# 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 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`

.

`Model.bounding_box`

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
... @property
... 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))
...
>>> model = Ellipsoid3D()
>>> model.bounding_box
((-4.0, 4.0), (-3.0, 3.0), (-2.0, 2.0))
```

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):

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

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.

### 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
np.random.seed(0)
nsrc = 100
model_params = [
dict(amplitude=np.random.uniform(.5, 1),
x_mean=np.random.uniform(0, imshape[1] - 1),
y_mean=np.random.uniform(0, imshape[0] - 1),
x_stddev=np.random.uniform(2, 6),
y_stddev=np.random.uniform(2, 6),
theta=np.random.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
plt.figure(figsize=(16, 7))
plt.subplots_adjust(left=.05, right=.97, bottom=.03, top=.97, wspace=0.15)
# Full model image
plt.subplot(121)
plt.imshow(full_image, origin='lower')
plt.title('Full Models\nTiming: {:.2f} seconds'.format(t_full), fontsize=16)
plt.xlabel('x')
plt.ylabel('y')
# Bounded model image with boxes overplotted
ax = plt.subplot(122)
plt.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)
ax.add_patch(r)
plt.title('Bounded Models\nTiming: {:.2f} seconds'.format(t_bb), fontsize=16)
plt.xlabel('x')
plt.ylabel('y')
# Difference image
plt.figure(figsize=(16, 8))
plt.subplot(111)
plt.imshow(diff, vmin=-max_err, vmax=max_err)
plt.colorbar(format='%.1e')
plt.title('Difference Image\nTotal Flux Err = {:.0e}'.format(
((flux - np.sum(bb_image)) / flux)))
plt.xlabel('x')
plt.ylabel('y')
plt.show()
```

### 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.

## 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 Details.