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Gammapy Models

This is an introduction and overview on how to work with models in Gammapy.

The sub-package gammapy.modeling contains all the functionality related to modeling and fitting data. This includes spectral, spatial and temporal model classes, as well as the fit and parameter API. We will cover the follwing topics in order:

1. Spectral Models

2. Spatial Models

3. SkyModel and SkyDiffuseCube

4. Model Lists and Serialisation

5. Implementing as Custom Model

The models follow a naming scheme which contains the category as a suffix to the class name. An overview of all the available models can be found in the :ref:model-gallery.

Note that there is a separate tutorial modeling that explains about gammapy.modeling, the Gammapy modeling and fitting framework. You have to read that to learn how to work with models in order to analyse data.

Setup

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

from astropy import units as u
from gammapy.maps import Map

Spectral Models

All models are imported from the gammapy.modeling.models namespace. Let’s start with a PowerLawSpectralModel:

from gammapy.modeling.models import PowerLawSpectralModel

pwl = PowerLawSpectralModel()
print(pwl)

PowerLawSpectralModel

   name     value        unit      min max frozen   error
--------- --------- -------------- --- --- ------ ---------
    index 2.000e+00                nan nan  False 0.000e+00
amplitude 1.000e-12 cm-2 s-1 TeV-1 nan nan  False 0.000e+00
reference 1.000e+00            TeV nan nan   True 0.000e+00

To get a list of all available spectral models you can import and print the spectral model registry or take a look at the model gallery:

from gammapy.modeling.models import SPECTRAL_MODELS

print(SPECTRAL_MODELS)

Registry
--------

  ConstantSpectralModel                  : ConstantSpectralModel
  CompoundSpectralModel                  : CompoundSpectralModel
  PowerLawSpectralModel                  : PowerLawSpectralModel
  PowerLaw2SpectralModel                 : PowerLaw2SpectralModel
  SmoothBrokenPowerLawSpectralModel      : SmoothBrokenPowerLawSpectralModel
  ExpCutoffPowerLawSpectralModel         : ExpCutoffPowerLawSpectralModel
  ExpCutoffPowerLaw3FGLSpectralModel     : ExpCutoffPowerLaw3FGLSpectralModel
  SuperExpCutoffPowerLaw3FGLSpectralModel: SuperExpCutoffPowerLaw3FGLSpectralModel
  SuperExpCutoffPowerLaw4FGLSpectralModel: SuperExpCutoffPowerLaw4FGLSpectralModel
  LogParabolaSpectralModel               : LogParabolaSpectralModel
  TemplateSpectralModel                  : TemplateSpectralModel
  GaussianSpectralModel                  : GaussianSpectralModel
  AbsorbedSpectralModel                  : AbsorbedSpectralModel
  NaimaSpectralModel                     : NaimaSpectralModel
  ScaleSpectralModel                     : ScaleSpectralModel

Spectral models all come with default parameters. Different parameter values can be passed on creation of the model, either as a string defining the value and unit or as an ~astropy.units.Quantity object directly:

amplitude = 1e-12 * u.Unit("TeV-1 cm-2 s-1")
pwl = PowerLawSpectralModel(amplitude=amplitude, index=2.2)

For convenience a str specifying the value and unit can be passed as well:

pwl = PowerLawSpectralModel(amplitude="2.7e-12 TeV-1 cm-2 s-1", index=2.2)
print(pwl)

PowerLawSpectralModel

   name     value        unit      min max frozen   error
--------- --------- -------------- --- --- ------ ---------
    index 2.200e+00                nan nan  False 0.000e+00
amplitude 2.700e-12 cm-2 s-1 TeV-1 nan nan  False 0.000e+00
reference 1.000e+00            TeV nan nan   True 0.000e+00

The model can be evaluated at given energies by calling the model instance:

energy = [1, 3, 10, 30] * u.TeV
dnde = pwl(energy)
print(dnde)

[2.70000000e-12 2.40822469e-13 1.70358483e-14 1.51948705e-15] 1 / (cm2 s TeV)

The returned quantity is a differential photon flux.

For spectral models you can computed in addition the integrated and energy flux in a given energy range:

flux = pwl.integral(emin=1 * u.TeV, emax=10 * u.TeV)
print(flux)

eflux = pwl.energy_flux(emin=1 * u.TeV, emax=10 * u.TeV)
print(eflux)

2.108034597491956e-12 1 / (cm2 s)
4.982075849517389e-12 TeV / (cm2 s)

This also works for a list or an array of integration boundaries:

energy = [1, 3, 10, 30] * u.TeV
flux = pwl.integral(emin=energy[:-1], emax=energy[1:])
print(flux)

[1.64794383e-12 4.60090769e-13 1.03978226e-13] 1 / (cm2 s)

In some cases it can be useful to find use the inverse of a spectral model, to find the energy at which a given flux is reached:

dnde = 2.7e-12 * u.Unit("TeV-1 cm-2 s-1")
energy = pwl.inverse(dnde)
print(energy)

1.0 TeV

As a convenience you can also plot any spectral model in a given energy range:

pwl.plot(energy_range=[1, 100] * u.TeV)

<matplotlib.axes._subplots.AxesSubplot at 0x11341d438>

Spatial Models

Spatial models are imported from the same gammapy.modeling.models namespace, let’s start with a GaussianSpatialModel:

from gammapy.modeling.models import GaussianSpatialModel

gauss = GaussianSpatialModel(lon_0="0 deg", lat_0="0 deg", sigma="0.2 deg")
print(gauss)

GaussianSpatialModel

 name   value   unit    min        max    frozen   error
----- --------- ---- ---------- --------- ------ ---------
lon_0 0.000e+00  deg        nan       nan  False 0.000e+00
lat_0 0.000e+00  deg -9.000e+01 9.000e+01  False 0.000e+00
sigma 2.000e-01  deg  0.000e+00       nan  False 0.000e+00
    e 0.000e+00       0.000e+00 1.000e+00   True 0.000e+00
  phi 0.000e+00  deg        nan       nan   True 0.000e+00

Again you can check the SPATIAL_MODELS registry to see which models are available or take a look at the model gallery.

from gammapy.modeling.models import SPATIAL_MODELS

print(SPATIAL_MODELS)

Registry
--------

  ConstantSpatialModel: ConstantSpatialModel
  TemplateSpatialModel: TemplateSpatialModel
  DiskSpatialModel    : DiskSpatialModel
  GaussianSpatialModel: GaussianSpatialModel
  PointSpatialModel   : PointSpatialModel
  ShellSpatialModel   : ShellSpatialModel

The default coordinate frame for all spatial models is "icrs", but the frame can be modified using the frame argument:

gauss = GaussianSpatialModel(
    lon_0="0 deg", lat_0="0 deg", sigma="0.2 deg", frame="galactic"
)

You can specify any valid ~astropy.coordinates frame. The center position of the model can be retrieved as a ~astropy.coordinates.SkyCoord object using SpatialModel.position:

print(gauss.position)

<SkyCoord (Galactic): (l, b) in deg
    (0., 0.)>

Spatial models can be evaluated again by calling the instance:

lon = [0, 0.1] * u.deg
lat = [0, 0.1] * u.deg

flux_per_omega = gauss(lon, lat)
print(flux_per_omega)

[13061.88470839 10172.60603928] 1 / sr

The returned quantity corresponds to a surface brightness. Spatial model can be also evaluated using gammapy.maps.Map and gammapy.maps.Geom objects:

m = Map.create(skydir=(0, 0), width=(1, 1), binsz=0.02, frame="galactic")
m.quantity = gauss.evaluate_geom(m.geom)
m.plot(add_cbar=True);

Again for convenience the model can be plotted directly:

gauss.plot(add_cbar=True);

All spatial models have an associated sky region to it e.g. to illustrate the extend of the model on a sky image. The returned object is an ~regions.SkyRegion object:

print(gauss.to_region())

Region: EllipseSkyRegion
center: <SkyCoord (Galactic): (l, b) in deg
    (0., 0.)>
width: 0.4 deg
height: 0.4 deg
angle: 0.0 deg

Now we can plot the region on an sky image:

# create and plot the model
gauss_elongated = GaussianSpatialModel(
    lon_0="0 deg", lat_0="0 deg", sigma="0.2 deg", e=0.7, phi="45 deg"
)
ax = gauss_elongated.plot(add_cbar=True)

# add region illustration
region = gauss_elongated.to_region()
region_pix = region.to_pixel(ax.wcs)
ax.add_artist(region_pix.as_artist());

The .to_region() method can also be useful to write e.g. ds9 region files using write_ds9 from the regions package:

from regions import write_ds9

regions = [gauss.to_region(), gauss_elongated.to_region()]

filename = "regions.reg"
write_ds9(regions, filename, coordsys="galactic", fmt=".4f", radunit="deg")

!cat regions.reg

# Region file format: DS9 astropy/regions
galactic
ellipse(0.0000,0.0000,0.2000,0.2000,0.0000)
ellipse(96.3373,-60.1886,0.1428,0.2000,45.0000)

SkyModel and SkyDiffuseCube

The gammapy.modeling.models.SkyModel class combines a spectral and a spatial model. It can be created from existing spatial and spectral model components:

from gammapy.modeling.models import SkyModel

model = SkyModel(spectral_model=pwl, spatial_model=gauss, name="my-source")
print(model)

SkyModel

  Name                      : my-source
  Datasets names            : None
  Spectral model type       : PowerLawSpectralModel
  Spatial  model type       : GaussianSpatialModel
  Temporal model type       : None
  Parameters:
    index                   :   2.200
    amplitude               :   2.70e-12  1 / (cm2 s TeV)
    reference    (frozen)   :   1.000  TeV
    lon_0                   :   0.000  deg
    lat_0                   :   0.000  deg
    sigma                   :   0.200  deg
    e            (frozen)   :   0.000
    phi          (frozen)   :   0.000  deg

It is good practice to specify a name for your sky model, so that you can access it later by name and have meaningful identifier you serilisation. If you don’t define a name, a unique random name is generated:

model_without_name = SkyModel(spectral_model=pwl, spatial_model=gauss)
print(model_without_name.name)

NmXTDyQY

The spectral and spatial component of the source model can be accessed using .spectral_model and .spatial_model:

model.spectral_model

<gammapy.modeling.models.spectral.PowerLawSpectralModel at 0x11340d048>

model.spatial_model

<gammapy.modeling.models.spatial.GaussianSpatialModel at 0x113763b38>

And can be used as you have seen already seen above:

model.spectral_model.plot(energy_range=[1, 10] * u.TeV);

In some cases (e.g. when doing a spectral analysis) there is only a spectral model associated with the source. So the spatial model is optional:

model_spectrum = SkyModel(spectral_model=pwl, name="source-spectrum")
print(model_spectrum)

SkyModel

  Name                      : source-spectrum
  Datasets names            : None
  Spectral model type       : PowerLawSpectralModel
  Spatial  model type       : None
  Temporal model type       : None
  Parameters:
    index                   :   2.200
    amplitude               :   2.70e-12  1 / (cm2 s TeV)
    reference    (frozen)   :   1.000  TeV

Additionally the gammapy.modeling.models.SkyDiffuseCube can be used to represent source models based on templates, where the spatial and energy axes are correlated. It can be created e.g. from an existing FITS file:

from gammapy.modeling.models import SkyDiffuseCube

diffuse = SkyDiffuseCube.read(
    "$GAMMAPY_DATA/fermi-3fhl-gc/gll_iem_v06_gc.fits.gz"
)
print(diffuse)

SkyDiffuseCube

  Name                      : gll_iem_v06_gc.fits
  Datasets names            : None
  Parameters:
    norm                    :   1.000
    tilt         (frozen)   :   0.000
    reference    (frozen)   :   1.000  TeV

Model Lists and Serialisation

In a typical analysis scenario a model consists of mutiple model components, or a “catalog” or “source library”. To handle this list of multiple model components, Gammapy has a Models class:

from gammapy.modeling.models import Models

models = Models([model, diffuse])
print(models)

Models

Component 0: SkyModel

  Name                      : my-source
  Datasets names            : None
  Spectral model type       : PowerLawSpectralModel
  Spatial  model type       : GaussianSpatialModel
  Temporal model type       : None
  Parameters:
    index                   :   2.200
    amplitude               :   2.70e-12  1 / (cm2 s TeV)
    reference    (frozen)   :   1.000  TeV
    lon_0                   :   0.000  deg
    lat_0                   :   0.000  deg
    sigma                   :   0.200  deg
    e            (frozen)   :   0.000
    phi          (frozen)   :   0.000  deg

Component 1: SkyDiffuseCube

  Name                      : gll_iem_v06_gc.fits
  Datasets names            : None
  Parameters:
    norm                    :   1.000
    tilt         (frozen)   :   0.000
    reference    (frozen)   :   1.000  TeV

Individual model components in the list can be accessed by their name:

print(models["my-source"])

SkyModel

  Name                      : my-source
  Datasets names            : None
  Spectral model type       : PowerLawSpectralModel
  Spatial  model type       : GaussianSpatialModel
  Temporal model type       : None
  Parameters:
    index                   :   2.200
    amplitude               :   2.70e-12  1 / (cm2 s TeV)
    reference    (frozen)   :   1.000  TeV
    lon_0                   :   0.000  deg
    lat_0                   :   0.000  deg
    sigma                   :   0.200  deg
    e            (frozen)   :   0.000
    phi          (frozen)   :   0.000  deg

Note:To make the access by name unambiguous, models are required to have a unique name, otherwise an error will be thrown.

To see which models are available you can use the .names attribute:

print(models.names)

['my-source', 'gll_iem_v06_gc.fits']

Note that a SkyModel object can be evaluated for a given longitude, latitude, and energy, but the Models object cannot. This Models container object will be assigned to Dataset or Datasets together with the data to be fitted as explained in other analysis tutorials (see for example the modeling notebook).

The Models class also has in place .append() and .extend() methods:

model_copy = model.copy(name="my-source-copy")
models.append(model_copy)

This list of models can be also serialised toa custom YAML based format:

models_yaml = models.to_yaml()
print(models_yaml)

components:
-   name: my-source
    type: SkyModel
    spectral:
        type: PowerLawSpectralModel
        parameters:
        - {name: index, value: 2.2, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: amplitude, value: 2.7e-12, unit: cm-2 s-1 TeV-1, min: .nan, max: .nan,
            frozen: false, error: 0}
        - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true,
            error: 0}
    spatial:
        type: GaussianSpatialModel
        frame: galactic
        parameters:
        - {name: lon_0, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: lat_0, value: 0.0, unit: deg, min: -90.0, max: 90.0, frozen: false,
            error: 0}
        - {name: sigma, value: 0.2, unit: deg, min: 0.0, max: .nan, frozen: false,
            error: 0}
        - {name: e, value: 0.0, unit: '', min: 0.0, max: 1.0, frozen: true, error: 0}
        - {name: phi, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: true, error: 0}
-   name: gll_iem_v06_gc.fits
    type: SkyDiffuseCube
    filename: $GAMMAPY_DATA/fermi-3fhl-gc/gll_iem_v06_gc.fits.gz
    parameters:
    - {name: norm, value: 1.0, unit: '', min: .nan, max: .nan, frozen: false, error: 0}
    - {name: tilt, value: 0.0, unit: '', min: .nan, max: .nan, frozen: true, error: 0}
    - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true,
        error: 0}
-   name: my-source-copy
    type: SkyModel
    spectral:
        type: PowerLawSpectralModel
        parameters:
        - {name: index, value: 2.2, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: amplitude, value: 2.7e-12, unit: cm-2 s-1 TeV-1, min: .nan, max: .nan,
            frozen: false, error: 0}
        - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true,
            error: 0}
    spatial:
        type: GaussianSpatialModel
        frame: galactic
        parameters:
        - {name: lon_0, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: lat_0, value: 0.0, unit: deg, min: -90.0, max: 90.0, frozen: false,
            error: 0}
        - {name: sigma, value: 0.2, unit: deg, min: 0.0, max: .nan, frozen: false,
            error: 0}
        - {name: e, value: 0.0, unit: '', min: 0.0, max: 1.0, frozen: true, error: 0}
        - {name: phi, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: true, error: 0}

The structure of the yaml files follows the structure of the python objects. The components listed correspond to the SkyModel and SkyDiffuseCube components of the Models. For each SkyModel we have informations about its name, type (corresponding to the tag attribute) and sub-mobels (i.e spectral model and eventually spatial model). Then the spatial and spectral models are defiend by their type and parameters. The parameters keys name/value/unit are mandatory, while the keys min/max/frozen are optionnals (so you can prepare shorter files).

If you want to write this list of models to disk and read it back later you can use:

models.write("models.yaml", overwrite=True)

models_read = Models.read("models.yaml")

Additionally the models can exported and imported togeter with the data using the Datasets.read() and Datasets.write() methods as shown in the analysis_mwl notebook.

Implementing a Custom Model

In order to add a user defined spectral model you have to create a SpectralModel subclass. This new model class should include:

• a tag used for serialization (it can be the same as the class name)

• an instantiation of each Parameter with their unit, default values and frozen status

• the evaluate function where the mathematical expression for the model is defined.

As an example we will use a PowerLawSpectralModel plus a Gaussian (with fixed width). First we define the new custom model class that we name MyCustomSpectralModel:

from gammapy.modeling.models import SpectralModel, Parameter


class MyCustomSpectralModel(SpectralModel):
    """My custom spectral model, parametrising a power law plus a Gaussian spectral line.

    Parameters
    ----------
    amplitude : `~astropy.units.Quantity`
        Amplitude of the spectra model.
    index : `~astropy.units.Quantity`
        Spectral index of the model.
    reference : `~astropy.units.Quantity`
        Reference energy of the power law.
    mean : `~astropy.units.Quantity`
        Mean value of the Gaussian.
    width : `~astropy.units.Quantity`
        Sigma width of the Gaussian line.

    """

    tag = "MyCustomSpectralModel"
    amplitude = Parameter("amplitude", "1e-12 cm-2 s-1 TeV-1", min=0)
    index = Parameter("index", 2, min=0)
    reference = Parameter("reference", "1 TeV", frozen=True)
    mean = Parameter("mean", "1 TeV", min=0)
    width = Parameter("width", "0.1 TeV", min=0, frozen=True)

    @staticmethod
    def evaluate(energy, index, amplitude, reference, mean, width):
        pwl = PowerLawSpectralModel.evaluate(
            energy=energy,
            index=index,
            amplitude=amplitude,
            reference=reference,
        )
        gauss = amplitude * np.exp(-((energy - mean) ** 2) / (2 * width ** 2))
        return pwl + gauss

It is good practice to also implement a docstring for the model, defining the parameters and also definig a tag, which specifies the name of the model for serialisation. Also note that gammapy assumes that all SpectralModel evaluate functions return a flux in unit of "cm-2 s-1 TeV-1" (or equivalent dimensions).

This model can now be used as any other spectral model in Gammapy:

my_custom_model = MyCustomSpectralModel(mean="3 TeV")
print(my_custom_model)

MyCustomSpectralModel

   name     value        unit         min    max frozen   error
--------- --------- -------------- --------- --- ------ ---------
amplitude 1.000e-12 cm-2 s-1 TeV-1 0.000e+00 nan  False 0.000e+00
    index 2.000e+00                0.000e+00 nan  False 0.000e+00
reference 1.000e+00            TeV       nan nan   True 0.000e+00
     mean 3.000e+00            TeV 0.000e+00 nan  False 0.000e+00
    width 1.000e-01            TeV 0.000e+00 nan   True 0.000e+00

my_custom_model.integral(1 * u.TeV, 10 * u.TeV)

$$1.1443958 \times 10^{-12} \; \mathrm{\frac{1}{s\,cm^{2}}}$$

my_custom_model.plot(energy_range=[1, 10] * u.TeV)

<matplotlib.axes._subplots.AxesSubplot at 0x113c178d0>

As a next step we can also register the custom model in the SPECTRAL_MODELS registry, so that it becomes available for serilisation:

SPECTRAL_MODELS.append(MyCustomSpectralModel)

model = SkyModel(spectral_model=my_custom_model, name="my-source")
models = Models([model])
models.write("my-custom-models.yaml", overwrite=True)

!cat my-custom-models.yaml

components:
-   name: my-source
    type: SkyModel
    spectral:
        type: MyCustomSpectralModel
        parameters:
        - {name: amplitude, value: 1.0e-12, unit: cm-2 s-1 TeV-1, min: 0.0, max: .nan,
            frozen: false, error: 0}
        - {name: index, value: 2.0, unit: '', min: 0.0, max: .nan, frozen: false,
            error: 0}
        - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true,
            error: 0}
        - {name: mean, value: 3.0, unit: TeV, min: 0.0, max: .nan, frozen: false,
            error: 0}
        - {name: width, value: 0.1, unit: TeV, min: 0.0, max: .nan, frozen: true,
            error: 0}
covariance: my-custom-models_covariance.dat

Similarly you can also create custom spatial models and add them to the SPATIAL_MODELS registry. In that case gammapy assumes that the evaluate function return a normalized quantity in “sr-1” such as the model integral over the whole sky is one.