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- Astropy v5.2.1 »
- Models and Fitting (`astropy.modeling`) »
- Fitting with constraints

# Fitting with constraints

`fitting` support constraints, however, different fitters support different types of constraints. The `supported_constraints` attribute shows the type of constraints supported by a specific fitter:

    >>> from astropy.modeling import fitting
    >>> fitting.LinearLSQFitter.supported_constraints
    ['fixed']
    >>> fitting.LevMarLSQFitter.supported_constraints
    ['fixed', 'tied', 'bounds']
    >>> fitting.SLSQPLSQFitter.supported_constraints
    ['bounds', 'eqcons', 'ineqcons', 'fixed', 'tied']

## Fixed Parameter Constraint

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 the corresponding term will be subtracted from the data before fitting a polynomial without that term to the result. For example, fixing `c0` in a polynomial model will fit a polynomial with the zero-th order term missing to the data minus that constant. The fixed coefficients and corresponding terms are restored to the fit polynomial and this is the polynomial returned from the fitter:

    >>> import numpy as np
    >>> rng = np.random.default_rng(seed=12345)
    >>> from astropy.modeling import models, fitting
    >>> x = np.arange(1, 10, .1)
    >>> p1 = models.Polynomial1D(2, c0=[1, 1], c1=[2, 2], c2=[3, 3],
    ...                          n_models=2)
    >>> p1
    <Polynomial1D(2, c0=[1., 1.], c1=[2., 2.], c2=[3., 3.], n_models=2)>
    >>> y = p1(x, model_set_axis=False)
    >>> n = (rng.standard_normal(y.size)).reshape(y.shape)
    >>> p1.c0.fixed = True
    >>> pfit = fitting.LinearLSQFitter()
    >>> new_model = pfit(p1, x, y + n)
    >>> print(new_model)
    Model: Polynomial1D
    Inputs: ('x',)
    Outputs: ('y',)
    Model set size: 2
    Degree: 2
    Parameters:
         c0         c1                 c2
        --- ------------------ ------------------
        1.0  2.072116176718454   2.99115839177437
        1.0 1.9818866652726403 3.0024208951927585

The syntax to fix the same parameter `c0` using an argument to the model instead of `p1.c0.fixed = True` would be:

    >>> p1 = models.Polynomial1D(2, c0=[1, 1], c1=[2, 2], c2=[3, 3],
    ...                          n_models=2, fixed={'c0': True})

## Bounded Constraints

Bounded fitting is supported through the `bounds` arguments to models or by setting `min` and `max` attributes on a parameter. Bounds for the `LevMarLSQFitter` 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 `SLSQPLSQFitter` optimization algorithm handles bounds internally.

## Tied Constraints

The `tied` constraint is often useful with Compound models. In this example we will read a spectrum from a file called `spec.txt` and fit Gaussians to the lines simultaneously while linking the flux of the OIII_1 and OIII_2 lines.

    import numpy as np
    from astropy.io import ascii
    from astropy.utils.data import get_pkg_data_filename
    from astropy.modeling import models, fitting
    fname = get_pkg_data_filename('data/spec.txt', package='astropy.modeling.tests')
    spec = ascii.read(fname)
    wave = spec['lambda']
    flux = spec['flux']

    # Use the rest wavelengths of known lines as initial values for the fit.

    Hbeta = 4862.721
    OIII_1 = 4958.911
    OIII_2 = 5008.239

    # Create Gaussian1D models for each of the Hbeta and OIII lines.

    h_beta = models.Gaussian1D(amplitude=34, mean=Hbeta, stddev=5)
    o3_2 = models.Gaussian1D(amplitude=170, mean=OIII_2, stddev=5)
    o3_1 = models.Gaussian1D(amplitude=57, mean=OIII_1, stddev=5)

    # Tie the ratio of the intensity of the two OIII lines.

    def tie_ampl(model):
        return model.amplitude_2 / 3.1

    o3_1.amplitude.tied = tie_ampl

    # Also tie the wavelength of the Hbeta line to the OIII wavelength.

    def tie_wave(model):
        return model.mean_0 * OIII_1 / Hbeta

    o3_1.mean.tied = tie_wave

    # Create a Polynomial model to fit the continuum.

    mean_flux = flux.mean()
    cont = np.where(flux > mean_flux, mean_flux, flux)
    linfitter = fitting.LinearLSQFitter()
    poly_cont = linfitter(models.Polynomial1D(1), wave, cont)

    # Create a compound model for the three lines and the continuum.

    hbeta_combo = h_beta + o3_1 + o3_2 + poly_cont

    # Fit all lines simultaneously -
    # this will need one iteration more than the default of 100.

    fitter = fitting.LevMarLSQFitter()
    fitted_model = fitter(hbeta_combo, wave, flux, maxiter=111)
    fitted_lines = fitted_model(wave)

    from matplotlib import pyplot as plt
    fig = plt.figure(figsize=(9, 6))
    p = plt.plot(wave, flux, label="data")
    p = plt.plot(wave, fitted_lines, 'r', label="fit")
    p = plt.legend()
    p = plt.xlabel("Wavelength")
    p = plt.ylabel("Flux")
    t = plt.text(4800, 70, 'Hbeta', rotation=90)
    t = plt.text(4900, 100, 'OIII_1', rotation=90)
    t = plt.text(4950, 180, 'OIII_2', rotation=90)
    plt.show()

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