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- Parameters

# Parameters

## Basics

Most models in this package are “parametric” in the sense that each subclass of `Model` represents an entire family of models, each member of which is distinguished by a fixed set of parameters that fit that model to some dependent and independent variable(s) (also referred to throughout the package as the outputs and inputs of the model).

Parameters are used in three different contexts within this package: Basic evaluation of models, fitting models to data, and providing information about individual models to users (including documentation).

Most subclasses of `Model`–specifically those implementing a specific physical or statistical model, have a fixed set of parameters that can be specified for instances of that model. There are a few classes of models (in particular polynomials) in which the number of parameters depends on some other property of the model (the degree in the case of polynomials).

Models maintain a list of parameter names, `param_names`. Single parameters are instances of `Parameter` which provides a proxy for the actual parameter values. Simple mathematical operations can be performed with them, but they also contain additional attributes specific to model parameters, such as any constraints on their values and documentation.

Parameter values may be scalars or array values. Some parameters are required by their very nature to be arrays (such as the transformation matrix for an `AffineTransformation2D`). In most other cases, however, array-valued parameters have no meaning specific to the model, and are simply combined with input arrays during model evaluation according to the standard Numpy broadcasting rules.

## Parameter constraints

`astropy.modeling` supports several types of parameter constraints. They are implemented as properties of `Parameter`, the class which defines all fittable parameters, and can be set on individual parameters or on model instances.

The `astropy.modeling.Parameter.fixed` constraint is boolean and indicates whether a parameter is kept “fixed” or “frozen” during fitting. For example, fixing the `stddev` of a `Gaussian1D` model means it will be excluded from the list of fitted parameters:

    >>> from astropy.modeling.models import Gaussian1D
    >>> g = Gaussian1D(amplitude=10.2, mean=2.3, stddev=1.2)
    >>> g.stddev.fixed
    False
    >>> g.stddev.fixed = True
    >>> g.stddev.fixed
    True

`astropy.modeling.Parameter.bounds` is a tuple of numbers setting minimum and maximum value for a parameter. `(None, None)` indicates the parameter values are not bound. `bounds` can be set also using the `min` and `max` properties. Assigning `None` to the corresponding property removes the bound on the parameter. For example, setting bounds on the `mean` value of a `Gaussian1D` model can be done either by setting `min` and `max`:

    >>> g.mean.bounds
    (None, None)
    >>> g.mean.min = 2.2
    >>> g.mean.bounds
    (2.2, None)
    >>> g.mean.max = 2.4
    >>> g.mean.bounds
    (2.2, 2.4)

or using the `bounds` property:

    >>> g.mean.bounds = (2.2, 2.4)

`astropy.modeling.Parameter.tied` is a user supplied callable which takes a model instance and returns a value for the parameter. It is most useful with setting constraints on compounds models, for example a ratio between two parameters (example).

Constraints can also be set when the model is initialized. For example:

    >>> g = Gaussian1D(amplitude=10.2, mean=2.3, stddev=1.2,
    ...                fixed={'stddev': True},
    ...                 bounds={'mean': (2.2, 2.4)})
    >>> g.stddev.fixed
    True
    >>> g.mean.bounds
    (2.2, 2.4)

## Parameter examples

- Model classes can be introspected directly to find out what parameters they accept:

      >>> from astropy.modeling import models
      >>> models.Gaussian1D.param_names
      ('amplitude', 'mean', 'stddev')

  The order of the items in the `param_names` list is relevant–this is the same order in which values for those parameters should be passed in when constructing an instance of that model:

      >>> g = models.Gaussian1D(1.0, 0.0, 0.1)
      >>> g
      <Gaussian1D(amplitude=1.0, mean=0.0, stddev=0.1)>

  However, parameters may also be given as keyword arguments (in any order):

      >>> g = models.Gaussian1D(mean=0.0, amplitude=2.0, stddev=0.2)
      >>> g
      <Gaussian1D(amplitude=2.0, mean=0.0, stddev=0.2)>

  So all that really matters is knowing the names (and meanings) of the parameters that each model accepts. More information about an individual model can also be obtained using the `help` built-in:

      >>> help(models.Gaussian1D)

- Some types of models can have different numbers of parameters depending on other properties of the model. In particular, the parameters of polynomial models are their coefficients, the number of which depends on the polynomial’s degree:

      >>> p1 = models.Polynomial1D(degree=3, c0=1.0, c1=0.0, c2=2.0, c3=3.0)
      >>> p1.param_names
      ('c0', 'c1', 'c2', 'c3')
      >>> p1
      <Polynomial1D(3, c0=1., c1=0., c2=2., c3=3.)>

  For the basic `Polynomial1D` class the parameters are named `c0` through `cN` where `N` is the degree of the polynomial. The above example represents the polynomial \(3x^3 + 2x^2 + 1\).

- Some models also have default values for one or more of their parameters. For polynomial models, for example, the default value of all coefficients is zero–this allows a polynomial instance to be created without specifying any of the coefficients initially:

      >>> p2 = models.Polynomial1D(degree=4)
      >>> p2
      <Polynomial1D(4, c0=0., c1=0., c2=0., c3=0., c4=0.)>

- Parameters can then be set/updated by accessing attributes on the model of the same names as the parameters:

      >>> p2.c4 = 1
      >>> p2.c2 = 3.5
      >>> p2.c0 = 2.0
      >>> p2
      <Polynomial1D(4, c0=2., c1=0., c2=3.5, c3=0., c4=1.)>

  This example now represents the polynomial \(x^4 + 3.5x^2 + 2\).

- It is possible to set the coefficients of a polynomial by passing the parameters in a dictionary, since all parameters can be provided as keyword arguments:

      >>> ch2 = models.Chebyshev2D(x_degree=2, y_degree=3)
      >>> coeffs = dict((name, [idx, idx + 10])
      ...               for idx, name in enumerate(ch2.param_names))
      >>> ch2 = models.Chebyshev2D(x_degree=2, y_degree=3, n_models=2,
      ...                          **coeffs)
      >>> ch2.param_sets
      array([[ 0., 10.],
             [ 1., 11.],
             [ 2., 12.],
             [ 3., 13.],
             [ 4., 14.],
             [ 5., 15.],
             [ 6., 16.],
             [ 7., 17.],
             [ 8., 18.],
             [ 9., 19.],
             [10., 20.],
             [11., 21.]])

- Or directly, using keyword arguments:

      >>> ch2 = models.Chebyshev2D(x_degree=2, y_degree=3,
      ...                          c0_0=[0, 10], c0_1=[3, 13],
      ...                          c0_2=[6, 16], c0_3=[9, 19],
      ...                          c1_0=[1, 11], c1_1=[4, 14],
      ...                          c1_2=[7, 17], c1_3=[10, 20,],
      ...                          c2_0=[2, 12], c2_1=[5, 15],
      ...                          c2_2=[8, 18], c2_3=[11, 21])

- Individual parameters values may be arrays of different sizes and shapes:

      >>> p3 = models.Polynomial1D(degree=2, c0=1.0, c1=[2.0, 3.0],
      ...                          c2=[[4.0, 5.0], [6.0, 7.0], [8.0, 9.0]])
      >>> p3(2.0)
      array([[21., 27.],
             [29., 35.],
             [37., 43.]])

  This is equivalent to evaluating the Numpy expression:

      >>> import numpy as np
      >>> c2 = np.array([[4.0, 5.0],
      ...                [6.0, 7.0],
      ...                [8.0, 9.0]])
      >>> c1 = np.array([2.0, 3.0])
      >>> c2 * 2.0**2 + c1 * 2.0 + 1.0
      array([[21., 27.],
             [29., 35.],
             [37., 43.]])

  Note that in most cases, when using array-valued parameters, the parameters must obey the standard broadcasting rules for Numpy arrays with respect to each other:

      >>> models.Polynomial1D(degree=2, c0=1.0, c1=[2.0, 3.0],
      ...                     c2=[4.0, 5.0, 6.0])
      Traceback (most recent call last):
      ...
      InputParameterError: Parameter u'c1' of shape (2,) cannot be broadcast
      with parameter u'c2' of shape (3,).  All parameter arrays must have
      shapes that are mutually compatible according to the broadcasting rules.