# Spectral analysis with energy-dependent directional cuts#

Perform a point like spectral analysis with energy dependent offset cut.

## Context#

As already explained in the Spectral analysis tutorial, the background is estimated fromthe field of view of the observation. In particular, the source and background events are counted within a circular ON region enclosing the source. The background to be subtracted is then estimated from one or more OFF regions with an expected background rate similar to the one in the ON region (i.e. from regions with similar acceptance).

Full-containment IRFs have no directional cut applied, when employed for a 1D analysis, it is required to apply a correction to the IRF accounting for flux leaking out of the ON region. This correction is typically obtained by integrating the PSF within the ON region.

When computing a point-like IRFs, a directional cut around the assumed source position is applied to the simulated events. For this IRF type, no PSF component is provided. The size of the ON and OFF regions used for the spectrum extraction should then reflect this cut, since a response computed within a specific region around the source is being provided.

The directional cut is typically an angular distance from the assumed source position, $$\theta$$. The gamma-astro-data-format specifications offer two different ways to store this information: * if the same $$\theta$$ cut is applied at all energies and offsets, a RAD_MAX keyword <https://gamma-astro-data-formats.readthedocs.io/en/latest/irfs/point_like/#rad-max>__ is added to the header of the data units containing IRF components. This should be used to define the size of the ON and OFF regions; * in case an energy- (and offset-) dependent :math: heta cut is applied, its values are stored in additional FITS data unit, named RAD_MAX_2D <https://gamma-astro-data-formats.readthedocs.io/en/latest/irfs/point_like/#rad-max-2d>__.

Gammapy provides a class to automatically read these values, RadMax2D, for both cases (fixed or energy-dependent :math: heta cut). In this notebook we will focus on how to perform a spectral extraction with a point-like IRF with an energy-dependent :math: heta cut. We remark that in this case a PointSkyRegion (and not a CircleSkyRegion) should be used to define the ON region. If a geometry based on a PointSkyRegion is fed to the spectra and the background Makers, the latter will automatically use the values stored in the RAD_MAX keyword / table for defining the size of the ON and OFF regions.

Beside the definition of the ON region during the data reduction, the remaining steps are identical to the other Spectral analysis tutorial., so we will not detail the data reduction steps already presented in the other tutorial.

Objective: perform the data reduction and analysis of 2 Crab Nebula observations of MAGIC and fit the resulting datasets.

## Introduction#

We load two MAGIC observations in the gammapy-data containing IRF component with a :math: heta cut.

We define the ON region, this time as a PointSkyRegion instead of a CircleSkyRegion, i.e. we specify only the center of our ON region. We create a RegionGeom adding to the region the estimated energy axis of the SpectrumDataset object we want to produce. The corresponding dataset maker will automatically use the :math: heta values in RadMax2D to set the appropriate ON region sizes (based on the offset on the observation and on the estimated energy binning).

In order to define the OFF regions it is recommended to use a WobbleRegionsFinder, that uses fixed positions for the OFF regions. In the different estimated energy bins we will have OFF regions centered at the same positions, but with changing size. As for the SpectrumDatasetMaker, the ReflectedRegionsBackgroundMaker will use the values in RadMax2D to define the sizes of the OFF regions.

Once the datasets with the ON and OFF counts are created, we can perform a 1D likelihood fit, exactly as illustrated in the previous example.

import astropy.units as u
from astropy.coordinates import SkyCoord
from regions import PointSkyRegion

# %matplotlib inline
import matplotlib.pyplot as plt


## Setup#

from IPython.display import display
from gammapy.data import DataStore
from gammapy.datasets import Datasets, SpectrumDataset
from gammapy.makers import (
ReflectedRegionsBackgroundMaker,
SpectrumDatasetMaker,
WobbleRegionsFinder,
)
from gammapy.maps import Map, MapAxis, RegionGeom
from gammapy.modeling import Fit
from gammapy.modeling.models import (
LogParabolaSpectralModel,
SkyModel,
create_crab_spectral_model,
)


## Check setup#

from gammapy.utils.check import check_tutorials_setup
from gammapy.visualization import plot_spectrum_datasets_off_regions

check_tutorials_setup()

System:

python_executable      : /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/bin/python
python_version         : 3.9.15
machine                : x86_64
system                 : Linux

Gammapy package:

version                : 1.0
path                   : /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/gammapy

Other packages:

numpy                  : 1.23.4
scipy                  : 1.9.3
astropy                : 5.1.1
regions                : 0.7
click                  : 8.1.3
yaml                   : 6.0
IPython                : 8.6.0
jupyterlab             : not installed
matplotlib             : 3.6.2
pandas                 : not installed
healpy                 : 1.16.1
iminuit                : 2.17.0
sherpa                 : 4.15.0
naima                  : 0.10.0
emcee                  : 3.1.3
corner                 : 2.2.1

Gammapy environment variables:

GAMMAPY_DATA           : /home/runner/work/gammapy-docs/gammapy-docs/gammapy-datasets/1.0


We load the two MAGIC observations of the Crab Nebula containing the RAD_MAX_2D table.

data_store = DataStore.from_dir("\$GAMMAPY_DATA/magic/rad_max/data")
observations = data_store.get_observations(required_irf="point-like")


A RadMax2D attribute, containing the RAD_MAX_2D table, is automatically loaded in the observation. As we can see from the IRF component axes, the table has a single offset value and 28 estimated energy values.

rad_max = observations["5029747"].rad_max

RadMax2D
--------

axes  : ['energy', 'offset']
shape : (20, 1)
ndim  : 2
unit  : deg
dtype : >f4


We can also plot the rad max value against the energy:

fig, ax = plt.subplots()

<AxesSubplot: xlabel='Energy [TeV]', ylabel='Rad max. [deg]'>


## Define the ON region#

To use the RAD_MAX_2D values to define the sizes of the ON and OFF regions **it is necessary to specify the ON region as a PointSkyRegion:

target_position = SkyCoord(ra=83.63, dec=22.01, unit="deg", frame="icrs")
on_region = PointSkyRegion(target_position)


## Run data reduction chain#

We begin with the configuration of the dataset maker classes:

# true and estimated energy axes
energy_axis = MapAxis.from_energy_bounds(
50, 1e5, nbin=5, per_decade=True, unit="GeV", name="energy"
)
energy_axis_true = MapAxis.from_energy_bounds(
10, 1e5, nbin=10, per_decade=True, unit="GeV", name="energy_true"
)

# geometry defining the ON region and SpectrumDataset based on it
geom = RegionGeom.create(region=on_region, axes=[energy_axis])

dataset_empty = SpectrumDataset.create(geom=geom, energy_axis_true=energy_axis_true)


The SpectrumDataset is now based on a geometry consisting of a single coordinate and an estimated energy axis. The SpectrumDatasetMaker and ReflectedRegionsBackgroundMaker will take care of producing ON and OFF with the proper sizes, automatically adopting the $$\theta$$ values in Observation.rad_max.

As explained in the introduction, we use a WobbleRegionsFinder, to determine the OFF positions. The parameter n_off_positions specifies the number of OFF regions to be considered.

dataset_maker = SpectrumDatasetMaker(
containment_correction=False, selection=["counts", "exposure", "edisp"]
)

# tell the background maker to use the WobbleRegionsFinder, let us use 1 off
region_finder = WobbleRegionsFinder(n_off_regions=3)
bkg_maker = ReflectedRegionsBackgroundMaker(region_finder=region_finder)

# use the energy threshold specified in the DL3 files

datasets = Datasets()

# create a counts map for visualisation later...
counts = Map.create(skydir=target_position, width=3)

for observation in observations:
dataset = dataset_maker.run(
dataset_empty.copy(name=str(observation.obs_id)), observation
)
counts.fill_events(observation.events)
dataset_on_off = bkg_maker.run(dataset, observation)
datasets.append(dataset_on_off)


Now we can plot the off regions and target positions on top of the counts map:

plt.figure()
ax = counts.plot(cmap="viridis")
geom.plot_region(ax=ax, kwargs_point={"color": "k", "marker": "*"})
plot_spectrum_datasets_off_regions(ax=ax, datasets=datasets)

<WCSAxesSubplot: >


## Fit spectrum#

We perform a joint likelihood fit of the two datasets.
For this particular datasets we select a fit range between $$80\,{\rm GeV}$$ and $$20\,{\rm TeV}$$.
e_min = 80 * u.GeV
e_max = 20 * u.TeV

for dataset in datasets:

spectral_model = LogParabolaSpectralModel(
amplitude=1e-12 * u.Unit("cm-2 s-1 TeV-1"),
alpha=2,
beta=0.1,
reference=1 * u.TeV,
)
model = SkyModel(spectral_model=spectral_model, name="crab")

datasets.models = [model]

fit = Fit()
result = fit.run(datasets=datasets)

# we make a copy here to compare it later
best_fit_model = model.copy()

/home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/numpy/core/fromnumeric.py:86: RuntimeWarning: overflow encountered in reduce
return ufunc.reduce(obj, axis, dtype, out, **passkwargs)


## Fit quality and model residuals#

We can access the results dictionary to see if the fit converged:

print(result)

OptimizeResult

backend    : minuit
success    : True
message    : Optimization terminated successfully.
nfev       : 213
total stat : 23.98

CovarianceResult

backend    : minuit
method     : hesse
success    : True
message    : Hesse terminated successfully.


and check the best-fit parameters

display(datasets.models.to_parameters_table())

model   type      name     value         unit      ... max frozen is_norm link
----- -------- --------- ---------- -------------- ... --- ------ ------- ----
crab spectral amplitude 4.2903e-11 cm-2 s-1 TeV-1 ... nan  False    True
crab spectral reference 1.0000e+00            TeV ... nan   True   False
crab spectral     alpha 2.5819e+00                ... nan  False   False
crab spectral      beta 1.9580e-01                ... nan  False   False


A simple way to inspect the model residuals is using the function plot_fit()

ax_spectrum, ax_residuals = datasets[0].plot_fit()
ax_spectrum.set_ylim(0.1, 120)

(0.1, 120)


For more ways of assessing fit quality, please refer to the dedicated modeling and fitting tutorial :doc:/tutorials/api/fitting tutorial.

## Compare against the literature#

Let us compare the spectrum we obtained against a previous measurement by MAGIC.

fig, ax = plt.subplots()
plot_kwargs = {
"energy_bounds": [0.08, 20] * u.TeV,
"sed_type": "e2dnde",
"yunits": u.Unit("TeV cm-2 s-1"),
"xunits": u.GeV,
"ax": ax,
}

crab_magic_lp = create_crab_spectral_model("magic_lp")

best_fit_model.spectral_model.plot(
ls="-", lw=1.5, color="crimson", label="best fit", **plot_kwargs
)
best_fit_model.spectral_model.plot_error(facecolor="crimson", alpha=0.4, **plot_kwargs)
crab_magic_lp.plot(ls="--", lw=1.5, color="k", label="MAGIC reference", **plot_kwargs)

ax.legend()
ax.set_ylim([1e-13, 1e-10])
plt.show()

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