===================== Line/Spectrum Fitting ===================== One of the primary tasks in spectroscopic analysis is fitting models of spectra. This concept is often applied mainly to line-fitting, but the same general approach applies to continuum fitting or even full-spectrum fitting. ``specutils`` provides conveniences that aim to leverage the general fitting framework of `astropy.modeling` to spectral-specific tasks. At a high level, this fitting takes the `~specutils.Spectrum1D` object and a list of `~astropy.modeling.Model` objects that have initial guesses for each of the parameters. these are used to create a compound model created from the model initial guesses. This model is then actually fit to the spectrum's ``flux``, yielding a single composite model result (which can be split back into its components if desired). Model (Line) Fitting -------------------- The generic model fitting machinery is well-suited to fitting spectral lines. The first step is to create a set of models with initial guesses as the parameters. To acheive better fits it may be wise to include a set of bounds for each parameter, but that is optional. .. note:: A method to make plausible initial guesses will be provided in a future version, but user defined initial guesses are required at present. Below are a series of examples of this sort of fitting. Simple Example ^^^^^^^^^^^^^^ Below is a simple example to demonstrate how to use the `~specutils.fitting.fit_lines` method to fit a spectrum to an Astropy model initial guess. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(0) x = np.linspace(0., 10., 200) y = 3 * np.exp(-0.5 * (x- 6.3)**2 / 0.8**2) y += np.random.normal(0., 0.2, x.shape) spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit the spectrum and calculate the fitted flux values (``y_fit``) g_init = models.Gaussian1D(amplitude=3.*u.Jy, mean=6.1*u.um, stddev=1.*u.um) g_fit = fit_lines(spectrum, g_init) y_fit = g_fit(x*u.um) # Plot the original spectrum and the fitted. plt.plot(x, y) plt.plot(x, y_fit) plt.title('Single fit peak') plt.grid('on') plt.legend('Original Spectrum', 'Specutils Fit Result') Simple Example with Different Units ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Similar fit example to above, but the Gaussian model initial guess has different units. The fit will convert the initial guess to the spectral units, fit and then output the fitted model in the spectrum units. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(0) x = np.linspace(0., 10., 200) y = 3 * np.exp(-0.5 * (x- 6.3)**2 / 0.8**2) y += np.random.normal(0., 0.2, x.shape) # Create the spectrum spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit the spectrum g_init = models.Gaussian1D(amplitude=3.*u.Jy, mean=61000*u.AA, stddev=10000.*u.AA) g_fit = fit_lines(spectrum, g_init) y_fit = g_fit(x*u.um) plt.plot(x, y) plt.plot(x, y_fit) plt.title('Single fit peak, different model units') plt.grid('on') Single Peak Fit Within a Window (Defined by Center) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Single peak fit with a window of ``2*u.um`` around the center of the mean of the model initial guess (so ``2*u.um`` around ``5.5*u.um``). .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(0) x = np.linspace(0., 10., 200) y = 3 * np.exp(-0.5 * (x- 6.3)**2 / 0.8**2) y += np.random.normal(0., 0.2, x.shape) # Create the spectrum spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit the spectrum g_init = models.Gaussian1D(amplitude=3.*u.Jy, mean=5.5*u.um, stddev=1.*u.um) g_fit = fit_lines(spectrum, g_init, window=2*u.um) y_fit = g_fit(x*u.um) plt.plot(x, y) plt.plot(x, y_fit) plt.title('Single fit peak window') plt.grid('on') Single Peak Fit Within a Window (Defined by Left and Right) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Single peak fit using spectral data *only* within the window ``6*u.um`` to ``7*u.um``, all other data will be ignored. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(0) x = np.linspace(0., 10., 200) y = 3 * np.exp(-0.5 * (x- 6.3)**2 / 0.8**2) y += np.random.normal(0., 0.2, x.shape) # Create the spectrum spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit the spectrum g_init = models.Gaussian1D(amplitude=3.*u.Jy, mean=5.5*u.um, stddev=1.*u.um) g_fit = fit_lines(spectrum, g_init, window=(6*u.um, 7*u.um)) y_fit = g_fit(x*u.um) plt.plot(x, y) plt.plot(x, y_fit) plt.title('Single fit peak window') plt.grid('on') Double Peak Fit ^^^^^^^^^^^^^^^ Double peak fit compound model initial guess in and compound model out. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(42) g1 = models.Gaussian1D(1, 4.6, 0.2) g2 = models.Gaussian1D(2.5, 5.5, 0.1) x = np.linspace(0, 10, 200) y = g1(x) + g2(x) + np.random.normal(0., 0.2, x.shape) # Create the spectrum to fit spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit the spectrum g1_init = models.Gaussian1D(amplitude=2.3*u.Jy, mean=5.6*u.um, stddev=0.1*u.um) g2_init = models.Gaussian1D(amplitude=1.*u.Jy, mean=4.4*u.um, stddev=0.1*u.um) g12_fit = fit_lines(spectrum, g1_init+g2_init) y_fit = g12_fit(x*u.um) plt.plot(x, y) plt.plot(x, y_fit) plt.title('Double Peak Fit') plt.grid('on') Double Peak Fit Within a Window ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Double peak fit using data in the spectrum from ``4.3*u.um`` to ``5.3*u.um``, only. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(42) g1 = models.Gaussian1D(1, 4.6, 0.2) g2 = models.Gaussian1D(2.5, 5.5, 0.1) x = np.linspace(0, 10, 200) y = g1(x) + g2(x) + np.random.normal(0., 0.2, x.shape) # Create the spectrum to fit spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit the spectrum g2_init = models.Gaussian1D(amplitude=1.*u.Jy, mean=4.7*u.um, stddev=0.2*u.um) g2_fit = fit_lines(spectrum, g2_init, window=(4.3*u.um, 5.3*u.um)) y_fit = g2_fit(x*u.um) plt.plot(x, y) plt.plot(x, y_fit) plt.title('Double Peak Fit Within a Window') plt.grid('on') Double Peak Fit Within Around a Center Window ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Double peak fit using data in the spectrum centered on ``4.7*u.um`` +/- ``0.3*u.um``. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(42) g1 = models.Gaussian1D(1, 4.6, 0.2) g2 = models.Gaussian1D(2.5, 5.5, 0.1) x = np.linspace(0, 10, 200) y = g1(x) + g2(x) + np.random.normal(0., 0.2, x.shape) # Create the spectrum to fit spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit the spectrum g2_init = models.Gaussian1D(amplitude=1.*u.Jy, mean=4.7*u.um, stddev=0.2*u.um) g2_fit = fit_lines(spectrum, g2_init, window=0.3*u.um) y_fit = g2_fit(x*u.um) plt.plot(x, y) plt.plot(x, y_fit) plt.title('Double Peak Fit Around a Center Window') plt.grid('on') Double Peak Fit - Two Separate Peaks ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Double peak fit where each model ``gl_init`` and ``gr_init`` is fit separately, each within ``0.2*u.um`` of the model's mean. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(42) g1 = models.Gaussian1D(1, 4.6, 0.2) g2 = models.Gaussian1D(2.5, 5.5, 0.1) x = np.linspace(0, 10, 200) y = g1(x) + g2(x) + np.random.normal(0., 0.2, x.shape) # Create the spectrum to fit spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit each peak gl_init = models.Gaussian1D(amplitude=1.*u.Jy, mean=4.8*u.um, stddev=0.2*u.um) gr_init = models.Gaussian1D(amplitude=2.*u.Jy, mean=5.3*u.um, stddev=0.2*u.um) gl_fit, gr_fit = fit_lines(spectrum, [gl_init, gr_init], window=0.2*u.um) yl_fit = gl_fit(x*u.um) yr_fit = gr_fit(x*u.um) plt.plot(x, y) plt.plot(x, yl_fit) plt.plot(x, yr_fit) plt.title('Double Peak - Two Models') plt.grid('on') Double Peak Fit - Two Separate Peaks With Two Windows ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Double peak fit where each model ``gl_init`` and ``gr_init`` is fit within the corresponding window. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(42) g1 = models.Gaussian1D(1, 4.6, 0.2) g2 = models.Gaussian1D(2.5, 5.5, 0.1) x = np.linspace(0, 10, 200) y = g1(x) + g2(x) + np.random.normal(0., 0.2, x.shape) # Create the spectrum to fit spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit each peak gl_init = models.Gaussian1D(amplitude=1.*u.Jy, mean=4.8*u.um, stddev=0.2*u.um) gr_init = models.Gaussian1D(amplitude=2.*u.Jy, mean=5.3*u.um, stddev=0.2*u.um) gl_fit, gr_fit = fit_lines(spectrum, [gl_init, gr_init], window=[(5.3*u.um, 5.8*u.um), (4.6*u.um, 5.3*u.um)]) yl_fit = gl_fit(x*u.um) yr_fit = gr_fit(x*u.um) plt.plot(x, y) plt.plot(x, yl_fit) plt.plot(x, yr_fit) plt.title('Double Peak - Two Models and Two Windows') plt.grid('on') Double Peak Fit - Exclude One Region ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Double peak fit where each model ``gl_init`` and ``gr_init`` is fit using all the data *except* between ``5.2*u.um`` and ``5.8*u.um``. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D, SpectralRegion from specutils.fitting import fit_lines # Create a simple spectrum with a Gaussian. np.random.seed(42) g1 = models.Gaussian1D(1, 4.6, 0.2) g2 = models.Gaussian1D(2.5, 5.5, 0.1) x = np.linspace(0, 10, 200) y = g1(x) + g2(x) + np.random.normal(0., 0.2, x.shape) # Create the spectrum to fit spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) # Fit each peak gl_init = models.Gaussian1D(amplitude=1.*u.Jy, mean=4.8*u.um, stddev=0.2*u.um) gl_fit = fit_lines(spectrum, gl_init, exclude_regions=[SpectralRegion(5.2*u.um, 5.8*u.um)]) yl_fit = gl_fit(x*u.um) plt.plot(x, y) plt.plot(x, yl_fit) plt.title('Double Peak - Single Models and Exclude Region') plt.grid('on') .. _specutils-continuum-fitting: Continuum Fitting ----------------- While the line-fitting machinery can be used to fit continuua at the same time as models, often it is convenient to subtract or normalize a spectrum by its continuum before other processing is done. ``specutils`` provides some convenience functions to perform exactly this task. An example is shown below. .. plot:: :include-source: :align: center import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models from astropy import units as u from specutils.spectra import Spectrum1D, SpectralRegion from specutils.fitting import fit_generic_continuum np.random.seed(0) x = np.linspace(0., 10., 200) y = 3 * np.exp(-0.5 * (x - 6.3)**2 / 0.1**2) y += np.random.normal(0., 0.2, x.shape) y_continuum = 3.2 * np.exp(-0.5 * (x - 5.6)**2 / 4.8**2) y += y_continuum spectrum = Spectrum1D(flux=y*u.Jy, spectral_axis=x*u.um) g1_fit = fit_generic_continuum(spectrum) y_continuum_fitted = g1_fit(x) plt.plot(x, y) plt.plot(x, y_continuum_fitted) plt.title('Continuum Fitting') plt.grid('on') Reference/API ------------- .. automodapi:: specutils.fitting :no-heading: