# 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 simultaneously fit Gaussians to the emission lines while linking their wavelengths and linking the flux of the [OIII] λ4959 line to the [OIII] λ5007 line.

import numpy as np
from astropy.io import ascii
from astropy.modeling import fitting, models
from astropy.utils.data import get_pkg_data_filename
from matplotlib import pyplot as plt

fname = get_pkg_data_filename("data/spec.txt", package="astropy.modeling.tests")
wave = spec["lambda"]
flux = spec["flux"]

# Use the (vacuum) rest wavelengths of known lines as initial values
# for the fit.
Hbeta = 4862.721
O3_4959 = 4960.295
O3_5007 = 5008.239

# Create Gaussian1D models for each of the H-beta and [OIII] lines.
hbeta_narrow = models.Gaussian1D(amplitude=20, mean=Hbeta, stddev=2)
o3_4959 = models.Gaussian1D(amplitude=70, mean=O3_4959, stddev=2)
o3_5007 = models.Gaussian1D(amplitude=180, mean=O3_5007, stddev=2)

# 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 four emission lines and the continuum.
model = hbeta_broad + hbeta_narrow + o3_4959 + o3_5007 + poly_cont

# Tie the ratio of the intensity of the two [OIII] lines.
def tie_o3_ampl(model):
return model.amplitude_3 / 2.98

o3_4959.amplitude.tied = tie_o3_ampl

# Tie the wavelengths of the two [OIII] lines
def tie_o3_wave(model):
return model.mean_3 * O3_4959 / O3_5007

o3_4959.mean.tied = tie_o3_wave

# Tie the wavelengths of the two (narrow and broad) H-beta lines
def tie_hbeta_wave1(model):
return model.mean_1

# Tie the wavelengths of the H-beta lines to the [OIII] 5007 line
def tie_hbeta_wave2(model):
return model.mean_3 * Hbeta / O3_5007

hbeta_narrow.mean.tied = tie_hbeta_wave2

# Simultaneously fit all the emission lines and continuum.
fitter = fitting.LevMarLSQFitter()
fitted_model = fitter(model, wave, flux)
fitted_lines = fitted_model(wave)

# Plot the data and the fitted model
fig = plt.figure(figsize=(9, 6))
plt.plot(wave, flux, label="Data")
plt.plot(wave, fitted_lines, color="C1", label="Fitted Model")
plt.legend(loc="upper left")
plt.xlabel("Wavelength (Angstrom)")
plt.ylabel("Flux")
plt.text(4860, 45, r"$H\beta$ (broad + narrow)", rotation=90)
plt.text(4958, 68, r"[OIII] $\lambda 4959$", rotation=90)
plt.text(4995, 140, r"[OIII] $\lambda 5007$", rotation=90)
plt.xlim(4700, 5100)
plt.show()