This is a fixed-text formatted version of a Jupyter notebook
Try online
You can contribute with your own notebooks in this GitHub repository.
Source files: models.ipynb | models.py
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:
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¶
[1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
[2]:
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:
[3]:
from gammapy.modeling.models import PowerLawSpectralModel
[4]:
pwl = PowerLawSpectralModel()
print(pwl)
PowerLawSpectralModel
name value error unit min max frozen
--------- --------- ----- -------------- --- --- ------
index 2.000e+00 nan nan nan False
amplitude 1.000e-12 nan cm-2 s-1 TeV-1 nan nan False
reference 1.000e+00 nan TeV nan nan True
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:
[5]:
from gammapy.modeling.models import SPECTRAL_MODELS
print(SPECTRAL_MODELS)
Registry
--------
ConstantSpectralModel
CompoundSpectralModel
PowerLawSpectralModel
PowerLaw2SpectralModel
SmoothBrokenPowerLawSpectralModel
ExpCutoffPowerLawSpectralModel
ExpCutoffPowerLaw3FGLSpectralModel
SuperExpCutoffPowerLaw3FGLSpectralModel
SuperExpCutoffPowerLaw4FGLSpectralModel
LogParabolaSpectralModel
TemplateSpectralModel
GaussianSpectralModel
AbsorbedSpectralModel
NaimaSpectralModel
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:
[6]:
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:
[7]:
pwl = PowerLawSpectralModel(amplitude="2.7e-12 TeV-1 cm-2 s-1", index=2.2)
print(pwl)
PowerLawSpectralModel
name value error unit min max frozen
--------- --------- ----- -------------- --- --- ------
index 2.200e+00 nan nan nan False
amplitude 2.700e-12 nan cm-2 s-1 TeV-1 nan nan False
reference 1.000e+00 nan TeV nan nan True
The model can be evaluated at given energies by calling the model instance:
[8]:
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:
[9]:
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:
[10]:
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:
[11]:
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:
[12]:
pwl.plot(energy_range=[1, 100] * u.TeV)
[12]:
<matplotlib.axes._subplots.AxesSubplot at 0x1c184a0b70>
Spatial Models¶
Spatial models are imported from the same gammapy.modeling.models namespace, let’s start with a GaussianSpatialModel:
[13]:
from gammapy.modeling.models import GaussianSpatialModel
[14]:
gauss = GaussianSpatialModel(lon_0="0 deg", lat_0="0 deg", sigma="0.2 deg")
print(gauss)
GaussianSpatialModel
name value error unit min max frozen
----- --------- ----- ---- ---------- --------- ------
lon_0 0.000e+00 nan deg nan nan False
lat_0 0.000e+00 nan deg -9.000e+01 9.000e+01 False
sigma 2.000e-01 nan deg 0.000e+00 nan False
e 0.000e+00 nan 0.000e+00 1.000e+00 True
phi 0.000e+00 nan deg nan nan True
Again you can check the SPATIAL_MODELS registry to see which models are available or take a look at the model gallery.
[15]:
from gammapy.modeling.models import SPATIAL_MODELS
print(SPATIAL_MODELS)
Registry
--------
ConstantSpatialModel
TemplateSpatialModel
DiskSpatialModel
GaussianSpatialModel
PointSpatialModel
ShellSpatialModel
The default coordinate frame for all spatial models is "icrs", but the frame can be modified using the frame argument:
[16]:
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:
[17]:
print(gauss.position)
<SkyCoord (Galactic): (l, b) in deg
(0., 0.)>
Spatial models can be evaluated again by calling the instance:
[18]:
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:
[19]:
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:
[20]:
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:
[21]:
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:
[22]:
# 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:
[23]:
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")
[24]:
!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:
[25]:
from gammapy.modeling.models import SkyModel
model = SkyModel(spectral_model=pwl, spatial_model=gauss, name="my-source")
print(model)
SkyModel
Name : my-source
Spectral model type : PowerLawSpectralModel
Spatial model type : GaussianSpatialModel
Temporal model type :
Parameters:
lon_0 : 0.000 deg
lat_0 : 0.000 deg
sigma : 0.200 deg
e (frozen) : 0.000
phi (frozen) : 0.000 deg
index : 2.200
amplitude : 2.70e-12 1 / (cm2 s TeV)
reference (frozen) : 1.000 TeV
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:
[26]:
model_without_name = SkyModel(spectral_model=pwl, spatial_model=gauss)
print(model_without_name.name)
uxWqU-Zd
The spectral and spatial component of the source model can be accessed using .spectral_model and .spatial_model:
[27]:
model.spectral_model
[27]:
<gammapy.modeling.models.spectral.PowerLawSpectralModel at 0x1c184878d0>
[28]:
model.spatial_model
[28]:
<gammapy.modeling.models.spatial.GaussianSpatialModel at 0x1c18619c50>
And can be used as you have seen already seen above:
[29]:
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:
[30]:
model_spectrum = SkyModel(spectral_model=pwl, name="source-spectrum")
print(model_spectrum)
SkyModel
Name : source-spectrum
Spectral model type : PowerLawSpectralModel
Spatial model type : None
Temporal model type :
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:
[31]:
from gammapy.modeling.models import SkyDiffuseCube
[32]:
diffuse = SkyDiffuseCube.read(
"$GAMMAPY_DATA/fermi-3fhl-gc/gll_iem_v06_gc.fits.gz"
)
print(diffuse)
SkyDiffuseCube
Name : gll_iem_v06_gc.fits
Parameters:
norm : 1.000
tilt (frozen) : 0.000
reference (frozen) : 1.000 TeV
WARNING: FITSFixedWarning: 'datfix' made the change 'Set DATE-REF to '1858-11-17' from MJD-REF'. [astropy.wcs.wcs]
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:
[33]:
from gammapy.modeling.models import Models
[34]:
models = Models([model, diffuse])
print(models)
Models
Component 0: SkyModel
Name : my-source
Spectral model type : PowerLawSpectralModel
Spatial model type : GaussianSpatialModel
Temporal model type :
Parameters:
lon_0 : 0.000 deg
lat_0 : 0.000 deg
sigma : 0.200 deg
e (frozen) : 0.000
phi (frozen) : 0.000 deg
index : 2.200
amplitude : 2.70e-12 1 / (cm2 s TeV)
reference (frozen) : 1.000 TeV
Component 1: SkyDiffuseCube
Name : gll_iem_v06_gc.fits
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:
[35]:
print(models["my-source"])
SkyModel
Name : my-source
Spectral model type : PowerLawSpectralModel
Spatial model type : GaussianSpatialModel
Temporal model type :
Parameters:
lon_0 : 0.000 deg
lat_0 : 0.000 deg
sigma : 0.200 deg
e (frozen) : 0.000
phi (frozen) : 0.000 deg
index : 2.200
amplitude : 2.70e-12 1 / (cm2 s TeV)
reference (frozen) : 1.000 TeV
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:
[36]:
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:
[37]:
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:
[38]:
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}
- {name: amplitude, value: 2.7e-12, unit: cm-2 s-1 TeV-1, min: .nan, max: .nan,
frozen: false}
- {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true}
spatial:
type: GaussianSpatialModel
frame: galactic
parameters:
- {name: lon_0, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: false}
- {name: lat_0, value: 0.0, unit: deg, min: -90.0, max: 90.0, frozen: false}
- {name: sigma, value: 0.2, unit: deg, min: 0.0, max: .nan, frozen: false}
- {name: e, value: 0.0, unit: '', min: 0.0, max: 1.0, frozen: true}
- {name: phi, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: true}
- 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}
- {name: tilt, value: 0.0, unit: '', min: .nan, max: .nan, frozen: true}
- {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true}
- name: my-source-copy
type: SkyModel
spectral:
type: PowerLawSpectralModel
parameters:
- {name: index, value: 2.2, unit: '', min: .nan, max: .nan, frozen: false}
- {name: amplitude, value: 2.7e-12, unit: cm-2 s-1 TeV-1, min: .nan, max: .nan,
frozen: false}
- {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true}
spatial:
type: GaussianSpatialModel
frame: galactic
parameters:
- {name: lon_0, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: false}
- {name: lat_0, value: 0.0, unit: deg, min: -90.0, max: 90.0, frozen: false}
- {name: sigma, value: 0.2, unit: deg, min: 0.0, max: .nan, frozen: false}
- {name: e, value: 0.0, unit: '', min: 0.0, max: 1.0, frozen: true}
- {name: phi, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: true}
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:
[39]:
models.write("models.yaml", overwrite=True)
[40]:
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:
[41]:
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:
[42]:
my_custom_model = MyCustomSpectralModel(mean="3 TeV")
print(my_custom_model)
MyCustomSpectralModel
name value error unit min max frozen
--------- --------- ----- -------------- --------- --- ------
amplitude 1.000e-12 nan cm-2 s-1 TeV-1 0.000e+00 nan False
index 2.000e+00 nan 0.000e+00 nan False
reference 1.000e+00 nan TeV nan nan True
mean 3.000e+00 nan TeV 0.000e+00 nan False
width 1.000e-01 nan TeV 0.000e+00 nan True
[43]:
my_custom_model.integral(1 * u.TeV, 10 * u.TeV)
[43]:
[44]:
my_custom_model.plot(energy_range=[1, 10] * u.TeV)
[44]:
<matplotlib.axes._subplots.AxesSubplot at 0x1c1b1e4240>
As a next step we can also register the custom model in the SPECTRAL_MODELS registry, so that it becomes available for serilisation:
[45]:
SPECTRAL_MODELS.append(MyCustomSpectralModel)
[46]:
model = SkyModel(spectral_model=my_custom_model, name="my-source")
models = Models([model])
models.write("my-custom-models.yaml", overwrite=True)
[47]:
!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}
- {name: index, value: 2.0, unit: '', min: 0.0, max: .nan, frozen: false}
- {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true}
- {name: mean, value: 3.0, unit: TeV, min: 0.0, max: .nan, frozen: false}
- {name: width, value: 0.1, unit: TeV, min: 0.0, max: .nan, frozen: true}
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.