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- Models And Fitting (`astropy.modeling`) »
- Fitting Models to Data

# Fitting Models to Data

This module provides wrappers, called Fitters, around some Numpy and Scipy fitting functions. All Fitters can be called as functions. They take an instance of `ParametricModel` as input and modify `parameters` attribute. The idea is to make this extensible and allow users to easily add other fitters.

Linear fitting is done using Numpy’s `lstsq` function. There are currently two non-linear fitters which use `leastsq` and `fmin_slsqp`.

The rules for passing input to fitters are:

- Non-linear fitters work only with single data sets.
- The linear fitter can fit a single input to multiple data sets creating multiple parameter sets. For example fitting a 2D model with input x, y arrays of shape (n, m) to a z array of shape (p, n, m), will set model.parameters.n_inputs to p, even if it was 1 when the model was created.
- Attempting to fit a model with multiple parameter sets to a single data set results in an error.

## Fitting examples

- Fitting a polynomial model to multiple data sets simultaneously:

      >>> from astropy.modeling import models, fitting
      >>> import numpy as np
      >>> p1 = models.Polynomial1D(3)
      >>> p1.c0 = 1
      >>> p1.c1 = 2
      >>> p1.parameters
      array([ 1.,  2.,  0.,  0.])
      >>> x = np.arange(10)
      >>> y = p1(x)
      >>> yy = np.array([y, y]).T
      >>> p2 = models.Polynomial1D(3, param_dim=2)
      >>> pfit = fitting.LinearLSQFitter()
      >>> new_model = pfit(p2, x, yy)
      >>> print(new_model.param_sets)
      [[  1.00000000e+00   1.00000000e+00]
       [  2.00000000e+00   2.00000000e+00]
       [  3.88335494e-16   3.88335494e-16]
       [ -2.99749607e-17  -2.99749607e-17]]

Fitters support constrained fitting.

- All fitters support fixed (frozen) parameters through the `fixed` argument to models or setting the `fixed` attribute directly on a parameter.

  For linear fitters, freezing a polynomial coefficient means that a polynomial without that term will be fitted to the data. For example, fixing `c0` in a polynomial model will fit a polynomial with the zero-th order term missing. However, the fixed value of the coefficient is used when evaluating the model:

      >>> x = np.arange(1, 10, .1)
      >>> p1 = models.Polynomial1D(2, param_dim=2)
      >>> p1.parameters = [1, 1, 2, 2, 3, 3]
      >>> p1.param_sets
      array([[ 1.,  1.],
             [ 2.,  2.],
             [ 3.,  3.]])
      >>> y = p1(x)
      >>> p1.c0.fixed = True
      >>> pfit = fitting.LinearLSQFitter()
      >>> new_model = pfit(p1, x, y)
      ...
      >>> new_model.param_sets
      array([[ 1.,          1.        ],
             [ 2.38641216,  2.38641216],
             [ 2.96827886,  2.96827886]])

- A parameter can be `tied` (linked to another parameter). This can be done in two ways:

      >>> def tiedfunc(g1):
      ...    mean = 3 * g1.stddev
      ...    return mean
      >>> g1 = models.Gaussian1D(amplitude=10., mean=3, stddev=.5,
      ...                        tied={'mean': tiedfunc})

  or:

      >>> g1 = models.Gaussian1D(amplitude=10., mean=3, stddev=.5)
      >>> g1.mean.tied = tiedfunc
      >>> gfit = fitting.NonLinearLSQFitter()

Bounded fitting is supported through the `bounds` arguments to models or by setting `min` and `max` attributes on a parameter. Bounds for the `NonLinearLSQFitter` are always exactly satisfied–if the value of the parameter is outside the fitting interval, it will be reset to the value at the bounds. The `SLSQPFitter` handles bounds internally.

- Different fitters support different types of constraints:

      >>> fitting.LinearLSQFitter.supported_constraints
      ['fixed']
      >>> fitting.NonLinearLSQFitter.supported_constraints
      ['fixed', 'tied', 'bounds']
      >>> fitting.SLSQPFitter.supported_constraints
      ['bounds', 'eqcons', 'ineqcons', 'fixed', 'tied']