# 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
np.random.seed(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) + np.random.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. 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.

from astropy.modeling import Fittable1DModel, Parameter

class Gaussian1D(Fittable1DModel):
inputs = ('x',)
outputs = ('y',)

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


The inputs and outputs class attributes must be tuples of strings 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 (in this case 'x' and 'y' respectively) are currently used for informational purposes only and have no requirements on them other than that they do not conflict with parameter names. 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. The lengths of these tuples are important for specifying the correct number of inputs and outputs. These attributes supersede the n_inputs and n_outputs attributes in older versions of this package.

There are two helpful base classes in the modeling package that can be used to avoid specifying inputs and outputs for most common models. These are Fittable1DModel and Fittable2DModel. For example, the real Gaussian1D model is actually a subclass of Fittable1DModel. This helps cut down on boilerplate by not having to specify 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((-(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]


Note that we did not have to define an __init__ method or a __call__ method for our model (this contrasts with Astropy versions 0.4.x and earlier). 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¶

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.

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

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.

# Defining New Fitter Classes¶

This section describes how to add a new nonlinear fitting algorithm to this package or write a user-defined fitter. In short, one needs to define an error function and a __call__ method and define the types of constraints which work with this fitter (if any).

The details are described below using scipy’s SLSQP algorithm as an example. The base class for all fitters is Fitter:

class SLSQPFitter(Fitter):
supported_constraints = ['bounds', 'eqcons', 'ineqcons', 'fixed',
'tied']

def __init__(self):
# Most currently defined fitters take no arguments in their
# __init__, but the option certainly exists for custom fitters
super().__init__()


All fitters take a model (their __call__ method modifies the model’s parameters) as their first argument.

Next, the error function takes a list of parameters returned by an iteration of the fitting algorithm and input coordinates, evaluates the model with them and returns some type of a measure for the fit. In the example the sum of the squared residuals is used as a measure of fitting.:

def objective_function(self, fps, *args):
model = args
meas = args[-1]
model.fitparams(fps)
res = self.model(*args[1:-1]) - meas
return np.sum(res**2)


The __call__ method performs the fitting. As a minimum it takes all coordinates as separate arguments. Additional arguments are passed as necessary:

def __call__(self, model, x, y , maxiter=MAXITER, epsilon=EPS):
if model.linear:
raise ModelLinearityException(
'Model is linear in parameters; '
'non-linear fitting methods should not be used.')
model_copy = model.copy()
init_values, _ = _model_to_fit_params(model_copy)
self.fitparams = optimize.fmin_slsqp(self.errorfunc, p0=init_values,
args=(y, x),
bounds=self.bounds,
eqcons=self.eqcons,
ineqcons=self.ineqcons)
return model_copy


## Defining a Plugin Fitter¶

astropy.modeling includes a plugin mechanism which allows fitters defined outside of astropy’s core to be inserted into the astropy.modeling.fitting namespace through the use of entry points. Entry points are references to importable objects. A tutorial on defining entry points can be found in setuptools’ documentation. Plugin fitters must to extend from the Fitter base class. For the fitter to be discovered and inserted into astropy.modeling.fitting the entry points must be inserted into the astropy.modeling entry point group

setup(
# ...
entry_points = {'astropy.modeling': 'PluginFitterName = fitter_module:PlugFitterClass'}
)


This would allow users to import the PlugFitterName through astropy.modeling.fitting by

from astropy.modeling.fitting import PlugFitterName


One project which uses this functionality is Saba and be can be used as a reference.

# Using a Custom Statistic Function¶

This section describes how to write a new fitter with a user-defined statistic function. The example below shows a specialized class which fits a straight line with uncertainties in both variables.

The following import statements are needed:

import numpy as np
from astropy.modeling.fitting import (_validate_model,
_fitter_to_model_params,
_model_to_fit_params, Fitter,
_convert_input)
from astropy.modeling.optimizers import Simplex


First one needs to define a statistic. This can be a function or a callable class.:

def chi_line(measured_vals, updated_model, x_sigma, y_sigma, x):
"""
Chi^2 statistic for fitting a straight line with uncertainties in x and
y.

Parameters
----------
measured_vals : array
updated_model : ~astropy.modeling.ParametricModel
model with parameters set by the current iteration of the optimizer
x_sigma : array
uncertainties in x
y_sigma : array
uncertainties in y

"""
model_vals = updated_model(x)
if x_sigma is None and y_sigma is None:
return np.sum((model_vals - measured_vals) ** 2)
elif x_sigma is not None and y_sigma is not None:
weights = 1 / (y_sigma ** 2 + updated_model.parameters ** 2 *
x_sigma ** 2)
return np.sum((weights * (model_vals - measured_vals)) ** 2)
else:
if x_sigma is not None:
weights = 1 / x_sigma ** 2
else:
weights = 1 / y_sigma ** 2
return np.sum((weights * (model_vals - measured_vals)) ** 2)


In general, to define a new fitter, all one needs to do is provide a statistic function and an optimizer. In this example we will let the optimizer be an optional argument to the fitter and will set the statistic to chi_line above:

class LineFitter(Fitter):
"""
Fit a straight line with uncertainties in both variables

Parameters
----------
optimizer : class or callable
one of the classes in optimizers.py (default: Simplex)
"""

def __init__(self, optimizer=Simplex):
self.statistic = chi_line
super().__init__(optimizer, statistic=self.statistic)


The last thing to define is the __call__ method:

def __call__(self, model, x, y, x_sigma=None, y_sigma=None, **kwargs):
"""
Fit data to this model.

Parameters
----------
model : ~astropy.modeling.core.ParametricModel
model to fit to x, y
x : array
input coordinates
y : array
input coordinates
x_sigma : array
uncertainties in x
y_sigma : array
uncertainties in y
kwargs : dict
optional keyword arguments to be passed to the optimizer

Returns
------
model_copy : ~astropy.modeling.core.ParametricModel
a copy of the input model with parameters set by the fitter

"""
model_copy = _validate_model(model,
self._opt_method.supported_constraints)

farg = _convert_input(x, y)
farg = (model_copy, x_sigma, y_sigma) + farg
p0, _ = _model_to_fit_params(model_copy)

fitparams, self.fit_info = self._opt_method(
self.objective_function, p0, farg, **kwargs)
_fitter_to_model_params(model_copy, fitparams)

return model_copy