.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "tutorials/analysis-1d/spectrum_simulation.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_spectrum_simulation.py: 1D spectrum simulation ====================== Simulate a number of spectral on-off observations of a source with a power-law spectral model using the CTA 1DC response and fit them with the assumed spectral model. Prerequisites ------------- - Knowledge of spectral extraction and datasets used in gammapy, see for instance the :doc:`spectral analysis tutorial ` Context ------- To simulate a specific observation, it is not always necessary to simulate the full photon list. For many uses cases, simulating directly a reduced binned dataset is enough: the IRFs reduced in the correct geometry are combined with a source model to predict an actual number of counts per bin. The latter is then used to simulate a reduced dataset using Poisson probability distribution. This can be done to check the feasibility of a measurement, to test whether fitted parameters really provide a good fit to the data etc. Here we will see how to perform a 1D spectral simulation of a CTAO observation, in particular, we will generate OFF observations following the template background stored in the CTAO IRFs. **Objective: simulate a number of spectral ON-OFF observations of a source with a power-law spectral model with CTAO using the CTA 1DC response, fit them with the assumed spectral model and check that the distribution of fitted parameters is consistent with the input values.** Proposed approach ----------------- We will use the following classes and functions: - `~gammapy.datasets.SpectrumDatasetOnOff` - `~gammapy.datasets.SpectrumDataset` - `~gammapy.irf.load_irf_dict_from_file` - `~gammapy.modeling.models.PowerLawSpectralModel` .. GENERATED FROM PYTHON SOURCE LINES 48-57 .. code-block:: Python import numpy as np import astropy.units as u from astropy.coordinates import Angle, SkyCoord from regions import CircleSkyRegion # %matplotlib inline import matplotlib.pyplot as plt .. GENERATED FROM PYTHON SOURCE LINES 58-61 Setup ----- .. GENERATED FROM PYTHON SOURCE LINES 61-70 .. code-block:: Python from IPython.display import display from gammapy.data import FixedPointingInfo, Observation, observatory_locations from gammapy.datasets import Datasets, SpectrumDataset, SpectrumDatasetOnOff from gammapy.irf import load_irf_dict_from_file from gammapy.makers import SpectrumDatasetMaker from gammapy.maps import MapAxis, RegionGeom from gammapy.modeling import Fit from gammapy.modeling.models import PowerLawSpectralModel, SkyModel .. GENERATED FROM PYTHON SOURCE LINES 71-73 Check setup ----------- .. GENERATED FROM PYTHON SOURCE LINES 73-78 .. code-block:: Python from gammapy.utils.check import check_tutorials_setup 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 79-89 Simulation of a single spectrum ------------------------------- To do a simulation, we need to define the observational parameters like the livetime, the offset, the assumed integration radius, the energy range to perform the simulation for and the choice of spectral model. We then use an in-memory observation which is convolved with the IRFs to get the predicted number of counts. This is Poisson fluctuated using the `fake()` to get the simulated counts for each observation. .. GENERATED FROM PYTHON SOURCE LINES 89-126 .. code-block:: Python # Define simulation parameters parameters livetime = 1 * u.h pointing_position = SkyCoord(0, 0, unit="deg", frame="galactic") # We want to simulate an observation pointing at a fixed position in the sky. # For this, we use the `FixedPointingInfo` class pointing = FixedPointingInfo( fixed_icrs=pointing_position.icrs, ) offset = 0.5 * u.deg # Reconstructed and true energy axis energy_axis = MapAxis.from_edges( np.logspace(-0.5, 1.0, 10), unit="TeV", name="energy", interp="log" ) energy_axis_true = MapAxis.from_edges( np.logspace(-1.2, 2.0, 31), unit="TeV", name="energy_true", interp="log" ) on_region_radius = Angle("0.11 deg") center = pointing_position.directional_offset_by( position_angle=0 * u.deg, separation=offset ) on_region = CircleSkyRegion(center=center, radius=on_region_radius) # Define spectral model - a simple Power Law in this case model_simu = PowerLawSpectralModel( index=3.0, amplitude=2.5e-12 * u.Unit("cm-2 s-1 TeV-1"), reference=1 * u.TeV, ) print(model_simu) # we set the sky model used in the dataset model = SkyModel(spectral_model=model_simu, name="source") .. rst-class:: sphx-glr-script-out .. code-block:: none PowerLawSpectralModel type name value unit error min max frozen link prior ---- --------- ---------- -------------- --------- --- --- ------ ---- ----- index 3.0000e+00 0.000e+00 nan nan False amplitude 2.5000e-12 cm-2 s-1 TeV-1 0.000e+00 nan nan False reference 1.0000e+00 TeV 0.000e+00 nan nan True .. GENERATED FROM PYTHON SOURCE LINES 127-129 Load the IRFs In this simulation, we use the CTA-1DC IRFs shipped with Gammapy. .. GENERATED FROM PYTHON SOURCE LINES 129-142 .. code-block:: Python irfs = load_irf_dict_from_file( "$GAMMAPY_DATA/cta-1dc/caldb/data/cta/1dc/bcf/South_z20_50h/irf_file.fits" ) location = observatory_locations["cta_south"] obs = Observation.create( pointing=pointing, livetime=livetime, irfs=irfs, location=location, ) print(obs) .. 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/core.py:2097: UnitsWarning: '1/s/MeV/sr' did not parse as fits unit: Numeric factor not supported by FITS If this is meant to be a custom unit, define it with 'u.def_unit'. To have it recognized inside a file reader or other code, enable it with 'u.add_enabled_units'. For details, see https://docs.astropy.org/en/latest/units/combining_and_defining.html warnings.warn(msg, UnitsWarning) Observation obs id : 0 tstart : 51544.00 tstop : 51544.04 duration : 3600.00 s pointing (icrs) : 266.4 deg, -28.9 deg deadtime fraction : 0.0% .. GENERATED FROM PYTHON SOURCE LINES 143-145 Simulate a spectra .. GENERATED FROM PYTHON SOURCE LINES 145-162 .. code-block:: Python # Make the SpectrumDataset geom = RegionGeom.create(region=on_region, axes=[energy_axis]) dataset_empty = SpectrumDataset.create( geom=geom, energy_axis_true=energy_axis_true, name="obs-0" ) maker = SpectrumDatasetMaker(selection=["exposure", "edisp", "background"]) dataset = maker.run(dataset_empty, obs) # Set the model on the dataset, and fake dataset.models = model dataset.fake(random_state=42) print(dataset) .. rst-class:: sphx-glr-script-out .. code-block:: none SpectrumDataset --------------- Name : obs-0 Total counts : 298 Total background counts : 22.29 Total excess counts : 275.71 Predicted counts : 303.66 Predicted background counts : 22.29 Predicted excess counts : 281.37 Exposure min : 2.53e+08 m2 s Exposure max : 1.77e+10 m2 s Number of total bins : 9 Number of fit bins : 9 Fit statistic type : cash Fit statistic value (-2 log(L)) : -1811.58 Number of models : 1 Number of parameters : 3 Number of free parameters : 2 Component 0: SkyModel Name : source Datasets names : None Spectral model type : PowerLawSpectralModel Spatial model type : Temporal model type : Parameters: index : 3.000 +/- 0.00 amplitude : 2.50e-12 +/- 0.0e+00 1 / (cm2 s TeV) reference (frozen): 1.000 TeV .. GENERATED FROM PYTHON SOURCE LINES 163-165 You can see that background counts are now simulated .. GENERATED FROM PYTHON SOURCE LINES 168-177 On-Off analysis ~~~~~~~~~~~~~~~ To do an on off spectral analysis, which is the usual science case, the standard would be to use `SpectrumDatasetOnOff`, which uses the acceptance to fake off-counts. Please also refer to the `Dataset simulations` section in the :doc:`/tutorials/analysis-1d/spectral_analysis_rad_max` tutorial, dealing with simulations based on observations of real off counts. .. GENERATED FROM PYTHON SOURCE LINES 177-185 .. code-block:: Python dataset_on_off = SpectrumDatasetOnOff.from_spectrum_dataset( dataset=dataset, acceptance=1, acceptance_off=5 ) dataset_on_off.fake(npred_background=dataset.npred_background()) print(dataset_on_off) .. rst-class:: sphx-glr-script-out .. code-block:: none SpectrumDatasetOnOff -------------------- Name : BscvzI8O Total counts : 302 Total background counts : 20.80 Total excess counts : 281.20 Predicted counts : 302.33 Predicted background counts : 20.96 Predicted excess counts : 281.37 Exposure min : 2.53e+08 m2 s Exposure max : 1.77e+10 m2 s Number of total bins : 9 Number of fit bins : 9 Fit statistic type : wstat Fit statistic value (-2 log(L)) : 11.95 Number of models : 1 Number of parameters : 3 Number of free parameters : 2 Component 0: SkyModel Name : source Datasets names : None Spectral model type : PowerLawSpectralModel Spatial model type : Temporal model type : Parameters: index : 3.000 +/- 0.00 amplitude : 2.50e-12 +/- 0.0e+00 1 / (cm2 s TeV) reference (frozen): 1.000 TeV Total counts_off : 104 Acceptance : 9 Acceptance off : 45 .. GENERATED FROM PYTHON SOURCE LINES 186-189 You can see that off counts are now simulated as well. We now simulate several spectra using the same set of observation conditions. .. GENERATED FROM PYTHON SOURCE LINES 191-205 .. code-block:: Python n_obs = 100 datasets = Datasets() for idx in range(n_obs): dataset_on_off.fake(random_state=idx, npred_background=dataset.npred_background()) dataset_fake = dataset_on_off.copy(name=f"obs-{idx}") dataset_fake.meta_table["OBS_ID"] = [idx] datasets.append(dataset_fake) table = datasets.info_table() display(table) .. rst-class:: sphx-glr-script-out .. code-block:: none name counts excess ... acceptance acceptance_off alpha ... ------ ------ ------ ... ---------- ----------------- ------------------- obs-0 317 298.6 ... 9.0 45.0 0.2 obs-1 275 253.0 ... 9.0 45.0 0.2 obs-2 293 272.4 ... 9.0 45.0 0.2 obs-3 280 257.6 ... 9.0 45.00000000000001 0.19999999999999998 obs-4 337 316.4 ... 9.0 45.0 0.2 obs-5 283 258.6 ... 9.0 45.0 0.2 obs-6 330 307.6 ... 9.0 44.99999999999999 0.20000000000000004 obs-7 283 257.2 ... 9.0 45.0 0.2 obs-8 308 284.6 ... 9.0 45.0 0.2 obs-9 299 278.6 ... 9.0 45.00000000000001 0.19999999999999998 ... ... ... ... ... ... ... obs-90 286 267.0 ... 9.0 44.99999999999999 0.20000000000000004 obs-91 285 259.8 ... 9.0 45.0 0.2 obs-92 313 289.4 ... 9.0 45.0 0.2 obs-93 302 283.2 ... 9.0 45.0 0.2 obs-94 322 300.2 ... 9.0 45.0 0.2 obs-95 305 280.6 ... 9.0 44.99999999999999 0.20000000000000004 obs-96 301 277.4 ... 9.0 45.0 0.2 obs-97 290 271.2 ... 9.0 45.0 0.2 obs-98 301 280.6 ... 9.0 45.0 0.2 obs-99 323 302.2 ... 9.0 45.0 0.2 Length = 100 rows .. GENERATED FROM PYTHON SOURCE LINES 206-209 Before moving on to the fit let’s have a look at the simulated observations. .. GENERATED FROM PYTHON SOURCE LINES 209-220 .. code-block:: Python fix, axes = plt.subplots(1, 3, figsize=(12, 4)) axes[0].hist(table["counts"]) axes[0].set_xlabel("Counts") axes[1].hist(table["counts_off"]) axes[1].set_xlabel("Counts Off") axes[2].hist(table["excess"]) axes[2].set_xlabel("excess") plt.show() .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_spectrum_simulation_001.png :alt: spectrum simulation :srcset: /tutorials/analysis-1d/images/sphx_glr_spectrum_simulation_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 221-223 Now, we fit each simulated spectrum individually .. GENERATED FROM PYTHON SOURCE LINES 225-240 .. code-block:: Python results = [] fit = Fit() for dataset in datasets: dataset.models = model.copy() result = fit.optimize(dataset) results.append( { "index": result.parameters["index"].value, "amplitude": result.parameters["amplitude"].value, } ) .. GENERATED FROM PYTHON SOURCE LINES 241-244 We take a look at the distribution of the fitted indices. This matches very well with the spectrum that we initially injected. .. GENERATED FROM PYTHON SOURCE LINES 244-254 .. code-block:: Python fig, ax = plt.subplots() index = np.array([_["index"] for _ in results]) ax.hist(index, bins=10, alpha=0.5) ax.axvline(x=model_simu.parameters["index"].value, color="red") ax.set_xlabel("Reconstructed spectral index") print(f"index: {index.mean()} += {index.std()}") plt.show() .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_spectrum_simulation_002.png :alt: spectrum simulation :srcset: /tutorials/analysis-1d/images/sphx_glr_spectrum_simulation_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none index: 3.0036925550381217 += 0.08081469527619482 .. GENERATED FROM PYTHON SOURCE LINES 255-265 Exercises --------- - Change the observation time to something longer or shorter. Does the observation and spectrum results change as you expected? - Change the spectral model, e.g. add a cutoff at 5 TeV, or put a steep-spectrum source with spectral index of 4.0 - Simulate spectra with the spectral model we just defined. How much observation duration do you need to get back the injected parameters? .. _sphx_glr_download_tutorials_analysis-1d_spectrum_simulation.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/spectrum_simulation.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: spectrum_simulation.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: spectrum_simulation.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: spectrum_simulation.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_