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Spectrum simulation for CTA

A quick example how to use the functions and classes in gammapy.spectrum in order to simulate and fit spectra.

We will simulate observations for the Cherenkov Telescope Array (CTA) first using a power law model without any background. Than we will add a power law shaped background component. The next part of the tutorial shows how to use user defined models for simulations and fitting.

We will use the following classes:

Setup

Same procedure as in every script …

[1]:

%matplotlib inline
import matplotlib.pyplot as plt

[2]:

import numpy as np
import astropy.units as u
from gammapy.spectrum import (
    SpectrumDatasetOnOff,
    SpectrumEvaluator,
    SpectrumDataset,
)
from gammapy.utils.fitting import Fit, Parameter
from gammapy.spectrum.models import PowerLaw
from gammapy.spectrum import models
from gammapy.irf import load_cta_irfs

Simulation of a single spectrum

To do a simulation, we need to define the observational parameters like the livetime, the offset, the energy range to perform the simulation for and the choice of spectral model. This will then be convolved with the IRFs, and Poission fluctuated, to get the simulated counts for each observation.

[3]:

# Define simulation parameters parameters
livetime = 1 * u.h
offset = 0.5 * u.deg
# Energy from 0.1 to 100 TeV with 10 bins/decade
energy = np.logspace(-1, 2, 31) * u.TeV

[4]:

# Define spectral model - a simple Power Law in this case
model_ref = PowerLaw(
    index=3.0,
    amplitude=2.5e-12 * u.Unit("cm-2 s-1 TeV-1"),
    reference=1 * u.TeV,
)
print(model_ref)

PowerLaw

Parameters:

           name     value   error      unit      min max frozen
        --------- --------- ----- -------------- --- --- ------
            index 3.000e+00   nan                nan nan  False
        amplitude 2.500e-12   nan cm-2 s-1 TeV-1 nan nan  False
        reference 1.000e+00   nan            TeV nan nan   True

Get and set the model parameters after initialising

The model parameters are stored in the Parameters object on the spectal model. Each model parameter is a Parameter instance. It has a value and a unit attribute, as well as a quantity property for convenience.

[5]:

print(model_ref.parameters)

Parameters

        index                   : 3.000
        amplitude               : 2.50e-12  1 / (cm2 s TeV)
        reference    (frozen)   : 1.000  TeV


[6]:

print(model_ref.parameters["index"])
model_ref.parameters["index"].value = 2.1
print(model_ref.parameters["index"])

Parameter(name='index', value=3.0, factor=3.0, scale=1.0, unit='', min=nan, max=nan, frozen=False)
Parameter(name='index', value=2.1, factor=2.1, scale=1.0, unit='', min=nan, max=nan, frozen=False)

[7]:

# Load IRFs
filename = (
    "$GAMMAPY_DATA/cta-1dc/caldb/data/cta/1dc/bcf/South_z20_50h/irf_file.fits"
)
cta_irf = load_cta_irfs(filename)

A quick look into the effective area and energy dispersion:

[8]:

aeff = cta_irf["aeff"].to_effective_area_table(offset=offset, energy=energy)
aeff.plot()
plt.loglog()
print(cta_irf["aeff"].data)

NDDataArray summary info
MapAxis

        name       : energy
        unit       : 'TeV'
        nbins      : 42
        node type  : edges
        edges min  : 1.3e-02 TeV
        edges max  : 2.0e+02 TeV
        interp     : log
MapAxis

        name       : offset
        unit       : 'deg'
        nbins      : 6
        node type  : edges
        edges min  : 0.0e+00 deg
        edges max  : 6.0e+00 deg
        interp     : lin
Data           : size =   252, min =  0.000 m2, max = 5371581.000 m2
../_images/notebooks_spectrum_simulation_13_1.png 
[9]:

edisp = cta_irf["edisp"].to_energy_dispersion(
    offset=offset, e_true=energy, e_reco=energy
)
edisp.plot_matrix()
print(edisp.data)

NDDataArray summary info
MapAxis

        name       : e_true
        unit       : 'TeV'
        nbins      : 30
        node type  : edges
        edges min  : 1.0e-01 TeV
        edges max  : 1.0e+02 TeV
        interp     : log
MapAxis

        name       : e_reco
        unit       : 'TeV'
        nbins      : 30
        node type  : edges
        edges min  : 1.0e-01 TeV
        edges max  : 1.0e+02 TeV
        interp     : log
Data           : size =   900, min =  0.000, max =  0.926
../_images/notebooks_spectrum_simulation_14_1.png 
[10]:

dataset = SpectrumDataset(
    aeff=aeff, edisp=edisp, model=model_ref, livetime=livetime, obs_id=0
)

dataset.fake(random_state=42)

[11]:

# Take a quick look at the simulated counts
dataset.counts.plot()

[11]:

<matplotlib.axes._subplots.AxesSubplot at 0x11c8d47f0>
../_images/notebooks_spectrum_simulation_16_1.png

Include Background

In this section we will include a background component. Furthermore, we will also simulate more than one observation and fit each one individually in order to get average fit results.

[12]:

# We assume a PowerLaw shape of the background as well
bkg_model = PowerLaw(
    index=2.5, amplitude=1e-11 * u.Unit("cm-2 s-1 TeV-1"), reference=1 * u.TeV
)

evaluator = SpectrumEvaluator(model=bkg_model, aeff=aeff, livetime=livetime)

npred_bkg = evaluator.compute_npred()

[13]:

dataset = SpectrumDatasetOnOff(
    aeff=aeff,
    edisp=edisp,
    model=model_ref,
    livetime=livetime,
    acceptance=1,
    acceptance_off=5,
)

[14]:

%%time
# Now simulate 30 indepenent spectra using the same set of observation conditions.
n_obs = 100
seeds = np.arange(n_obs)

datasets = []

for idx in range(n_obs):
    dataset.fake(random_state=idx, background_model=npred_bkg)
    datasets.append(dataset.copy())

CPU times: user 160 ms, sys: 2.67 ms, total: 163 ms
Wall time: 162 ms

Before moving on to the fit let’s have a look at the simulated observations.

[15]:

n_on = [dataset.counts.data.sum() for dataset in datasets]
n_off = [dataset.counts_off.data.sum() for dataset in datasets]
excess = [dataset.excess.data.sum() for dataset in datasets]

fix, axes = plt.subplots(1, 3, figsize=(12, 4))
axes[0].hist(n_on)
axes[0].set_xlabel("n_on")
axes[1].hist(n_off)
axes[1].set_xlabel("n_off")
axes[2].hist(excess)
axes[2].set_xlabel("excess");
../_images/notebooks_spectrum_simulation_22_0.png

Now, we fit each simulated spectrum individually

[16]:

%%time
results = []
for dataset in datasets:
    dataset.model = model_ref.copy()
    fit = Fit([dataset])
    result = fit.optimize()
    results.append(
        {
            "index": result.parameters["index"].value,
            "amplitude": result.parameters["amplitude"].value,
        }
    )

CPU times: user 3.78 s, sys: 9.86 ms, total: 3.79 s
Wall time: 3.79 s

We take a look at the distribution of the fitted indices. This matches very well with the spectrum that we initially injected, index=2.1

[17]:

index = np.array([_["index"] for _ in results])
plt.hist(index, bins=10, alpha=0.5)
plt.axvline(x=model_ref.parameters["index"].value, color="red")
print("spectral index: {:.2f} +/- {:.2f}".format(index.mean(), index.std()))

spectral index: 2.11 +/- 0.08
../_images/notebooks_spectrum_simulation_26_1.png

Adding a user defined model

Many spectral models in gammapy are subclasses of SpectralModel. The list of available models is shown below.

[18]:

models.SpectralModel.__subclasses__()

[18]:

[gammapy.spectrum.models.ConstantModel,
 gammapy.spectrum.models.CompoundSpectralModel,
 gammapy.spectrum.models.PowerLaw,
 gammapy.spectrum.models.PowerLaw2,
 gammapy.spectrum.models.ExponentialCutoffPowerLaw,
 gammapy.spectrum.models.ExponentialCutoffPowerLaw3FGL,
 gammapy.spectrum.models.PLSuperExpCutoff3FGL,
 gammapy.spectrum.models.PLSuperExpCutoff4FGL,
 gammapy.spectrum.models.LogParabola,
 gammapy.spectrum.models.TableModel,
 gammapy.spectrum.models.ScaleModel,
 gammapy.spectrum.models.AbsorbedSpectralModel,
 gammapy.spectrum.models.NaimaModel,
 gammapy.spectrum.models.SpectralGaussian,
 gammapy.spectrum.models.SpectralLogGaussian,
 gammapy.spectrum.crab.MeyerCrabModel]

This section shows how to add a user defined spectral model.

To do that you need to subclass SpectralModel. All SpectralModel subclasses need to have an __init__ function, which sets up the Parameters of the model and a static function called evaluate where the mathematical expression for the model is defined.

As an example we will use a PowerLaw plus a Gaussian (with fixed width).

[19]:

class UserModel(models.SpectralModel):
    def __init__(self, index, amplitude, reference, mean, width):
        super().__init__(
            [
                Parameter("index", index, min=0),
                Parameter("amplitude", amplitude, min=0),
                Parameter("reference", reference, frozen=True),
                Parameter("mean", mean, min=0),
                Parameter("width", width, min=0, frozen=True),
            ]
        )

    @staticmethod
    def evaluate(energy, index, amplitude, reference, mean, width):
        pwl = models.PowerLaw.evaluate(
            energy=energy,
            index=index,
            amplitude=amplitude,
            reference=reference,
        )
        gauss = amplitude * np.exp(-(energy - mean) ** 2 / (2 * width ** 2))
        return pwl + gauss

[20]:

model = UserModel(
    index=2,
    amplitude=1e-12 * u.Unit("cm-2 s-1 TeV-1"),
    reference=1 * u.TeV,
    mean=5 * u.TeV,
    width=0.2 * u.TeV,
)
print(model)

UserModel

Parameters:

           name     value   error      unit         min    max frozen
        --------- --------- ----- -------------- --------- --- ------
            index 2.000e+00   nan                0.000e+00 nan  False
        amplitude 1.000e-12   nan cm-2 s-1 TeV-1 0.000e+00 nan  False
        reference 1.000e+00   nan            TeV       nan nan   True
             mean 5.000e+00   nan            TeV 0.000e+00 nan  False
            width 2.000e-01   nan            TeV 0.000e+00 nan   True

[21]:

energy_range = [1, 10] * u.TeV
model.plot(energy_range=energy_range);
../_images/notebooks_spectrum_simulation_32_0.png

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?

What next?

In this tutorial we simulated and analysed the spectrum of source using CTA prod 2 IRFs.

If you’d like to go further, please see the other tutorial notebooks.