# Defining New Model Classes#

This document describes how to add a model to the package or to create a user-defined model. In short, one needs to define all model parameters and write a function which evaluates the model, that is, computes the mathematical function that implements the model. If the model is fittable, a function to compute the derivatives with respect to parameters is required if a linear fitting algorithm is to be used and optional if a non-linear fitter is to be used.

## Basic custom models#

For most cases, the `custom_model` decorator provides an easy way to make a new `Model` class from an existing Python callable. The following example demonstrates how to set up a model consisting of two Gaussians:

```import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling.models import custom_model
from astropy.modeling.fitting import LevMarLSQFitter

# Define model
@custom_model
def sum_of_gaussians(x, amplitude1=1., mean1=-1., sigma1=1.,
amplitude2=1., mean2=1., sigma2=1.):
return (amplitude1 * np.exp(-0.5 * ((x - mean1) / sigma1)**2) +
amplitude2 * np.exp(-0.5 * ((x - mean2) / sigma2)**2))

# Generate fake data
rng = np.random.default_rng(0)
x = np.linspace(-5., 5., 200)
m_ref = sum_of_gaussians(amplitude1=2., mean1=-0.5, sigma1=0.4,
amplitude2=0.5, mean2=2., sigma2=1.0)
y = m_ref(x) + rng.normal(0., 0.1, x.shape)

# Fit model to data
m_init = sum_of_gaussians()
fit = LevMarLSQFitter()
m = fit(m_init, x, y)

# Plot the data and the best fit
plt.plot(x, y, 'o', color='k')
plt.plot(x, m(x))
```

This decorator also supports setting a model’s `fit_deriv` as well as creating models with more than one inputs. Note that when creating a model from a function with multiple outputs, the keyword argument `n_outputs` must be set to the number of outputs of the function. It can also be used as a normal factory function (for example `SumOfGaussians = custom_model(sum_of_gaussians)`) rather than as a decorator. See the `custom_model` documentation for more examples.

## A step by step definition of a 1-D Gaussian model#

The example described in Basic custom models can be used for most simple cases, but the following section describes how to construct model classes in general. Defining a full model class may be desirable, for example, to provide more specialized parameters, or to implement special functionality not supported by the basic `custom_model` factory function.

The details are explained below with a 1-D Gaussian model as an example. There are two base classes for models. If the model is fittable, it should inherit from `FittableModel`; if not it should subclass `Model`.

If the model takes parameters they should be specified as class attributes in the model’s class definition using the `Parameter` descriptor. All arguments to the Parameter constructor are optional, and may include a default value for that parameter, a text description of the parameter (useful for `help` and documentation generation), as well default constraints and custom getters/setters for the parameter value. It is also possible to define a “validator” method for each parameter, enabling custom code to check whether that parameter’s value is valid according to the model definition (for example if it must be non-negative). See the example in `Parameter.validator` for more details. Note, that if pickling the model is important the validator function should be assigned directly to the instance `Parameter._validator` instead of using the decorator.

```from astropy.modeling import Fittable1DModel, Parameter

class Gaussian1D(Fittable1DModel):
n_inputs = 1
n_outputs = 1

amplitude = Parameter()
mean = Parameter()
stddev = Parameter()
```

The `n_inputs` and `n_outputs` class attributes must be integers indicating the number of independent variables that are input to evaluate the model, and the number of outputs it returns. The labels of the inputs and outputs, `inputs` and `outputs`, are generated automatically. It is possible to overwrite the default ones by assigning the desired values in the class `__init__` method, after calling `super`. `outputs` and `inputs` must be tuples of strings with length `n_outputs` and `n_inputs` respectively. Outputs may have the same labels as inputs (eg. `inputs = ('x', 'y')` and `outputs = ('x', 'y')`). However, inputs must not conflict with each other (eg. `inputs = ('x', 'x')` is incorrect) and likewise for outputs.

There are two helpful base classes in the modeling package that can be used to avoid specifying `n_inputs` and `n_outputs` for most common models. These are `Fittable1DModel` and `Fittable2DModel`. For example, the actual `Gaussian1D` model is a subclass of `Fittable1DModel`. This helps cut down on boilerplate by not having to specify `n_inputs`, `n_outputs`, `inputs` and `outputs` for many models (follow the link to Gaussian1D to see its source code, for example).

Fittable models can be linear or nonlinear in a regression sense. The default value of the `linear` attribute is `False`. Linear models should define the `linear` class attribute as `True`. Because this model is non-linear we can stick with the default.

Models which inherit from `Fittable1DModel` have the `Model._separable` property already set to `True`. All other models should define this property to indicate the Model Separability.

Next, provide methods called `evaluate` to evaluate the model and `fit_deriv`, to compute its derivatives with respect to parameters. These may be normal methods, `classmethod`, or `staticmethod`, though the convention is to use `staticmethod` when the function does not depend on any of the object’s other attributes (i.e., it does not reference `self`) or any of the class’s other attributes as in the case of `classmethod`. The evaluation method takes all input coordinates as separate arguments and all of the model’s parameters in the same order they would be listed by `param_names`.

For this example:

```@staticmethod
def evaluate(x, amplitude, mean, stddev):
return amplitude * np.exp((-(1 / (2. * stddev**2)) * (x - mean)**2))
```

It should be made clear that the `evaluate` method must be designed to take the model’s parameter values as arguments. This may seem at odds with the fact that the parameter values are already available via attribute of the model (eg. `model.amplitude`). However, passing the parameter values directly to `evaluate` is a more efficient way to use it in many cases, such as fitting.

Users of your model would not generally use `evaluate` directly. Instead they create an instance of the model and call it on some input. The `__call__` method of models uses `evaluate` internally, but users do not need to be aware of it. The default `__call__` implementation also handles details such as checking that the inputs are correctly formatted and follow Numpy’s broadcasting rules before attempting to evaluate the model.

Like `evaluate`, the `fit_deriv` method takes as input all coordinates and all parameter values as arguments. There is an option to compute numerical derivatives for nonlinear models in which case the `fit_deriv` method should be `None`:

```@staticmethod
def fit_deriv(x, amplitude, mean, stddev):
d_amplitude = np.exp(- 0.5 / stddev**2 * (x - mean)**2)
d_mean = (amplitude *
np.exp(- 0.5 / stddev**2 * (x - mean)**2) *
(x - mean) / stddev**2)
d_stddev = (2 * amplitude *
np.exp(- 0.5 / stddev**2 * (x - mean)**2) *
(x - mean)**2 / stddev**3)
return [d_amplitude, d_mean, d_stddev]
```

Note that we did not have to define an `__init__` method or a `__call__` method for our model. For most models the `__init__` follows the same pattern, taking the parameter values as positional arguments, followed by several optional keyword arguments (constraints, etc.). The modeling framework automatically generates an `__init__` for your class that has the correct calling signature (see for yourself by calling `help(Gaussian1D.__init__)` on the example model we just defined).

There are cases where it might be desirable to define a custom `__init__`. For example, the `Gaussian2D` model takes an optional `cov_matrix` argument which can be used as an alternative way to specify the x/y_stddev and theta parameters. This is perfectly valid so long as the `__init__` determines appropriate values for the actual parameters and then calls the super `__init__` with the standard arguments. Schematically this looks something like:

```def __init__(self, amplitude, x_mean, y_mean, x_stddev=None,
y_stddev=None, theta=None, cov_matrix=None, **kwargs):
# The **kwargs here should be understood as other keyword arguments
# accepted by the basic Model.__init__ (such as constraints)
if cov_matrix is not None:
# Set x/y_stddev and theta from the covariance matrix
x_stddev = ...
y_stddev = ...
theta = ...

# Don't pass on cov_matrix since it doesn't mean anything to the base
# class
super().__init__(amplitude, x_mean, y_mean, x_stddev, y_stddev, theta,
**kwargs)
```

### Full example#

```import numpy as np
from astropy.modeling import Fittable1DModel, Parameter

class Gaussian1D(Fittable1DModel):
amplitude = Parameter()
mean = Parameter()
stddev = Parameter()

@staticmethod
def evaluate(x, amplitude, mean, stddev):
return amplitude * np.exp((-(1 / (2. * stddev**2)) * (x - mean)**2))

@staticmethod
def fit_deriv(x, amplitude, mean, stddev):
d_amplitude = np.exp((-(1 / (stddev**2)) * (x - mean)**2))
d_mean = (2 * amplitude *
np.exp((-(1 / (stddev**2)) * (x - mean)**2)) *
(x - mean) / (stddev**2))
d_stddev = (2 * amplitude *
np.exp((-(1 / (stddev**2)) * (x - mean)**2)) *
((x - mean)**2) / (stddev**3))
return [d_amplitude, d_mean, d_stddev]
```

## A full example of a LineModel#

This example demonstrates one other optional feature for model classes, which is an inverse. An `inverse` implementation should be a `property` that returns a new model instance (not necessarily of the same class as the model being inverted) that computes the inverse of that model, so that for some model instance with an inverse, ```model.inverse(model(*input)) == input```.

```import numpy as np
from astropy.modeling import Fittable1DModel, Parameter

class LineModel(Fittable1DModel):
slope = Parameter()
intercept = Parameter()
linear = True

@staticmethod
def evaluate(x, slope, intercept):
return slope * x + intercept

@staticmethod
def fit_deriv(x, slope, intercept):
d_slope = x
d_intercept = np.ones_like(x)
return [d_slope, d_intercept]

@property
def inverse(self):
new_slope = self.slope ** -1
new_intercept = -self.intercept / self.slope
return LineModel(slope=new_slope, intercept=new_intercept)
```

Note

The above example is essentially equivalent to the built-in `Linear1D` model.