flux')
plt.legend(loc=2)

(Source code, png, hires.png, pdf)

As shown above, once instantiated, the fitter class can be used as a function that takes the initial model (t_init or g_init) and the data values (x and y), and returns a fitted model (t or g).

Simple 2-D model fitting

Similarly to the 1-D example, we can create a simulated 2-D data dataset, and fit a polynomial model to it. This could be used for example to fit the background in an image.

import warnings
import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling import models, fitting

# Generate fake data
np.random.seed(0)
y, x = np.mgrid[:128, :128]
z = 2. * x ** 2 - 0.5 * x ** 2 + 1.5 * x * y - 1.
z += np.random.normal(0., 0.1, z.shape) * 50000.

# Fit the data using astropy.modeling
p_init = models.Polynomial2D(degree=2)
fit_p = fitting.LevMarLSQFitter()

with warnings.catch_warnings():
    # Ignore model linearity warning from the fitter
    warnings.simplefilter('ignore')
    p = fit_p(p_init, x, y, z)

# Plot the data with the best-fit model
plt.figure(figsize=(8, 2.5))
plt.subplot(1, 3, 1)
plt.imshow(z, origin='lower', interpolation='nearest', vmin=-1e4, vmax=5e4)
plt.title("Data")
plt.subplot(1, 3, 2)
plt.imshow(p(x, y), origin='lower', interpolation='nearest', vmin=-1e4,
           vmax=5e4)
plt.title("Model")
plt.subplot(1, 3, 3)
plt.imshow(z - p(x, y), origin='lower', interpolation='nearest', vmin=-1e4,
           vmax=5e4)
plt.title("Residual")

(Source code, png, hires.png, pdf)

A list of models is provided in the Reference/API section. The fitting framework includes many useful features that are not demonstrated here, such as weighting of datapoints, fixing or linking parameters, and placing lower or upper limits on parameters. For more information on these, take a look at the Fitting Models to Data documentation.

Model sets

In some cases it is necessary to describe many models of the same type but with different sets of parameter values. This could be done simply by instantiating as many instances of a Model as are needed. But that can be inefficient for a large number of models. To that end, all model classes in astropy.modeling can also be used to represent a model set which is a collection of models of the same type, but with different values for their parameters.

To instantiate a model set, use argument n_models=N where N is the number of models in the set when constructing the model. The value of each parameter must be a list or array of length N, such that each item in the array corresponds to one model in the set:

>>> g = models.Gaussian1D(amplitude=[1, 2], mean=[0, 0],
...                       stddev=[0.1, 0.2], n_models=2)
>>> print(g)
Model: Gaussian1D
Inputs: ('x',)
Outputs: ('y',)
Model set size: 2
Parameters:
    amplitude mean stddev
    --------- ---- ------
          1.0  0.0    0.1
          2.0  0.0    0.2

This is equivalent to two Gaussians with the parameters amplitude=1, mean=0, stddev=0.1 and amplitude=2, mean=0, stddev=0.2 respectively. When printing the model the parameter values are displayed as a table, with each row corresponding to a single model in the set.

The number of models in a model set can be determined using the len builtin:

>>> len(g)
2

Single models have a length of 1, and are not considered a model set as such.

When evaluating a model set, by default the input must be the same length as the number of models, with one input per model:

>>> g([0, 0.1])
array([ 1.        ,  1.76499381])

The result is an array with one result per model in the set. It is also possible to broadcast a single value to all models in the set:

>>> g(0)
array([ 1.,  2.])

Model sets are used primarily for fitting, allowing a large number of models of the same type to be fitted simultaneously (and independently from each other) to some large set of inputs. For example, fitting a polynomial to the time response of each pixel in a data cube. This can greatly speed up the fitting process, especially for linear models.

Compound models

New in version 1.0: This feature is experimental and expected to see significant further development, but the basic usage is stable and expected to see wide use.

While the Astropy modeling package makes it very easy to define new models either from existing functions, or by writing a Model subclass, an additional way to create new models is by combining them using arithmetic expressions. This works with models built into Astropy, and most user-defined models as well. For example, it is possible to create a superposition of two Gaussians like so:

>>> from astropy.modeling import models
>>> g1 = models.Gaussian1D(1, 0, 0.2)
>>> g2 = models.Gaussian1D(2.5, 0.5, 0.1)
>>> g1_plus_2 = g1 + g2

The resulting object g1_plus_2 is itself a new model. Evaluating, say, g1_plus_2(0.25) is the same as evaluating g1(0.25) + g2(0.25):

>>> g1_plus_2(0.25)  
0.5676756958301329
>>> g1_plus_2(0.25) == g1(0.25) + g2(0.25)
True

This model can be further combined with other models in new expressions. It is also possible to define entire new model classes using arithmetic expressions of other model classes. This allows general compound models to be created without specifying any parameter values up front. This more advanced usage is explained in more detail in the compound model documentation.

These new compound models can also be fitted to data, like most other models:

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

# Generate fake data
np.random.seed(42)
g1 = models.Gaussian1D(1, 0, 0.2)
g2 = models.Gaussian1D(2.5, 0.5, 0.1)
x = np.linspace(-1, 1, 200)
y = g1(x) + g2(x) + np.random.normal(0., 0.2, x.shape)

# Now to fit the data create a new superposition with initial
# guesses for the parameters:
gg_init = models.Gaussian1D(1, 0, 0.1) + models.Gaussian1D(2, 0.5, 0.1)
fitter = fitting.SLSQPLSQFitter()
gg_fit = fitter(gg_init, x, y)

# Plot the data with the best-fit model
plt.figure(figsize=(8,5))
plt.plot(x, y, 'ko')
plt.plot(x, gg_fit(x))
plt.xlabel('Position')
plt.ylabel('Flux')

(Source code, png, hires.png, pdf)

This works for 1-D models, 2-D models, and combinations thereof, though there are some complexities involved in correctly matching up the inputs and outputs of all models used to build a compound model. You can learn more details in the Compound Models documentation.

Using astropy.modeling

• Instantiating and Evaluating Models
• Parameters
• Fitting Models to Data
• Compound Models
• Defining New Model Classes
• Defining New Fitter Classes
• Using a Custom Statistic Function
• Efficient Model Rendering with Bounding Boxes
• Algorithms

Reference/API

astropy.modeling Package

This subpackage provides a framework for representing models and performing model evaluation and fitting. It supports 1D and 2D models and fitting with parameter constraints. It has some predefined models and fitting routines.

Functions

custom_model(*args, **kwargs)

Create a model from a user defined function.

Classes

Fittable1DModel	Base class for one-dimensional fittable models.
Fittable2DModel	Base class for two-dimensional fittable models.
FittableModel	Base class for models that can be fitted using the built-in fitting algorithms.
InputParameterError	Used for incorrect input parameter values and definitions.
LabeledInput(*args, **kwargs)	Deprecated since version 1.0.
Model	Base class for all models.
ModelDefinitionError	Used for incorrect models definitions
Parameter([name, description, default, ...])	Wraps individual parameters.
SerialCompositeModel	Deprecated since version 1.0.
SummedCompositeModel	Deprecated since version 1.0.

Class Inheritance Diagram

astropy.modeling.mappings Module

Special models useful for complex compound models where control is needed over which outputs from a source model are mapped to which inputs of a target model.

Classes

Mapping	Allows inputs to be reordered, duplicated or dropped.
Identity	Returns inputs unchanged.

Class Inheritance Diagram

astropy.modeling.functional_models Module

Mathematical models.

Functions

custom_model_1d(*args, **kwargs)

Deprecated since version 1.0.

Classes

AiryDisk2D	Two dimensional Airy disk model.
Box1D	One dimensional Box model.
Box2D	Two dimensional Box model.
Const1D	One dimensional Constant model.
Const2D	Two dimensional Constant model.
Disk2D	Two dimensional radial symmetric Disk model.
Ellipse2D	A 2D Ellipse model.
Gaussian1D	One dimensional Gaussian model.
Gaussian2D	Two dimensional Gaussian model.
GaussianAbsorption1D	One dimensional Gaussian absorption line model.
Linear1D	One dimensional Line model.
Lorentz1D	One dimensional Lorentzian model.
MexicanHat1D	One dimensional Mexican Hat model.
MexicanHat2D	Two dimensional symmetric Mexican Hat model.
Moffat1D	One dimensional Moffat model.
Moffat2D	Two dimensional Moffat model.
Redshift	One dimensional redshift model.
Ring2D	Two dimensional radial symmetric Ring model.
Scale	Multiply a model by a factor.
Sersic1D	One dimensional Sersic surface brightness profile.
Sersic2D	Two dimensional Sersic surface brightness profile.
Shift	Shift a coordinate.
Sine1D	One dimensional Sine model.
Trapezoid1D	One dimensional Trapezoid model.
TrapezoidDisk2D	Two dimensional circular Trapezoid model.
Voigt1D	One dimensional model for the Voigt profile.

Class Inheritance Diagram

astropy.modeling.powerlaws Module

Power law model variants

Classes

BrokenPowerLaw1D	One dimensional power law model with a break.
ExponentialCutoffPowerLaw1D	One dimensional power law model with an exponential cutoff.
LogParabola1D	One dimensional log parabola model (sometimes called curved power law).
PowerLaw1D	One dimensional power law model.

Class Inheritance Diagram

astropy.modeling.polynomial Module

This module contains predefined polynomial models.

Classes

Chebyshev1D	1D Chebyshev polynomial of the 1st kind.
Chebyshev2D	2D Chebyshev polynomial of the 1st kind.
InverseSIP	Inverse Simple Imaging Polynomial
Legendre1D	1D Legendre polynomial.
Legendre2D	Legendre 2D polynomial.
Polynomial1D	1D Polynomial model.
Polynomial2D	2D Polynomial model.
SIP	Simple Imaging Polynomial (SIP) model.
OrthoPolynomialBase	This is a base class for the 2D Chebyshev and Legendre models.
PolynomialModel	Base class for polynomial models.

Class Inheritance Diagram

astropy.modeling.projections Module

Implements projections–particularly sky projections defined in WCS Paper II [R45].

All angles are set and and displayed in degrees but internally computations are performed in radians.

References

[R45]

Calabretta, M.R., Greisen, E.W., 2002, A&A, 395, 1077 (Paper II)

Classes

Projection	Base class for all sky projections.
Pix2SkyProjection	Base class for all Pix2Sky projections.
Sky2PixProjection	Base class for all Sky2Pix projections.
Zenithal	Base class for all Zenithal projections.
Cylindrical	Base class for Cylindrical projections.
PseudoCylindrical	Base class for pseudocylindrical projections.
Conic	Base class for conic projections.
PseudoConic	Base class for pseudoconic projections.
QuadCube	Base class for quad cube projections.
HEALPix	Base class for HEALPix projections.
AffineTransformation2D	Perform an affine transformation in 2 dimensions.
Pix2Sky_ZenithalPerspective	Zenithal perspective projection - pixel to sky.
Sky2Pix_ZenithalPerspective	Zenithal perspective projection - sky to pixel.
Pix2Sky_SlantZenithalPerspective	Slant zenithal perspective projection - pixel to sky.
Sky2Pix_SlantZenithalPerspective	Zenithal perspective projection - sky to pixel.
Pix2Sky_Gnomonic	Gnomonic projection - pixel to sky.
Sky2Pix_Gnomonic	Gnomonic Projection - sky to pixel.
Pix2Sky_Stereographic	Stereographic Projection - pixel to sky.
Sky2Pix_Stereographic	Stereographic Projection - sky to pixel.
Pix2Sky_SlantOrthographic	Slant orthographic projection - pixel to sky.
Sky2Pix_SlantOrthographic	Slant orthographic projection - sky to pixel.
Pix2Sky_ZenithalEquidistant	Zenithal equidistant projection - pixel to sky.
Sky2Pix_ZenithalEquidistant	Zenithal equidistant projection - sky to pixel.
Pix2Sky_ZenithalEqualArea	Zenithal equidistant projection - pixel to sky.
Sky2Pix_ZenithalEqualArea	Zenithal equidistant projection - sky to pixel.
Pix2Sky_Airy	Airy projection - pixel to sky.
Sky2Pix_Airy	Airy - sky to pixel.
Pix2Sky_CylindricalPerspective	Cylindrical perspective - pixel to sky.
Sky2Pix_CylindricalPerspective	Cylindrical Perspective - sky to pixel.
Pix2Sky_CylindricalEqualArea	Cylindrical equal area projection - pixel to sky.
Sky2Pix_CylindricalEqualArea	Cylindrical equal area projection - sky to pixel.
Pix2Sky_PlateCarree	Plate carrée projection - pixel to sky.
Sky2Pix_PlateCarree	Plate carrée projection - sky to pixel.
Pix2Sky_Mercator	Mercator - pixel to sky.
Sky2Pix_Mercator	Mercator - sky to pixel.
Pix2Sky_SansonFlamsteed	Sanson-Flamsteed projection - pixel to sky.
Sky2Pix_SansonFlamsteed	Sanson-Flamsteed projection - sky to pixel.
Pix2Sky_Parabolic	Parabolic projection - pixel to sky.
Sky2Pix_Parabolic	Parabolic projection - sky to pixel.
Pix2Sky_Molleweide	Molleweide’s projection - pixel to sky.
Sky2Pix_Molleweide	Molleweide’s projection - sky to pixel.
Pix2Sky_HammerAitoff	Hammer-Aitoff projection - pixel to sky.
Sky2Pix_HammerAitoff	Hammer-Aitoff projection - sky to pixel.
Pix2Sky_ConicPerspective	Colles’ conic perspective projection - pixel to sky.
Sky2Pix_ConicPerspective	Colles’ conic perspective projection - sky to pixel.
Pix2Sky_ConicEqualArea	Alber’s conic equal area projection - pixel to sky.
Sky2Pix_ConicEqualArea	Alber’s conic equal area projection - sky to pixel.
Pix2Sky_ConicEquidistant	Conic equidistant projection - pixel to sky.
Sky2Pix_ConicEquidistant	Conic equidistant projection - sky to pixel.
Pix2Sky_ConicOrthomorphic	Conic orthomorphic projection - pixel to sky.
Sky2Pix_ConicOrthomorphic	Conic orthomorphic projection - sky to pixel.
Pix2Sky_BonneEqualArea	Bonne’s equal area pseudoconic projection - pixel to sky.
Sky2Pix_BonneEqualArea	Bonne’s equal area pseudoconic projection - sky to pixel.
Pix2Sky_Polyconic	Polyconic projection - pixel to sky.
Sky2Pix_Polyconic	Polyconic projection - sky to pixel.
Pix2Sky_TangentialSphericalCube	Tangential spherical cube projection - pixel to sky.
Sky2Pix_TangentialSphericalCube	Tangential spherical cube projection - sky to pixel.
Pix2Sky_COBEQuadSphericalCube	COBE quadrilateralized spherical cube projection - pixel to sky.
Sky2Pix_COBEQuadSphericalCube	COBE quadrilateralized spherical cube projection - sky to pixel.
Pix2Sky_QuadSphericalCube	Quadrilateralized spherical cube projection - pixel to sky.
Sky2Pix_QuadSphericalCube	Quadrilateralized spherical cube projection - sky to pixel.
Pix2Sky_HEALPix	HEALPix - pixel to sky.
Sky2Pix_HEALPix	HEALPix projection - sky to pixel.
Pix2Sky_HEALPixPolar	HEALPix polar, aka “butterfly” projection - pixel to sky.
Sky2Pix_HEALPixPolar	HEALPix polar, aka “butterfly” projection - pixel to sky.
Pix2Sky_AZP	alias of Pix2Sky_ZenithalPerspective
Sky2Pix_AZP	alias of Sky2Pix_ZenithalPerspective
Pix2Sky_SZP	alias of Pix2Sky_SlantZenithalPerspective
Sky2Pix_SZP	alias of Sky2Pix_SlantZenithalPerspective
Pix2Sky_TAN	alias of Pix2Sky_Gnomonic
Sky2Pix_TAN	alias of Sky2Pix_Gnomonic
Pix2Sky_STG	alias of Pix2Sky_Stereographic
Sky2Pix_STG	alias of Sky2Pix_Stereographic
Pix2Sky_SIN	alias of Pix2Sky_SlantOrthographic
Sky2Pix_SIN	alias of Sky2Pix_SlantOrthographic
Pix2Sky_ARC	alias of Pix2Sky_ZenithalEquidistant
Sky2Pix_ARC	alias of Sky2Pix_ZenithalEquidistant
Pix2Sky_ZEA	alias of Pix2Sky_ZenithalEqualArea
Sky2Pix_ZEA	alias of Sky2Pix_ZenithalEqualArea
Pix2Sky_AIR	alias of Pix2Sky_Airy
Sky2Pix_AIR	alias of Sky2Pix_Airy
Pix2Sky_CYP	alias of Pix2Sky_CylindricalPerspective
Sky2Pix_CYP	alias of Sky2Pix_CylindricalPerspective
Pix2Sky_CEA	alias of Pix2Sky_CylindricalEqualArea
Sky2Pix_CEA	alias of Sky2Pix_CylindricalEqualArea
Pix2Sky_CAR	alias of Pix2Sky_PlateCarree
Sky2Pix_CAR	alias of Sky2Pix_PlateCarree
Pix2Sky_MER	alias of Pix2Sky_Mercator
Sky2Pix_MER	alias of Sky2Pix_Mercator
Pix2Sky_SFL	alias of Pix2Sky_SansonFlamsteed
Sky2Pix_SFL	alias of Sky2Pix_SansonFlamsteed
Pix2Sky_PAR	alias of Pix2Sky_Parabolic
Sky2Pix_PAR	alias of Sky2Pix_Parabolic
Pix2Sky_MOL	alias of Pix2Sky_Molleweide
Sky2Pix_MOL	alias of Sky2Pix_Molleweide
Pix2Sky_AIT	alias of Pix2Sky_HammerAitoff
Sky2Pix_AIT	alias of Sky2Pix_HammerAitoff
Pix2Sky_COP	alias of Pix2Sky_ConicPerspective
Sky2Pix_COP	alias of Sky2Pix_ConicPerspective
Pix2Sky_COE	alias of Pix2Sky_ConicEqualArea
Sky2Pix_COE	alias of Sky2Pix_ConicEqualArea
Pix2Sky_COD	alias of Pix2Sky_ConicEquidistant
Sky2Pix_COD	alias of Sky2Pix_ConicEquidistant
Pix2Sky_COO	alias of Pix2Sky_ConicOrthomorphic
Sky2Pix_COO	alias of Sky2Pix_ConicOrthomorphic
Pix2Sky_BON	alias of Pix2Sky_BonneEqualArea
Sky2Pix_BON	alias of Sky2Pix_BonneEqualArea
Pix2Sky_PCO	alias of Pix2Sky_Polyconic
Sky2Pix_PCO	alias of Sky2Pix_Polyconic
Pix2Sky_TSC	alias of Pix2Sky_TangentialSphericalCube
Sky2Pix_TSC	alias of Sky2Pix_TangentialSphericalCube
Pix2Sky_CSC	alias of Pix2Sky_COBEQuadSphericalCube
Sky2Pix_CSC	alias of Sky2Pix_COBEQuadSphericalCube
Pix2Sky_QSC	alias of Pix2Sky_QuadSphericalCube
Sky2Pix_QSC	alias of Sky2Pix_QuadSphericalCube
Pix2Sky_HPX	alias of Pix2Sky_HEALPix
Sky2Pix_HPX	alias of Sky2Pix_HEALPix
Pix2Sky_XPH	alias of Pix2Sky_HEALPixPolar
Sky2Pix_XPH	alias of Sky2Pix_HEALPixPolar

Class Inheritance Diagram

astropy.modeling.rotations Module

Implements rotations, including spherical rotations as defined in WCS Paper II [R46]

RotateNative2Celestial and RotateCelestial2Native follow the convention in WCS Paper II to rotate to/from a native sphere and the celestial sphere.

The user interface sets and displays angles in degrees but the values are stored internally in radians. This is managed through the parameter setters/getters.

References

[R46]

Calabretta, M.R., Greisen, E.W., 2002, A&A, 395, 1077 (Paper II)

Classes

RotateCelestial2Native	Transform from Celestial to Native to Spherical Coordinates.
RotateNative2Celestial	Transform from Native to Celestial Spherical Coordinates.
Rotation2D	Perform a 2D rotation given an angle in degrees.
EulerAngleRotation	Implements Euler angle intrinsic rotations.

Class Inheritance Diagram

astropy.modeling.fitting Module

This module implements classes (called Fitters) which combine optimization algorithms (typically from scipy.optimize) with statistic functions to perform fitting. Fitters are implemented as callable classes. In addition to the data to fit, the __call__ method takes an instance of FittableModel as input, and returns a copy of the model with its parameters determined by the optimizer.

Optimization algorithms, called “optimizers” are implemented in optimizers and statistic functions are in statistic. The goal is to provide an easy to extend framework and allow users to easily create new fitters by combining statistics with optimizers.

There are two exceptions to the above scheme. LinearLSQFitter uses Numpy’s lstsq function. LevMarLSQFitter uses leastsq which combines optimization and statistic in one implementation.

Classes

LinearLSQFitter()	A class performing a linear least square fitting.
LevMarLSQFitter()	Levenberg-Marquardt algorithm and least squares statistic.
SLSQPLSQFitter()	SLSQP optimization algorithm and least squares statistic.
SimplexLSQFitter()	Simplex algorithm and least squares statistic.
JointFitter(models, jointparameters, initvals)	Fit models which share a parameter.
Fitter(optimizer, statistic)	Base class for all fitters.

Class Inheritance Diagram

astropy.modeling.optimizers Module

Optimization algorithms used in fitting.

Classes

Optimization(opt_method)	Base class for optimizers.
SLSQP()	Sequential Least Squares Programming optimization algorithm.
Simplex()	Neald-Mead (downhill simplex) algorithm [1].

Class Inheritance Diagram

astropy.modeling.statistic Module

Statistic functions used in fitting.

Functions

leastsquare(measured_vals, updated_model, ...)

Least square statistic with optional weights.