.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "tutorials/analysis-1d/spectral_analysis_rad_max.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_tutorials_analysis-1d_spectral_analysis_rad_max.py: Spectral analysis with energy-dependent directional cuts ======================================================== Perform a point like spectral analysis with energy dependent offset cut. Prerequisites ------------- - Understanding the basic data reduction performed in the :doc:`/tutorials/analysis-1d/spectral_analysis` tutorial. - understanding the difference between a `point-like `__ and a `full-enclosure `__ IRF. Context ------- As already explained in the :doc:`/tutorials/analysis-1d/spectral_analysis` tutorial, the background is estimated from the 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, :math:`\theta`. The `gamma-astro-data-format `__ specifications offer two different ways to store this information: * if the same :math:`\theta` cut is applied at all energies and offsets, a `RAD_MAX `__ keyword 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-dependent (and offset-dependent) :math:`\theta` cut is applied, its values are stored in additional `FITS` data unit, named `RAD_MAX_2D `__. `Gammapy` provides a class to automatically read these values, `~gammapy.irf.RadMax2D`, for both cases (fixed or energy-dependent :math:`\theta` cut). In this notebook we will focus on how to perform a spectral extraction with a point-like IRF with an energy-dependent :math:`\theta` cut. We remark that in this case a `~regions.PointSkyRegion` (and not a `~regions.CircleSkyRegion`) should be used to define the ON region. If a geometry based on a `~regions.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 :doc:`/tutorials/analysis-1d/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:`\theta` cut. We define the ON region, this time as a `~regions.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 `~gammapy.datasets.SpectrumDataset` object we want to produce. The corresponding dataset maker will automatically use the :math:`\theta` values in `~gammapy.irf.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 `~gammapy.makers.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 `~gammapy.makers.SpectrumDatasetMaker`, the `~gammapy.makers.ReflectedRegionsBackgroundMaker` will use the values in `~gammapy.irf.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. .. GENERATED FROM PYTHON SOURCE LINES 102-110 .. code-block:: Python import astropy.units as u from astropy.coordinates import SkyCoord from regions import PointSkyRegion # %matplotlib inline import matplotlib.pyplot as plt .. GENERATED FROM PYTHON SOURCE LINES 111-116 Setup ----- As usual, we’ll start with some setup … .. GENERATED FROM PYTHON SOURCE LINES 116-133 .. code-block:: Python from IPython.display import display from gammapy.data import DataStore from gammapy.datasets import Datasets, SpectrumDataset from gammapy.makers import ( ReflectedRegionsBackgroundMaker, SafeMaskMaker, 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, ) .. GENERATED FROM PYTHON SOURCE LINES 134-136 Check setup ----------- .. GENERATED FROM PYTHON SOURCE LINES 136-142 .. code-block:: Python from gammapy.utils.check import check_tutorials_setup from gammapy.visualization import plot_spectrum_datasets_off_regions check_tutorials_setup() .. rst-class:: sphx-glr-script-out .. code-block:: none System: python_executable : /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/bin/python python_version : 3.9.20 machine : x86_64 system : Linux Gammapy package: version : 1.3 path : /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/gammapy Other packages: numpy : 1.26.4 scipy : 1.13.1 astropy : 5.2.2 regions : 0.8 click : 8.1.7 yaml : 6.0.2 IPython : 8.18.1 jupyterlab : not installed matplotlib : 3.9.2 pandas : not installed healpy : 1.17.3 iminuit : 2.30.1 sherpa : 4.16.1 naima : 0.10.0 emcee : 3.1.6 corner : 2.2.3 ray : 2.39.0 Gammapy environment variables: GAMMAPY_DATA : /home/runner/work/gammapy-docs/gammapy-docs/gammapy-datasets/1.3 .. GENERATED FROM PYTHON SOURCE LINES 143-149 Load data --------- We load the two MAGIC observations of the Crab Nebula containing the `RAD_MAX_2D` table. .. GENERATED FROM PYTHON SOURCE LINES 149-154 .. code-block:: Python data_store = DataStore.from_dir("$GAMMAPY_DATA/magic/rad_max/data") observations = data_store.get_observations(required_irf="point-like") .. GENERATED FROM PYTHON SOURCE LINES 155-160 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. .. GENERATED FROM PYTHON SOURCE LINES 160-165 .. code-block:: Python rad_max = observations["5029747"].rad_max print(rad_max) .. rst-class:: sphx-glr-script-out .. code-block:: none RadMax2D -------- axes : ['energy', 'offset'] shape : (20, 1) ndim : 2 unit : deg dtype : >f4 .. GENERATED FROM PYTHON SOURCE LINES 166-168 We can also plot the rad max value against the energy: .. GENERATED FROM PYTHON SOURCE LINES 168-174 .. code-block:: Python fig, ax = plt.subplots() rad_max.plot_rad_max_vs_energy(ax=ax) plt.show() .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_001.png :alt: spectral analysis rad max :srcset: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 175-182 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 `~regions.PointSkyRegion`: .. GENERATED FROM PYTHON SOURCE LINES 182-187 .. code-block:: Python target_position = SkyCoord(ra=83.63, dec=22.01, unit="deg", frame="icrs") on_region = PointSkyRegion(target_position) .. GENERATED FROM PYTHON SOURCE LINES 188-193 Run data reduction chain ------------------------ We begin with the configuration of the dataset maker classes: .. GENERATED FROM PYTHON SOURCE LINES 193-208 .. code-block:: Python # 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) .. GENERATED FROM PYTHON SOURCE LINES 209-219 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 :math:`\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. .. GENERATED FROM PYTHON SOURCE LINES 219-231 .. code-block:: Python dataset_maker = SpectrumDatasetMaker( containment_correction=False, selection=["counts", "exposure", "edisp"] ) # tell the background maker to use the WobbleRegionsFinder, let us use 3 off region_finder = WobbleRegionsFinder(n_off_regions=3) bkg_maker = ReflectedRegionsBackgroundMaker(region_finder=region_finder) # use the energy threshold specified in the DL3 files safe_mask_masker = SafeMaskMaker(methods=["aeff-default"]) .. GENERATED FROM PYTHON SOURCE LINES 232-247 .. code-block:: Python 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) dataset_on_off = safe_mask_masker.run(dataset_on_off, observation) datasets.append(dataset_on_off) .. GENERATED FROM PYTHON SOURCE LINES 248-251 Now we can plot the off regions and target positions on top of the counts map: .. GENERATED FROM PYTHON SOURCE LINES 251-258 .. code-block:: Python ax = counts.plot(cmap="viridis") geom.plot_region(ax=ax, kwargs_point={"color": "k", "marker": "*"}) plot_spectrum_datasets_off_regions(ax=ax, datasets=datasets) plt.show() .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_002.png :alt: spectral analysis rad max :srcset: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/gammapy/visualization/datasets.py:84: UserWarning: Setting the 'color' property will override the edgecolor or facecolor properties. handle = Patch(**plot_kwargs) .. GENERATED FROM PYTHON SOURCE LINES 259-266 Fit spectrum ------------ | We perform a joint likelihood fit of the two datasets. | For this particular datasets we select a fit range between :math:`80\,{\rm GeV}` and :math:`20\,{\rm TeV}`. .. GENERATED FROM PYTHON SOURCE LINES 266-290 .. code-block:: Python e_min = 80 * u.GeV e_max = 20 * u.TeV for dataset in datasets: dataset.mask_fit = dataset.counts.geom.energy_mask(e_min, e_max) 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() .. rst-class:: sphx-glr-script-out .. code-block:: none /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/numpy/core/fromnumeric.py:88: RuntimeWarning: overflow encountered in reduce return ufunc.reduce(obj, axis, dtype, out, **passkwargs) .. GENERATED FROM PYTHON SOURCE LINES 291-294 Fit quality and model residuals ------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 297-299 We can access the results dictionary to see if the fit converged: .. GENERATED FROM PYTHON SOURCE LINES 299-303 .. code-block:: Python print(result) .. rst-class:: sphx-glr-script-out .. code-block:: none OptimizeResult backend : minuit method : migrad success : True message : Optimization terminated successfully. nfev : 213 total stat : 23.98 CovarianceResult backend : minuit method : hesse success : True message : Hesse terminated successfully. .. GENERATED FROM PYTHON SOURCE LINES 304-306 and check the best-fit parameters .. GENERATED FROM PYTHON SOURCE LINES 306-310 .. code-block:: Python display(datasets.models.to_parameters_table()) .. rst-class:: sphx-glr-script-out .. code-block:: none model type name value unit ... min max frozen link prior ----- ---- --------- ---------- -------------- ... --- --- ------ ---- ----- crab amplitude 4.2903e-11 cm-2 s-1 TeV-1 ... nan nan False crab reference 1.0000e+00 TeV ... nan nan True crab alpha 2.5819e+00 ... nan nan False crab beta 1.9580e-01 ... nan nan False .. GENERATED FROM PYTHON SOURCE LINES 311-314 A simple way to inspect the model residuals is using the function `~SpectrumDataset.plot_fit()` .. GENERATED FROM PYTHON SOURCE LINES 314-319 .. code-block:: Python ax_spectrum, ax_residuals = datasets[0].plot_fit() ax_spectrum.set_ylim(0.1, 120) plt.show() .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_003.png :alt: spectral analysis rad max :srcset: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 320-323 For more ways of assessing fit quality, please refer to the dedicated :doc:`/tutorials/api/fitting` tutorial. .. GENERATED FROM PYTHON SOURCE LINES 326-333 Compare against the literature ------------------------------ Let us compare the spectrum we obtained against a `previous measurement by MAGIC `__. .. GENERATED FROM PYTHON SOURCE LINES 333-355 .. code-block:: Python 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() .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_004.png :alt: spectral analysis rad max :srcset: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 356-364 Dataset simulations ------------------- A common way to check if a fit is biased is to simulate multiple datasets with the obtained best fit model, and check the distribution of the fitted parameters. Here, we show how to perform one such simulation assuming the measured off counts provide a good distribution of the background. .. GENERATED FROM PYTHON SOURCE LINES 364-383 .. code-block:: Python dataset_simulated = datasets.stack_reduce().copy(name="simulated_ds") simulated_model = best_fit_model.copy(name="simulated") dataset_simulated.models = simulated_model dataset_simulated.fake( npred_background=dataset_simulated.counts_off * dataset_simulated.alpha ) dataset_simulated.peek() plt.show() # The important thing to note here is that while this samples the on-counts, the off counts are # not sampled. If you have multiple measurements of the off counts, they should be used. # Alternatively, you can try to create a parametric model of the background. result = fit.run(datasets=[dataset_simulated]) print(result.models.to_parameters_table()) # sphinx_gallery_thumbnail_number = 4 .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_005.png :alt: Counts, Exposure, Energy Dispersion :srcset: /tutorials/analysis-1d/images/sphx_glr_spectral_analysis_rad_max_005.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/astropy/units/quantity.py:673: RuntimeWarning: invalid value encountered in divide result = super().__array_ufunc__(function, method, *arrays, **kwargs) model type name value unit ... min max frozen link prior --------- ---- --------- ---------- -------------- ... --- --- ------ ---- ----- simulated amplitude 4.2035e-11 cm-2 s-1 TeV-1 ... nan nan False simulated reference 1.0000e+00 TeV ... nan nan True simulated alpha 2.7087e+00 ... nan nan False simulated beta 2.9291e-01 ... nan nan False .. _sphx_glr_download_tutorials_analysis-1d_spectral_analysis_rad_max.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/gammapy/gammapy-webpage/v1.3?urlpath=lab/tree/notebooks/1.3/tutorials/analysis-1d/spectral_analysis_rad_max.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: spectral_analysis_rad_max.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: spectral_analysis_rad_max.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: spectral_analysis_rad_max.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_