\n",
"\n",
"**This is a fixed-text formatted version of a Jupyter notebook**\n",
"\n",
"- Try online [![Binder](https://mybinder.org/badge.svg)](https://mybinder.org/v2/gh/gammapy/gammapy-webpage/v0.15?urlpath=lab/tree/spectrum_simulation.ipynb)\n",
"- You can contribute with your own notebooks in this\n",
"[GitHub repository](https://github.com/gammapy/gammapy/tree/master/tutorials).\n",
"- **Source files:**\n",
"[spectrum_simulation.ipynb](../_static/notebooks/spectrum_simulation.ipynb) |\n",
"[spectrum_simulation.py](../_static/notebooks/spectrum_simulation.py)\n",
"
\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Spectrum simulation for CTA\n",
"\n",
"A quick example how to use the functions and classes in `~gammapy.spectrum` in order to simulate and fit spectra. \n",
"\n",
"We will simulate observations for CTA first using a power law model without any background.\n",
"Then we will add a power law shaped background component.\n",
"The next part of the tutorial shows how to use user defined models for simulations and fitting.\n",
"\n",
"We will use the following classes:\n",
"\n",
"* `~gammapy.spectrum.SpectrumDatasetOnOff`\n",
"* `~gammapy.spectrum.SpectrumDataset`\n",
"* `~gammapy.irf.load_cta_irfs`\n",
"* `~gammapy.modeling.models.PowerLawSpectralModel`"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Setup\n",
"\n",
"Same procedure as in every script ..."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import astropy.units as u\n",
"from astropy.coordinates import SkyCoord, Angle\n",
"from regions import CircleSkyRegion\n",
"from gammapy.spectrum import (\n",
" SpectrumDatasetOnOff,\n",
" SpectrumDataset,\n",
" SpectrumDatasetMaker,\n",
")\n",
"from gammapy.modeling import Fit, Parameter\n",
"from gammapy.modeling.models import (\n",
" PowerLawSpectralModel,\n",
" SpectralModel,\n",
" SkyModel,\n",
")\n",
"from gammapy.irf import load_cta_irfs\n",
"from gammapy.data import Observation\n",
"from gammapy.maps import MapAxis"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Simulation of a single spectrum\n",
"\n",
"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 Poission fluctuated using the `fake()` to get the simulated counts for each observation. "
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"# Define simulation parameters parameters\n",
"livetime = 1 * u.h\n",
"pointing = SkyCoord(0, 0, unit=\"deg\", frame=\"galactic\")\n",
"offset = 0.5 * u.deg\n",
"# Reconstructed and true energy axis\n",
"energy_axis = MapAxis.from_edges(\n",
" np.logspace(-0.5, 1.0, 10), unit=\"TeV\", name=\"energy\", interp=\"log\"\n",
")\n",
"energy_axis_true = MapAxis.from_edges(\n",
" np.logspace(-1.2, 2.0, 31), unit=\"TeV\", name=\"energy\", interp=\"log\"\n",
")\n",
"\n",
"on_region_radius = Angle(\"0.11 deg\")\n",
"on_region = CircleSkyRegion(center=pointing, radius=on_region_radius)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"PowerLawSpectralModel\n",
"\n",
" name value error unit min max frozen\n",
"--------- --------- ----- -------------- --- --- ------\n",
" index 3.000e+00 nan nan nan False\n",
"amplitude 2.500e-12 nan cm-2 s-1 TeV-1 nan nan False\n",
"reference 1.000e+00 nan TeV nan nan True\n"
]
}
],
"source": [
"# Define spectral model - a simple Power Law in this case\n",
"model_simu = PowerLawSpectralModel(\n",
" index=3.0,\n",
" amplitude=2.5e-12 * u.Unit(\"cm-2 s-1 TeV-1\"),\n",
" reference=1 * u.TeV,\n",
")\n",
"print(model_simu)\n",
"# we set the sky model used in the dataset\n",
"model = SkyModel(spectral_model=model_simu)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"# Load the IRFs\n",
"# In this simulation, we use the CTA-1DC irfs shipped with gammapy.\n",
"irfs = load_cta_irfs(\n",
" \"$GAMMAPY_DATA/cta-1dc/caldb/data/cta/1dc/bcf/South_z20_50h/irf_file.fits\"\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Info for OBS_ID = 1\n",
"- Pointing pos: RA 266.40 deg / Dec -28.94 deg\n",
"- Livetime duration: 3600.0 s\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"WARNING: AstropyDeprecationWarning: The truth value of a Quantity is ambiguous. In the future this will raise a ValueError. [astropy.units.quantity]\n"
]
}
],
"source": [
"obs = Observation.create(pointing=pointing, livetime=livetime, irfs=irfs)\n",
"print(obs)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"# Make the SpectrumDataset\n",
"dataset_empty = SpectrumDataset.create(\n",
" e_reco=energy_axis.edges, e_true=energy_axis_true.edges, region=on_region\n",
")\n",
"maker = SpectrumDatasetMaker(selection=[\"aeff\", \"edisp\", \"background\"])\n",
"dataset = maker.run(dataset_empty, obs)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"SpectrumDataset\n",
"\n",
" Name : 1 \n",
"\n",
" Total counts : 16 \n",
" Total predicted counts : nan\n",
" Total background counts : 22.35\n",
"\n",
" Effective area min : 8.16e+04 m2\n",
" Effective area max : 5.08e+06 m2\n",
"\n",
" Livetime : 3.60e+03 s\n",
"\n",
" Number of total bins : 9 \n",
" Number of fit bins : 9 \n",
"\n",
" Fit statistic type : cash\n",
" Fit statistic value (-2 log(L)) : nan\n",
"\n",
" Number of parameters : 0\n",
" Number of free parameters : 0\n",
"\n",
"\n"
]
}
],
"source": [
"# Set the model on the dataset, and fake\n",
"dataset.model = model\n",
"dataset.fake(random_state=42)\n",
"print(dataset)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You can see that backgound counts are now simulated"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### OnOff analysis\n",
"\n",
"To do `OnOff` spectral analysis, which is the usual science case, the standard would be to use `SpectrumDatasetOnOff`, which uses the acceptance to fake off-counts "
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"SpectrumDatasetOnOff\n",
"\n",
" Name : \n",
"\n",
" Total counts : 289 \n",
" Total predicted counts : 297.02\n",
" Total off counts : 123.00\n",
"\n",
" Total background counts : 24.60\n",
"\n",
" Effective area min : 8.16e+04 m2\n",
" Effective area max : 5.08e+06 m2\n",
"\n",
" Livetime : 1.00e+00 h\n",
"\n",
" Number of total bins : 9 \n",
" Number of fit bins : 9 \n",
"\n",
" Fit statistic type : wstat\n",
" Fit statistic value (-2 log(L)) : 10.22\n",
"\n",
" Number of parameters : 3\n",
" Number of free parameters : 2\n",
"\n",
" Model type : SkyModels\n",
" Acceptance mean: : 1.0\n",
"\n"
]
}
],
"source": [
"dataset_onoff = SpectrumDatasetOnOff(\n",
" aeff=dataset.aeff,\n",
" edisp=dataset.edisp,\n",
" models=model,\n",
" livetime=livetime,\n",
" acceptance=1,\n",
" acceptance_off=5,\n",
")\n",
"dataset_onoff.fake(background_model=dataset.background)\n",
"print(dataset_onoff)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You can see that off counts are now simulated as well. We now simulate several spectra using the same set of observation conditions."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 220 ms, sys: 4.87 ms, total: 225 ms\n",
"Wall time: 228 ms\n"
]
}
],
"source": [
"%%time\n",
"\n",
"n_obs = 100\n",
"datasets = []\n",
"\n",
"for idx in range(n_obs):\n",
" dataset_onoff.fake(random_state=idx, background_model=dataset.background)\n",
" dataset_onoff.name = f\"obs_{idx}\"\n",
" datasets.append(dataset_onoff.copy())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Before moving on to the fit let's have a look at the simulated observations."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
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\n",
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