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# Models

## Basics

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:

    >>> import numpy as np
    >>> 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, bounds=(1.1754943508222875e-38, None))

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.

### Instantiating and Evaluating Models

In general, models are instantiated by supplying the parameter values that define that instance of the model to the constructor, as demonstrated in the section on Parameters.

Additionally, a `Model` instance may represent a single model with one set of parameters, or a Model set consisting of a set of parameters each representing a different parameterization of the same parametric model. For example, you may instantiate a single Gaussian model with one mean, standard deviation, and amplitude. Or you may create a set of N Gaussians, each one of which would be evaluated on, for example, a different plane in an image cube.

For example, a single Gaussian model may be instantiated with all scalar parameters:

    >>> from astropy.modeling.models import Gaussian1D
    >>> g = Gaussian1D(amplitude=1, mean=0, stddev=1)
    >>> g
    <Gaussian1D(amplitude=1., mean=0., stddev=1.)>

The newly created model instance `g` now works like a Gaussian function with the specific parameters. It takes a single input:

    >>> g.inputs
    ('x',)
    >>> g(x=0)
    1.0

The model can also be called without explicitly using keyword arguments:

    >>> g(0)
    1.0

Or a set of Gaussians may be instantiated by passing multiple parameter values:

    >>> from astropy.modeling.models import Gaussian1D
    >>> gset = Gaussian1D(amplitude=[1, 1.5, 2],
    ...                   mean=[0, 1, 2],
    ...                   stddev=[1., 1., 1.],
    ...                   n_models=3)
    >>> print(gset)
    Model: Gaussian1D
    Inputs: ('x',)
    Outputs: ('y',)
    Model set size: 3
    Parameters:
        amplitude mean stddev
        --------- ---- ------
              1.0  0.0    1.0
              1.5  1.0    1.0
              2.0  2.0    1.0

This model also works like a Gaussian function. The three models in the model set can be evaluated on the same input:

    >>> gset(1.)
    array([0.60653066, 1.5       , 1.21306132])

or on `N=3` inputs:

    >>> gset([1, 2, 3])
    array([0.60653066, 0.90979599, 1.21306132])

For a comprehensive example of fitting a model set see Fitting Model Sets.

### Model inverses

All models have a `Model.inverse` property which may, for some models, return a new model that is the analytic inverse of the model it is attached to. For example:

    >>> from astropy.modeling.models import Linear1D
    >>> linear = Linear1D(slope=0.8, intercept=1.0)
    >>> linear.inverse
    <Linear1D(slope=1.25, intercept=-1.25)>

The inverse of a model will always be a fully instantiated model in its own right, and so can be evaluated directly like:

    >>> linear.inverse(2.0)
    1.25

It is also possible to assign a custom inverse to a model. This may be useful, for example, in cases where a model does not have an analytic inverse, but may have an approximate inverse that was computed numerically and is represented by another model. This works even if the target model has a default analytic inverse–in this case the default is overridden with the custom inverse:

    >>> from astropy.modeling.models import Polynomial1D
    >>> linear.inverse = Polynomial1D(degree=1, c0=-1.25, c1=1.25)
    >>> linear.inverse
    <Polynomial1D(1, c0=-1.25, c1=1.25)>

If a custom inverse has been assigned to a model, it can be deleted with `del model.inverse`. This resets the inverse to its default (if one exists). If a default does not exist, accessing `model.inverse` raises a `NotImplementedError`. For example polynomial models do not have a default inverse:

    >>> del linear.inverse
    >>> linear.inverse
    <Linear1D(slope=1.25, intercept=-1.25)>
    >>> p = Polynomial1D(degree=2, c0=1.0, c1=2.0, c2=3.0)
    >>> p.inverse
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
      File "astropy\modeling\core.py", line 796, in inverse
        raise NotImplementedError("An analytical inverse transform has not "
    NotImplementedError: No analytical or user-supplied inverse transform
    has been implemented for this model.

One may certainly compute an inverse and assign it to a polynomial model though.

Note

When assigning a custom inverse to a model no validation is performed to ensure that it is actually an inverse or even approximate inverse. So assign custom inverses at your own risk.

### Bounding Boxes

#### Efficient Model Rendering with Bounding Boxes

All `Model` subclasses have a `bounding_box` attribute that can be used to set the limits over which the model is significant. This greatly improves the efficiency of evaluation when the input range is much larger than the characteristic width of the model itself. For example, to create a sky model image from a large survey catalog, each source should only be evaluated over the pixels to which it contributes a significant amount of flux. This task can otherwise be computationally prohibitive on an average CPU.

The `Model.render` method can be used to evaluate a model on an output array, or input coordinate arrays, limiting the evaluation to the `bounding_box` region if it is set. This function will also produce postage stamp images of the model if no other input array is passed. To instead extract postage stamps from the data array itself, see 2D Cutout Images.

#### Using the standard Bounding Box

For basic usage, see `Model.bounding_box`. By default no `bounding_box` is set, except on model subclasses where a `bounding_box` property or method is explicitly defined. The default is then the minimum rectangular region symmetric about the position that fully contains the model. If the model does not have a finite extent, the containment criteria are noted in the documentation. For example, see `Gaussian2D.bounding_box`.

Warning

Accessing the `Model.bounding_box` property when it has not been set, or does not have a default will result in a `NotImplementedError`. If this behavior is undesirable, then one can instead use the `Model.get_bounding_box` method instead. This method will return the bounding box if one exists (by setting or default) otherwise it will return `None` instead of raising an error.

A `Model.bounding_box` default can be set by the user to any callable. This is particularly useful for models created with `custom_model` or as a `CompoundModel`:

    >>> from astropy.modeling import custom_model
    >>> def ellipsoid(x, y, z, x0=0, y0=0, z0=0, a=2, b=3, c=4, amp=1):
    ...     rsq = ((x - x0) / a) ** 2 + ((y - y0) / b) ** 2 + ((z - z0) / c) ** 2
    ...     val = (rsq < 1) * amp
    ...     return val
    ...
    >>> class Ellipsoid3D(custom_model(ellipsoid)):
    ...     # A 3D ellipsoid model
    ...     def bounding_box(self):
    ...         return ((self.z0 - self.c, self.z0 + self.c),
    ...                 (self.y0 - self.b, self.y0 + self.b),
    ...                 (self.x0 - self.a, self.x0 + self.a))
    ...
    >>> model1 = Ellipsoid3D()
    >>> model1.bounding_box
    ModelBoundingBox(
        intervals={
            x0: Interval(lower=-2.0, upper=2.0)
            x1: Interval(lower=-3.0, upper=3.0)
            x2: Interval(lower=-4.0, upper=4.0)
        }
        model=Ellipsoid3D(inputs=('x0', 'x1', 'x2'))
        order='C'
    )

By default models are evaluated on any inputs. By passing a flag they can be evaluated only on inputs within the bounding box. For inputs outside of the bounding_box a `fill_value` is returned (`np.nan` by default):

    >>> model1(-5, 1, 1)
    0.0
    >>> model1(-5, 1, 1, with_bounding_box=True)
    nan
    >>> model1(-5, 1, 1, with_bounding_box=True, fill_value=-1)
    -1.0

`Model.bounding_box` can be set on any model instance via the usage of the property setter. For example for a single input model one needs to only set a tuple of the lower and upper bounds

    >>> from astropy.modeling.models import Polynomial1D
    >>> model2 = Polynomial1D(2)
    >>> model2.bounding_box = (-1, 1)
    >>> model2.bounding_box
    ModelBoundingBox(
        intervals={
            x: Interval(lower=-1, upper=1)
        }
        model=Polynomial1D(inputs=('x',))
        order='C'
    )
    >>> model2(-2)
    0.0
    >>> model2(-2, with_bounding_box=True)
    nan
    >>> model2(-2, with_bounding_box=True, fill_value=47)
    47.0

For multi-input models, `Model.bounding_box` can be set on any model instance by specifying a tuple of lower/upper bound tuples

    >>> from astropy.modeling.models import Polynomial2D
    >>> model3 = Polynomial2D(2)
    >>> model3.bounding_box = ((-2, 2), (-1, 1))
    >>> model3.bounding_box
    ModelBoundingBox(
        intervals={
            x: Interval(lower=-1, upper=1)
            y: Interval(lower=-2, upper=2)
        }
        model=Polynomial2D(inputs=('x', 'y'))
        order='C'
    )
    >>> model3(-2, 0)
    0.0
    >>> model3(-2, 0, with_bounding_box=True)
    nan
    >>> model3(-2, 0, with_bounding_box=True, fill_value=7)
    7.0

Note that if one wants to directly recover the tuple used to formulate a bounding box, then one can use the `ModelBoundingBox.bounding_box()` method

    >>> model1.bounding_box.bounding_box()
    ((np.float64(-4.0), np.float64(4.0)), (np.float64(-3.0), np.float64(3.0)), (np.float64(-2.0), np.float64(2.0)))
    >>> model2.bounding_box.bounding_box()
    (-1, 1)
    >>> model3.bounding_box.bounding_box()
    ((-2, 2), (-1, 1))

Warning

When setting multi-dimensional bounding boxes it is important to remember that by default the tuple of tuples is assumed to be `'C'` ordered, which means that the bound tuples will be ordered in the reverse order to their respective input order. That is if the inputs are in the order `('x', 'y', 'z')` then the bounds will need to be listed in `('z', 'y', 'x')` order.

The if one does not want to work directly with the default `'C'` ordered bounding boxes. It is possible to use the alternate `'F'` ordering, which orders the bounding box tuple in the same order as the inputs. To do this one can use the `bind_bounding_box` function, and passing the `order='F'` keyword argument

    >>> from astropy.modeling import bind_bounding_box
    >>> model4 = Polynomial2D(3)
    >>> bind_bounding_box(model4, ((-1, 1), (-2, 2)), order='F')
    >>> model4.bounding_box
    ModelBoundingBox(
        intervals={
            x: Interval(lower=-1, upper=1)
            y: Interval(lower=-2, upper=2)
        }
        model=Polynomial2D(inputs=('x', 'y'))
        order='F'
    )
    >>> model4(-2, 0)
    0.0
    >>> model4(-2, 0, with_bounding_box=True)
    nan
    >>> model4(-2, 0, with_bounding_box=True, fill_value=12)
    12.0
    >>> model4.bounding_box.bounding_box()
    ((-1, 1), (-2, 2))
    >>> model4.bounding_box.bounding_box(order='C')
    ((-2, 2), (-1, 1))

Warning

Currently when combining models the bounding boxes of components are combined only when joining models with the `&` operator. For the other operators bounding boxes for compound models must be assigned explicitly. A future release will determine the appropriate bounding box for a compound model where possible.

#### Using the Compound Bounding Box

Sometimes it is useful to have multiple bounding boxes for the same model, which are selectable when the model is evaluated. In this case, one should consider using a `CompoundBoundingBox`.

A common use case for this may be if the model has a single “discrete” selector input (for example `'slit_id'`), which among other things, determines what bounding box should be applied to the other inputs. To do this one needs to first define a dictionary of bounding box tuples, with dictionary keys being the specific values of the selector input corresponding to that specific bounding box

    >>> from astropy.modeling.models import Shift, Identity
    >>> model1 = Shift(1) & Shift(2) & Identity(1)
    >>> model1.inputs = ('x', 'y', 'slit_id')
    >>> bboxes = {
    ...     0: ((0, 1), (1, 2)),
    ...     1: ((2, 3), (3, 4))
    ... }

In order for the compound bounding box to function one must specify a list of selector arguments, where the elements of this list are tuples of the input’s name and whether or not the bounding box should be applied to the selector argument or not. In this case, it makes sense for the selector argument to be ignored

    >>> from astropy.modeling.core import bind_compound_bounding_box
    >>> selector_args = [('slit_id', True)]
    >>> bind_compound_bounding_box(model1, bboxes, selector_args, order='F')
    >>> model1.bounding_box
    CompoundBoundingBox(
        bounding_boxes={
            (0,) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=0, upper=1)
                        y: Interval(lower=1, upper=2)
                    }
                    ignored=['slit_id']
                    model=CompoundModel(inputs=('x', 'y', 'slit_id'))
                    order='F'
                )
            (1,) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=2, upper=3)
                        y: Interval(lower=3, upper=4)
                    }
                    ignored=['slit_id']
                    model=CompoundModel(inputs=('x', 'y', 'slit_id'))
                    order='F'
                )
        }
        selector_args = SelectorArguments(
                Argument(name='slit_id', ignore=True)
            )
    )
    >>> model1(0.5, 1.5, 0, with_bounding_box=True)
    (1.5, 3.5, 0.0)
    >>> model1(0.5, 1.5, 1, with_bounding_box=True)
    (np.float64(nan), np.float64(nan), np.float64(nan))

Multiple selector arguments can also be used, in this case the keys of the dictionary of bounding boxes need to be specified as tuples of values

    >>> model2 = Shift(1) & Shift(2) & Identity(2)
    >>> model2.inputs = ('x', 'y', 'slit_x', 'slit_y')
    >>> bboxes = {
    ...     (0, 0): ((0, 1), (1, 2)),
    ...     (0, 1): ((2, 3), (3, 4)),
    ...     (1, 0): ((4, 5), (5, 6)),
    ...     (1, 1): ((6, 7), (7, 8)),
    ... }
    >>> selector_args = [('slit_x', True), ('slit_y', True)]
    >>> bind_compound_bounding_box(model2, bboxes, selector_args, order='F')
    >>> model2.bounding_box
    CompoundBoundingBox(
        bounding_boxes={
            (0, 0) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=0, upper=1)
                        y: Interval(lower=1, upper=2)
                    }
                    ignored=['slit_x', 'slit_y']
                    model=CompoundModel(inputs=('x', 'y', 'slit_x', 'slit_y'))
                    order='F'
                )
            (0, 1) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=2, upper=3)
                        y: Interval(lower=3, upper=4)
                    }
                    ignored=['slit_x', 'slit_y']
                    model=CompoundModel(inputs=('x', 'y', 'slit_x', 'slit_y'))
                    order='F'
                )
            (1, 0) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=4, upper=5)
                        y: Interval(lower=5, upper=6)
                    }
                    ignored=['slit_x', 'slit_y']
                    model=CompoundModel(inputs=('x', 'y', 'slit_x', 'slit_y'))
                    order='F'
                )
            (1, 1) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=6, upper=7)
                        y: Interval(lower=7, upper=8)
                    }
                    ignored=['slit_x', 'slit_y']
                    model=CompoundModel(inputs=('x', 'y', 'slit_x', 'slit_y'))
                    order='F'
                )
        }
        selector_args = SelectorArguments(
                Argument(name='slit_x', ignore=True)
                Argument(name='slit_y', ignore=True)
            )
    )
    >>> model2(0.5, 1.5, 0, 0, with_bounding_box=True)
    (1.5, 3.5, 0.0, 0.0)
    >>> model2(0.5, 1.5, 1, 1, with_bounding_box=True)
    (np.float64(nan), np.float64(nan), np.float64(nan), np.float64(nan))

Note that one can also specify the ordering for all the bounding boxes comprising the compound bounding using the `order` keyword argument.

Another use case for this maybe a if one wants to use multiple bounding boxes for the same model, where the user chooses the bounding box when evaluating the model. In this case, one must still choose a selector argument as a fall back default for bounding box selection; however, this argument should not be ignored by the bounding box:

    >>> from astropy.modeling.models import Polynomial2D
    >>> from astropy.modeling import bind_compound_bounding_box
    >>> model = Polynomial2D(3)
    >>> bboxes = {
    ...     0: ((0, 1), (1, 2)),
    ...     1: ((2, 3), (3, 4))
    ... }
    >>> selector_args = [('x', False)]
    >>> bind_compound_bounding_box(model, bboxes, selector_args, order='F')
    >>> model.bounding_box
        CompoundBoundingBox(
        bounding_boxes={
            (0,) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=0, upper=1)
                        y: Interval(lower=1, upper=2)
                    }
                    model=Polynomial2D(inputs=('x', 'y'))
                    order='F'
                )
            (1,) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=2, upper=3)
                        y: Interval(lower=3, upper=4)
                    }
                    model=Polynomial2D(inputs=('x', 'y'))
                    order='F'
                )
        }
        selector_args = SelectorArguments(
                Argument(name='x', ignore=False)
            )
    )

For the user to select the bounding box on evaluation, instead of specifying, `with_bounding_box=True` as the keyword argument; the user instead specifies `with_bounding_box=<bounding_key>`

    >>> model(0.5, 1.5, with_bounding_box=0)
    0.0
    >>> model(0.5, 1.5, with_bounding_box=1)
    nan

#### Ignoring Inputs in Bounding Boxes

Both `standard bounding box` and `CompoundBoundingBox` support ignoring specific inputs from enforcement by the bounding box. Effectively, for multi-dimensional models one can define bounding boxes so that bounds are only applied to a subset of the model’s inputs rather than the default of enforcing a bound of some kind on every input. Note that use of this feature is equivalent to defining the bounds for an input to be `[-np.inf, np.inf]`.

Warning

The `ignored` input feature is not available when constructing/adding bounding boxes to models using tuples and the property interface. That is one cannot ignore inputs when setting bounding boxes using `model.bounding_box = (-1, 1)`. This feature is only available via the methods `bind_bounding_box` and `bind_compound_bounding_box`.

Ignoring inputs for a bounding box can be achieved via passing a list of the input name strings to be ignored to the `ignored` keyword argument in any of the main bounding box interfaces.

    >>> from astropy.modeling.models import Polynomial1D
    >>> from astropy.modeling import bind_bounding_box
    >>> model1 = Polynomial2D(3)
    >>> bind_bounding_box(model1, {'x': (-1, 1)}, ignored=['y'])
    >>> model1.bounding_box
    ModelBoundingBox(
        intervals={
            x: Interval(lower=-1, upper=1)
        }
        ignored=['y']
        model=Polynomial2D(inputs=('x', 'y'))
        order='C'
    )
    >>> model1(-2, 0, with_bounding_box=True)
    nan
    >>> model1(0, 300, with_bounding_box=True)
    0.0

Similarly, the ignored inputs will be applied to all of the bounding boxes contained within a compound bounding box.

    >>> from astropy.modeling import bind_compound_bounding_box
    >>> model2 = Polynomial2D(3)
    >>> bboxes = {
    ...     0: {'x': (0, 1)},
    ...     1: {'x': (1, 2)}
    ... }
    >>> selector_args = [('x', False)]
    >>> bind_compound_bounding_box(model2, bboxes, selector_args, ignored=['y'], order='F')
    >>> model2.bounding_box
        CompoundBoundingBox(
        bounding_boxes={
            (0,) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=0, upper=1)
                    }
                    ignored=['y']
                    model=Polynomial2D(inputs=('x', 'y'))
                    order='F'
                )
            (1,) = ModelBoundingBox(
                    intervals={
                        x: Interval(lower=1, upper=2)
                    }
                    ignored=['y']
                    model=Polynomial2D(inputs=('x', 'y'))
                    order='F'
                )
        }
        selector_args = SelectorArguments(
                Argument(name='x', ignore=False)
            )
    )
    >>> model2(0.5, 300, with_bounding_box=0)
    0.0
    >>> model2(0.5, 300, with_bounding_box=1)
    nan

### Efficient evaluation with `Model.render()`

When a model is evaluated over a range much larger than the model itself, it may be prudent to use the `Model.render` method if efficiency is a concern. The `render` method can be used to evaluate the model on an array of the same dimensions. `model.render()` can be called with no arguments to return a “postage stamp” of the bounding box region.

In this example, we generate a 300x400 pixel image of 100 2D Gaussian sources. For comparison, the models are evaluated both with and without using bounding boxes. By using bounding boxes, the evaluation speed increases by approximately a factor of 10 with negligible loss of information.

    import numpy as np
    from time import time
    from astropy.modeling import models
    import matplotlib.pyplot as plt
    from matplotlib.patches import Rectangle

    imshape = (300, 400)
    y, x = np.indices(imshape)

    # Generate random source model list
    rng = np.random.default_rng(0)
    nsrc = 100
    model_params = [
        dict(amplitude=rng.uniform(.5, 1),
             x_mean=rng.uniform(0, imshape[1] - 1),
             y_mean=rng.uniform(0, imshape[0] - 1),
             x_stddev=rng.uniform(2, 6),
             y_stddev=rng.uniform(2, 6),
             theta=rng.uniform(0, 2 * np.pi))
        for _ in range(nsrc)]

    model_list = [models.Gaussian2D(**kwargs) for kwargs in model_params]

    # Render models to image using bounding boxes
    bb_image = np.zeros(imshape)
    t_bb = time()
    for model in model_list:
        model.render(bb_image)
    t_bb = time() - t_bb

    # Render models to image using full evaluation
    full_image = np.zeros(imshape)
    t_full = time()
    for model in model_list:
        model.bounding_box = None
        model.render(full_image)
    t_full = time() - t_full

    flux = full_image.sum()
    diff = (full_image - bb_image)
    max_err = diff.max()

    # Plots
    fig, axs = plt.subplots(figsize=(16, 7), ncols=2)
    fig.subplots_adjust(left=.05, right=.97, bottom=.03, top=.97, wspace=0.15)

    # Full model image
    ax1 = axs[0]
    ax1.imshow(full_image, origin='lower')
    ax1.set_title(f'Full Models\nTiming: {t_full:.2f} seconds', fontsize=16)
    ax1.set(xlabel='x', ylabel='y')

    # Bounded model image with boxes overplotted
    ax2 = axs[1]
    ax2.imshow(bb_image, origin='lower')
    for model in model_list:
        del model.bounding_box  # Reset bounding_box to its default
        dy, dx = np.diff(model.bounding_box).flatten()
        pos = (model.x_mean.value - dx / 2, model.y_mean.value - dy / 2)
        r = Rectangle(pos, dx, dy, edgecolor='w', facecolor='none', alpha=.25)
        ax2.add_patch(r)
    ax2.set_title(f'Bounded Models\nTiming: {t_bb:.2f} seconds', fontsize=16)
    ax2.set(xlabel='x', ylabel='y')

    # Difference image
    fig2, ax = plt.subplots(figsize=(16, 8))
    im = ax.imshow(diff, vmin=-max_err, vmax=max_err)
    fig2.colorbar(im, format='%.1e')
    ax.set_title(f'Difference Image\nTotal Flux Err = {((flux - np.sum(bb_image)) / flux):.0e}')
    ax.set(xlabel='x', ylabel='y')
    plt.show()

(`png`, `svg`, `pdf`)

(`png`, `svg`, `pdf`)

### Model Separability

Simple models have a boolean `Model.separable` property. It indicates whether the outputs are independent and is essential for computing the separability of compound models using the `is_separable()` function. Having a separable compound model means that it can be decomposed into independent models, which in turn is useful in many applications. For example, it may be easier to define inverses using the independent parts of a model than the entire model. In other cases, tools using Generalized World Coordinate System (GWCS), can be more flexible and take advantage of separable spectral and spatial transforms.

If a custom subclass of `Model` needs to override the computation of its separability it can implement the `_calculate_separability_matrix` method which should return the separability matrix for that model.

## Model Sets

In some cases it is useful 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:

    >>> from astropy.modeling import models
    >>> 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 input value to all models in the set:

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

Or when the input is an array:

    >>> x = np.array([[0, 0, 0], [0.1, 0.1, 0.1]])
    >>> print(x)
    [[0.  0.  0. ]
     [0.1 0.1 0.1]]
    >>> g(x)
    array([[1.        , 1.        , 1.        ],
           [1.76499381, 1.76499381, 1.76499381]])

Internally the shape of the inputs, outputs, and parameter values is controlled by an attribute - `model_set_axis`. In the above case `model_set_axis=0`:

    >>> g.model_set_axis
    0

This indicates that elements along the 0-th axis will be passed as inputs to individual models. Sometimes it may be useful to pass inputs along a different axis, for example the 1st axis:

    >>> x = np.array([[0, 0, 0], [0.1, 0.1, 0.1]]).T
    >>> print(x)
    [[0.  0.1]
     [0.  0.1]
     [0.  0.1]]

Because there are two models in this model set and we are passing three inputs along the 0th axis, evaluation will fail:

    >>> g(x)
    Traceback (most recent call last):
    ...
    ValueError: Input argument 'x' does not have the correct dimensions in
    model_set_axis=0 for a model set with n_models=2.

There are two ways to get around this. `model_set_axis` can be passed in when the model is evaluated:

    >>> g(x, model_set_axis=1)
    array([[1.        , 1.76499381],
           [1.        , 1.76499381],
           [1.        , 1.76499381]])

Or when the model is initialized:

    >>> g = models.Gaussian1D(amplitude=[[1, 2]], mean=[[0, 0]],
    ...                       stddev=[[0.1, 0.2]], n_models=2,
    ...                       model_set_axis=1)
    >>> g(x)
    array([[1.        , 1.76499381],
           [1.        , 1.76499381],
           [1.        , 1.76499381]])

Note that in the latter case, the shape of the individual parameters has changed to 2D because now the parameters are defined along the 1st axis.

The value of `model_set_axis` is either an integer number, representing the axis along which the different parameter sets and inputs are defined, or a boolean of value `False`, in which case it indicates all model sets should use the same inputs on evaluation. For example, the above model has a value of 1 for `model_set_axis`. If `model_set_axis=False` is passed the two models will be evaluated on the same input:

    >>> g.model_set_axis
    1
    >>> result = g(x, model_set_axis=False)
    >>> result
    array([[[1.        , 0.60653066],
            [2.        , 1.76499381]],

           [[1.        , 0.60653066],
            [2.        , 1.76499381]],

           [[1.        , 0.60653066],
            [2.        , 1.76499381]]])
    >>> result[: , 0]
    array([[1.        , 0.60653066],
           [1.        , 0.60653066],
           [1.        , 0.60653066]])
    >>> result[: , 1]
    array([[2.        , 1.76499381],
           [2.        , 1.76499381],
           [2.        , 1.76499381]])

Currently model sets are most useful for fitting a set of linear models (example) 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, such as fitting a polynomial to the time response of each pixel in a data cube. This can greatly speed up the fitting process. The speed-up is due to solving the set of equations to find the exact solution. Nonlinear models, which require an iterative algorithm, cannot be currently fit using model sets. Model sets of nonlinear models can only be evaluated.

When fitting model sets it is important that data arrays are passed to the fitter in the correct shape. The shape depends on the `model_set_axis` attribute of the model to be fit. The rule is that the index of the dependent variable that corresponds to a model set should be along the `model_set_axis` dimension. For example, for a 1D model set with 3 models with `model_set_axis == 1` the shape of `y` should be (x, 3):

    >>> import numpy as np
    >>> from astropy.modeling.models import Polynomial1D
    >>> from astropy.modeling.fitting import LinearLSQFitter
    >>> fitter = LinearLSQFitter()
    >>> x = np.arange(4)
    >>> y = np.array([2*x+1, x+4, x]).T
    >>> print(y)
    [[1 4 0]
     [3 5 1]
     [5 6 2]
     [7 7 3]]
    >>> print(y.shape)
    (4, 3)
    >>> m = Polynomial1D(1, n_models=3, model_set_axis=1)
    >>> mfit = fitter(m, x, y)

For 2D models with 3 models and `model_set_axis = 0` the shape of `z` should be (3, x, y):

    >>> import numpy as np
    >>> from astropy.modeling.models import Polynomial2D
    >>> from astropy.modeling.fitting import LinearLSQFitter
    >>> fitter = LinearLSQFitter()
    >>> x = np.arange(8).reshape(2, 4)
    >>> y = x
    >>> z = np.asarray([2 * x + 1, x + 4, x + 3])
    >>> print(z.shape)
    (3, 2, 4)
    >>> m = Polynomial2D(1, n_models=3, model_set_axis=0)
    >>> mfit = fitter(m, x, y, z)

## Model Serialization (Writing a Model to a File)

Models are serializable using the ASDF format. This can be useful in many contexts, one of which is the implementation of a Generalized World Coordinate System (GWCS).

Serializing a model to disk is possible by assigning the object to `AsdfFile.tree`:

    >>> from asdf import AsdfFile
    >>> from astropy.modeling import models
    >>> rotation = models.Rotation2D(angle=23.7)
    >>> f = AsdfFile()
    >>> f.tree['model'] = rotation
    >>> f.write_to('rotation.asdf')

To read the file and create the model:

    >>> import asdf
    >>> with asdf.open('rotation.asdf') as f:
    ...     model = f.tree['model']
    >>> print(model)
    Model: Rotation2D
    Inputs: ('x', 'y')
    Outputs: ('x', 'y')
    Model set size: 1
    Parameters:
        angle
        -----
         23.7

Compound models can also be serialized. Please note that some model attributes (e.g `meta`, `tied` parameter constraints used in fitting), as well as model sets are not yet serializable. For more information on serialization of models, see asdf-astropy:asdf-astropy.