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3D simulation and fitting¶

This tutorial shows how to do a 3D map-based simulation and fit.

For a tutorial on how to do a 3D map analyse of existing data, see the analysis_3d tutorial.

This can be useful to do a performance / sensitivity study, or to evaluate the capabilities of Gammapy or a given analysis method. Note that is is a binned simulation as is e.g. done also in Sherpa for Chandra, not an event sampling and anbinned analysis as is done e.g. in the Fermi ST or ctools.

Imports and versions¶

[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 gammapy.maps import WcsGeom, MapAxis, WcsNDMap, Map
from gammapy.spectrum.models import PowerLaw
from gammapy.image.models import SkyGaussian
from gammapy.cube.models import SkyModel, SkyModels, BackgroundModel
from gammapy.cube import MapDataset, PSFKernel
from gammapy.cube import make_map_exposure_true_energy, make_map_background_irf
from gammapy.utils.fitting import Fit
from gammapy.data import FixedPointingInfo

[3]:

!gammapy info --no-envvar --no-dependencies --no-system


Gammapy package:

version                : 0.11



Simulate¶

[4]:

filename = (
"$GAMMAPY_DATA/cta-1dc/caldb/data/cta/1dc/bcf/South_z20_50h/irf_file.fits" ) irfs = load_cta_irfs(filename)  [5]:  # Define sky model to simulate the data spatial_model = SkyGaussian(lon_0="0.2 deg", lat_0="0.1 deg", sigma="0.3 deg") spectral_model = PowerLaw( index=3, amplitude="1e-11 cm-2 s-1 TeV-1", reference="1 TeV" ) sky_model = SkyModel( spatial_model=spatial_model, spectral_model=spectral_model ) print(sky_model)  SkyModel Parameters: name value error unit min max frozen --------- --------- ----- -------------- ---------- --------- ------ lon_0 2.000e-01 nan deg -1.800e+02 1.800e+02 False lat_0 1.000e-01 nan deg -9.000e+01 9.000e+01 False sigma 3.000e-01 nan deg 0.000e+00 nan False index 3.000e+00 nan nan nan False amplitude 1.000e-11 nan cm-2 s-1 TeV-1 nan nan False reference 1.000e+00 nan TeV nan nan True  [6]:  # Define map geometry axis = MapAxis.from_edges( np.logspace(-1.0, 1.0, 10), unit="TeV", name="energy", interp="log" ) geom = WcsGeom.create( skydir=(0, 0), binsz=0.02, width=(5, 4), coordsys="GAL", axes=[axis] )  [7]:  # Define some observation parameters # We read in the pointing info from one of the 1dc event list files as an example pointing = FixedPointingInfo.read( "$GAMMAPY_DATA/cta-1dc/data/baseline/gps/gps_baseline_110380.fits"
)

livetime = 1 * u.hour
offset_max = 2 * u.deg
offset = Angle("2 deg")

[8]:

exposure = make_map_exposure_true_energy(
)

[9]:

background = make_map_background_irf(
pointing=pointing, ontime=livetime, bkg=irfs["bkg"], geom=geom
)

WARNING: Tried to get polar motions for times after IERS data is valid. Defaulting to polar motion from the 50-yr mean for those. This may affect precision at the 10s of arcsec level [astropy.coordinates.builtin_frames.utils]

[10]:

psf = irfs["psf"].to_energy_dependent_table_psf(theta=offset)
psf_kernel = PSFKernel.from_table_psf(psf, geom, max_radius=0.3 * u.deg)
psf_kernel.psf_kernel_map.sum_over_axes().plot(stretch="log");

[11]:

energy = axis.edges * axis.unit
edisp = irfs["edisp"].to_energy_dispersion(
offset, e_reco=energy, e_true=energy
)
edisp.plot_matrix();


Now we have to compute npred maps, i.e. “predicted counts per pixel” given the model and the observation infos: exposure, background, PSF and EDISP. For this we use the MapDataset object:

[12]:

background_model = BackgroundModel(background)
dataset = MapDataset(
model=sky_model,
exposure=exposure,
background_model=background_model,
psf=psf_kernel,
edisp=edisp,
)

[13]:

npred = dataset.npred()

[14]:

npred.sum_over_axes().plot(add_cbar=True);

[15]:

# This one line is the core of how to simulate data when
# using binned simulation / analysis: you Poisson fluctuate
# npred to obtain simulated observed counts.
# Compute counts as a Poisson fluctuation
rng = np.random.RandomState(seed=42)
counts = rng.poisson(npred.data)
counts_map = WcsNDMap(geom, counts)

[16]:

counts_map.sum_over_axes().plot();


Fit¶

Now let’s analyse the simulated data. Here we just fit it again with the same model we had before, but you could do any analysis you like here, e.g. fit a different model, or do a region-based analysis, …

[17]:

# Define sky model to fit the data
spatial_model = SkyGaussian(lon_0="0.1 deg", lat_0="0.1 deg", sigma="0.5 deg")
spectral_model = PowerLaw(
index=2, amplitude="1e-11 cm-2 s-1 TeV-1", reference="1 TeV"
)
model = SkyModel(spatial_model=spatial_model, spectral_model=spectral_model)
print(model)

SkyModel

Parameters:

name     value   error      unit         min        max    frozen
--------- --------- ----- -------------- ---------- --------- ------
lon_0 1.000e-01   nan            deg -1.800e+02 1.800e+02  False
lat_0 1.000e-01   nan            deg -9.000e+01 9.000e+01  False
sigma 5.000e-01   nan            deg  0.000e+00       nan  False
index 2.000e+00   nan                       nan       nan  False
amplitude 1.000e-11   nan cm-2 s-1 TeV-1        nan       nan  False
reference 1.000e+00   nan            TeV        nan       nan   True

[18]:

# We do not want to fit the background in this case, so we will freeze the parameters
background_model.parameters["norm"].value = 1.0
background_model.parameters["norm"].frozen = True
background_model.parameters["tilt"].frozen = True

print(background_model)

BackgroundModel

Parameters:

name     value   error unit    min    max frozen
--------- --------- ----- ---- --------- --- ------
norm 1.000e+00   nan      0.000e+00 nan   True
tilt 0.000e+00   nan            nan nan   True
reference 1.000e+00   nan  TeV       nan nan   True

[19]:

dataset = MapDataset(
model=model,
exposure=exposure,
counts=counts_map,
background_model=background_model,
psf=psf_kernel,
edisp=edisp,
)

[20]:

%%time
fit = Fit(dataset)
result = fit.run(optimize_opts={"print_level": 1})


 FCN = 233997.84912191165 TOTAL NCALL = 214 NCALLS = 214 EDM = 4.7430748660357885e-06 GOAL EDM = 1e-05 UP = 1.0
 Valid Valid Param Accurate Covar PosDef Made PosDef True True True True False Hesse Fail HasCov Above EDM Reach calllim False True False False
 + Name Value Hesse Error Minos Error- Minos Error+ Limit- Limit+ Fixed? 0 par_000_lon_0 18.5861 0.881898 -18000 18000 No 1 par_001_lat_0 0.876832 0.0874735 -900 900 No 2 par_002_sigma 2.96346 0.0609912 0 No 3 par_003_index 3.05558 0.0305672 No 4 par_004_amplitude 0.905963 0.046131 No

CPU times: user 16.2 s, sys: 374 ms, total: 16.6 s
Wall time: 8.33 s


True model:

[21]:

print(sky_model)

SkyModel

Parameters:

name     value   error      unit         min        max    frozen
--------- --------- ----- -------------- ---------- --------- ------
lon_0 2.000e-01   nan            deg -1.800e+02 1.800e+02  False
lat_0 1.000e-01   nan            deg -9.000e+01 9.000e+01  False
sigma 3.000e-01   nan            deg  0.000e+00       nan  False
index 3.000e+00   nan                       nan       nan  False
amplitude 1.000e-11   nan cm-2 s-1 TeV-1        nan       nan  False
reference 1.000e+00   nan            TeV        nan       nan   True


Best-fit model:

[22]:

print(model)

SkyModel

Parameters:

name     value   error      unit         min        max    frozen
--------- --------- ----- -------------- ---------- --------- ------
lon_0 1.859e-01   nan            deg -1.800e+02 1.800e+02  False
lat_0 8.768e-02   nan            deg -9.000e+01 9.000e+01  False
sigma 2.963e-01   nan            deg  0.000e+00       nan  False
index 3.056e+00   nan                       nan       nan  False
amplitude 9.060e-12   nan cm-2 s-1 TeV-1        nan       nan  False
reference 1.000e+00   nan            TeV        nan       nan   True


To get the errors on the model, we can check the covariance table:

[23]:

result.parameters.covariance_to_table()

[23]:

Table length=9
namelon_0lat_0sigmaindexamplitudereferencenormtilt
str9float64float64float64float64float64float64float64float64
lon_07.777e-053.533e-07-1.008e-07-4.077e-064.958e-170.000e+000.000e+000.000e+00
lat_03.533e-077.652e-051.832e-063.182e-064.912e-170.000e+000.000e+000.000e+00
sigma-1.008e-071.832e-063.720e-05-5.426e-077.917e-160.000e+000.000e+000.000e+00
index-4.077e-063.182e-06-5.426e-079.344e-04-1.196e-140.000e+000.000e+000.000e+00
amplitude4.958e-174.912e-177.917e-16-1.196e-142.128e-250.000e+000.000e+000.000e+00
reference0.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+00
norm0.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+00
tilt0.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+00
reference0.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+000.000e+00
[24]:

# Or, to see the value of and error on an individual parameter, say index:
print(result.parameters["index"].value, result.parameters.error("index"))

3.055576702983643 0.03056715401514759

[25]:

# TODO: show e.g. how to make a residual image