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Source files: spectrum_simulation.ipynb | spectrum_simulation.py

Spectrum simulation¶

Prerequisites¶

Knowledge of spectral extraction and datasets used in gammapy, see for instance the 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 CTA observation, in particular, we will generate OFF observations following the template background stored in the CTA IRFs.

Objective: simulate a number of spectral ON-OFF observations of a source with a power-law spectral model with CTA 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:

Setup¶

[1]:

%matplotlib inline
import matplotlib.pyplot as plt

[2]:

import numpy as np
import astropy.units as u
from astropy.coordinates import SkyCoord, Angle
from regions import CircleSkyRegion
from gammapy.datasets import SpectrumDatasetOnOff, SpectrumDataset, Datasets
from gammapy.makers import SpectrumDatasetMaker
from gammapy.modeling import Fit
from gammapy.modeling.models import (
    PowerLawSpectralModel,
    SkyModel,
)
from gammapy.irf import load_cta_irfs
from gammapy.data import Observation
from gammapy.maps import MapAxis

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 Poission fluctuated using the fake() to get the simulated counts for each observation.

[3]:

# Define simulation parameters parameters
livetime = 1 * u.h

pointing = SkyCoord(0, 0, unit="deg", frame="galactic")
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.directional_offset_by(
    position_angle=0 * u.deg, separation=offset
)
on_region = CircleSkyRegion(center=center, radius=on_region_radius)

[4]:

# 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")

PowerLawSpectralModel

   name     value         unit      min max frozen   error
--------- ---------- -------------- --- --- ------ ---------
    index 3.0000e+00                nan nan  False 0.000e+00
amplitude 2.5000e-12 cm-2 s-1 TeV-1 nan nan  False 0.000e+00
reference 1.0000e+00            TeV nan nan   True 0.000e+00

[5]:

# Load the IRFs
# In this simulation, we use the CTA-1DC irfs shipped with gammapy.
irfs = load_cta_irfs(
    "$GAMMAPY_DATA/cta-1dc/caldb/data/cta/1dc/bcf/South_z20_50h/irf_file.fits"
)

Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)

[6]:

obs = Observation.create(pointing=pointing, livetime=livetime, irfs=irfs)
print(obs)

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%

[7]:

# Make the SpectrumDataset
dataset_empty = SpectrumDataset.create(
    e_reco=energy_axis, e_true=energy_axis_true, region=on_region, name="obs-0"
)
maker = SpectrumDatasetMaker(selection=["exposure", "edisp", "background"])

dataset = maker.run(dataset_empty, obs)

[8]:

# Set the model on the dataset, and fake
dataset.models = model
dataset.fake(random_state=42)
print(dataset)

SpectrumDataset
---------------

  Name                            : obs-0

  Total counts                    : 298
  Total background counts         : 22.32
  Total excess counts             : 275.68

  Predicted counts                : 303.69
  Predicted background counts     : 22.32
  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
      amplitude               :   2.50e-12  1 / (cm2 s TeV)
      reference    (frozen)   :   1.000  TeV

You can see that backgound counts are now simulated

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

[9]:

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)

SpectrumDatasetOnOff
--------------------

  Name                            : obs-0

  Total counts                    : 310
  Total off counts                : 117.00
  Total background counts         : 23.40
  Total excess counts             : 286.60

  Predicted counts                : 304.91
  Predicted background counts     : 23.54
  Predicted excess counts         : 281.37

  Exposure min                    : 2.53e+08 m2 s
  Exposure max                    : 1.77e+10 m2 s

  Acceptance mean                 : 1.000
  Acceptance off                  : 45.000

  Number of total bins            : 9
  Number of fit bins              : 9

  Fit statistic type              : wstat
  Fit statistic value (-2 log(L)) : 3.69

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

You can see that off counts are now simulated as well. We now simulate several spectra using the same set of observation conditions.

[10]:

%%time

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)

CPU times: user 562 ms, sys: 9.67 ms, total: 572 ms
Wall time: 571 ms

[11]:

table = datasets.info_table()
table

[11]:

Table length=100

name	counts	background	excess	sqrt_ts	npred	npred_background	npred_signal	exposure_min	exposure_max	livetime	ontime	counts_rate	background_rate	excess_rate	n_bins	n_fit_bins	stat_type	stat_sum	counts_off	acceptance	acceptance_off	alpha
								m2 s	m2 s	s	s	1 / s	1 / s	1 / s
str6	float64	float64	float64	float64	float64	float64	float64	float64	float64	float64	float64	float64	float64	float64	int64	int64	str5	float64	int64	float64	float64	float64
obs-0	317.0	18.400000000000002	298.6	27.08240194504324	300.0237217925293	18.653546302364763	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08805555555555555	0.005111111111111111	0.08294444444444445	9	9	wstat	9.917001109717983	92	9.0	44.99999999999999	0.2
obs-1	275.0	22.0	253.0	23.76785365487285	302.95276320606564	21.582587715901177	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.0763888888888889	0.006111111111111111	0.07027777777777777	9	9	wstat	8.81866082855619	110	9.0	45.0	0.2
obs-2	293.0	20.6	272.4	25.17110555404655	301.8397348729255	20.469559382761037	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08138888888888889	0.005722222222222222	0.07566666666666666	9	9	wstat	7.359111002998746	103	9.0	45.0	0.2
obs-3	280.0	22.4	257.6	23.982951737405376	303.4557139026035	22.085538412438968	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.07777777777777778	0.006222222222222222	0.07155555555555557	9	9	wstat	11.832983489177316	112	9.0	45.0	0.2
obs-4	337.0	20.6	316.4	27.682709945184747	302.2812841625807	20.911108672416212	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.09361111111111112	0.005722222222222222	0.08788888888888888	9	9	wstat	17.867103727171312	103	9.0	45.0	0.2
obs-5	283.0	24.400000000000002	258.6	23.727154782347895	305.3838138459026	24.013638355738106	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.07861111111111112	0.006777777777777778	0.07183333333333335	9	9	wstat	8.005196180438258	122	9.0	44.99999999999999	0.2
obs-6	330.0	22.400000000000006	307.6	26.889184475727866	304.1716823126791	22.8015068225146	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.09166666666666666	0.006222222222222224	0.08544444444444445	9	9	wstat	10.098766444429558	112	9.0	44.999999999999986	0.2
obs-7	283.0	26.000000000000004	257.0	23.389178133235443	307.02186335244875	25.651687862284245	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.07861111111111112	0.007222222222222224	0.07138888888888889	9	9	wstat	4.310899779189119	130	9.0	44.99999999999999	0.2
obs-8	308.0	23.400000000000002	284.6	25.42049273328333	304.8351782861283	23.46500279596388	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08555555555555555	0.006500000000000001	0.07905555555555556	9	9	wstat	4.142947213516996	117	9.0	44.99999999999999	0.2
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
obs-90	286.0	19.000000000000004	267.0	25.131221887043438	300.26600271111903	18.895827220954608	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.07944444444444444	0.005277777777777779	0.07416666666666667	9	9	wstat	6.724388228060566	95	9.0	44.99999999999999	0.2
obs-91	285.0	25.200000000000003	259.8	23.67754591931069	306.2191485971411	24.848973106976626	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.07916666666666666	0.007000000000000001	0.07216666666666667	9	9	wstat	14.577940826871759	126	9.0	44.99999999999999	0.2
obs-92	313.0	23.6	289.4	25.664935420194176	305.058327340992	23.688151850827467	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08694444444444445	0.006555555555555556	0.08038888888888888	9	9	wstat	6.304646568308098	118	9.0	45.0	0.2
obs-93	302.0	18.8	283.2	26.123867522605497	300.12595138715096	18.75577589698646	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08388888888888889	0.005222222222222223	0.07866666666666666	9	9	wstat	5.86503429029057	94	9.0	45.0	0.2
obs-94	322.0	22.0	300.0	26.5292481657426	303.66420796523397	22.294032475069507	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08944444444444444	0.006111111111111111	0.08333333333333333	9	9	wstat	10.005883760873052	110	9.0	45.0	0.2
obs-95	305.0	24.600000000000005	280.4	24.98804632088198	305.8930204465737	24.522844956409255	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08472222222222223	0.0068333333333333345	0.07788888888888888	9	9	wstat	5.627799171046288	123	9.0	44.99999999999999	0.2
obs-96	301.0	23.6	277.4	24.969845969421428	305.0399592645493	23.669783774384797	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08361111111111111	0.006555555555555556	0.07705555555555554	9	9	wstat	5.51177203120533	118	9.0	45.0	0.2
obs-97	290.0	18.8	271.2	25.417982194454826	300.0933776554593	18.723202165294804	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08055555555555556	0.005222222222222223	0.07533333333333334	9	9	wstat	5.667231272491558	94	9.0	45.0	0.2
obs-98	301.0	20.400000000000002	280.6	25.687832964675007	301.756148559105	20.38597306894053	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08361111111111111	0.005666666666666667	0.07794444444444446	9	9	wstat	7.14214115301812	102	9.0	44.99999999999999	0.2
obs-99	323.0	20.8	302.2	26.85707842376871	302.4766917650425	21.106516274878086	281.37017549016446	252718170.97287515	17719697919.59926	3600.0	3600.0	0.08972222222222222	0.005777777777777778	0.08394444444444445	9	9	wstat	5.059623929851605	104	9.0	45.0	0.2

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

[12]:

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");

../_images/tutorials_spectrum_simulation_19_0.png

Now, we fit each simulated spectrum individually

[13]:

%%time
results = []

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

CPU times: user 11.7 s, sys: 129 ms, total: 11.8 s
Wall time: 12.1 s

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

[14]:

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

index: 3.0036666673409944 += 0.08075110145690628

../_images/tutorials_spectrum_simulation_23_1.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?

[ ]: