This is a fixed-text formatted version of a Jupyter notebook

Flux point fitting in Gammapy

Prerequisites

Context

Some high level studies do not rely on reduced datasets with their IRFs but directly on higher level products such as flux points. This is not ideal because flux points already contain some hypothesis for the underlying spectral shape and the uncertainties they carry are usually simplified (e.g. symmetric gaussian errors). Yet, this is an efficient way to combine heterogeneous data.

Objective: fit spectral models to combined Fermi-LAT and IACT flux points.

Proposed approach

Here we will load, the spectral points from Fermi-LAT and TeV catalogs and fit them with various spectral models to find the best representation of the wide band spectrum.

The central class we’re going to use for this example analysis is:

In addition we will work with the following data classes:

And the following spectral model classes:

Setup

Let us start with the usual IPython notebook and Python imports:

[1]:

%matplotlib inline

[2]:

import numpy as np
from astropy import units as u
from gammapy.modeling.models import (
    PowerLawSpectralModel,
    ExpCutoffPowerLawSpectralModel,
    LogParabolaSpectralModel,
    SkyModel,
)
from gammapy.spectrum import FluxPointsDataset, FluxPoints
from gammapy.catalog import SOURCE_CATALOGS
from gammapy.modeling import Fit

Load spectral points

For this analysis we choose to work with the source ‘HESS J1507-622’ and the associated Fermi-LAT sources ‘3FGL J1506.6-6219’ and ‘3FHL J1507.9-6228e’. We load the source catalogs, and then access source of interest by name:

[3]:

catalog_3fgl = SOURCE_CATALOGS["3fgl"]()
catalog_3fhl = SOURCE_CATALOGS["3fhl"]()
catalog_gammacat = SOURCE_CATALOGS["gamma-cat"]()

[4]:

source_fermi_3fgl = catalog_3fgl["3FGL J1506.6-6219"]
source_fermi_3fhl = catalog_3fhl["3FHL J1507.9-6228e"]
source_gammacat = catalog_gammacat["HESS J1507-622"]

The corresponding flux points data can be accessed with .flux_points attribute:

[5]:

flux_points_gammacat = source_gammacat.flux_points
flux_points_gammacat.table

[5]:
Table length=6
e_refdndednde_errndnde_errp
TeV1 / (cm2 s TeV)1 / (cm2 s TeV)1 / (cm2 s TeV)
float32float32float32float32
0.86092.29119e-128.70543e-138.95502e-13
1.561516.98172e-132.20354e-132.30407e-13
2.763751.69062e-136.7587e-147.18838e-14
4.89167.72925e-142.40132e-142.60749e-14
9.988581.03253e-145.06315e-155.64195e-15
27.04037.44987e-165.72089e-167.25999e-16

In the Fermi-LAT catalogs, integral flux points are given. Currently the flux point fitter only works with differential flux points, so we apply the conversion here.

[6]:

flux_points_3fgl = source_fermi_3fgl.flux_points.to_sed_type(
    sed_type="dnde", model=source_fermi_3fgl.spectral_model()
)
flux_points_3fhl = source_fermi_3fhl.flux_points.to_sed_type(
    sed_type="dnde", model=source_fermi_3fhl.spectral_model()
)

Finally we stack the flux points into a single gammapy.spectrum.FluxPoints object and drop the upper limit values, because currently we can’t handle them in the fit:

[7]:

# Stack flux point tables
flux_points = FluxPoints.stack(
    [flux_points_gammacat, flux_points_3fhl, flux_points_3fgl]
)

t = flux_points.table
t["dnde_err"] = 0.5 * (t["dnde_errn"] + t["dnde_errp"])

# Remove upper limit points, where `dnde_errn = nan`
is_ul = np.isfinite(t["dnde_err"])
flux_points = FluxPoints(t[is_ul])
flux_points

[7]:

FluxPoints(sed_type='dnde', n_points=14)

Power Law Fit

First we start with fitting a simple gammapy.modeling.models.PowerLawSpectralModel.

[8]:

pwl = PowerLawSpectralModel(
    index=2, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV"
)
model = SkyModel(spectral_model=pwl)

After creating the model we run the fit by passing the 'flux_points' and 'model' objects:

[9]:

dataset_pwl = FluxPointsDataset(model, flux_points)
fitter = Fit([dataset_pwl])
result_pwl = fitter.run()

And print the result:

[10]:

print(result_pwl)

OptimizeResult

        backend    : minuit
        method     : minuit
        success    : True
        message    : Optimization terminated successfully.
        nfev       : 40
        total stat : 28.29


[11]:

print(pwl)

PowerLawSpectralModel

   name     value   error      unit      min max frozen
--------- --------- ----- -------------- --- --- ------
    index 1.985e+00   nan                nan nan  False
amplitude 1.283e-12   nan cm-2 s-1 TeV-1 nan nan  False
reference 1.000e+00   nan            TeV nan nan   True

Finally we plot the data points and the best fit model:

[12]:

ax = flux_points.plot(energy_power=2)
pwl.plot(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)

# assign covariance for plotting
pwl.parameters.covariance = result_pwl.parameters.covariance

pwl.plot_error(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)
ax.set_ylim(1e-13, 1e-11);
../_images/notebooks_sed_fitting_22_0.png

Exponential Cut-Off Powerlaw Fit

Next we fit an gammapy.modeling.models.ExpCutoffPowerLawSpectralModel law to the data.

[13]:

ecpl = ExpCutoffPowerLawSpectralModel(
    index=1.8,
    amplitude="2e-12 cm-2 s-1 TeV-1",
    reference="1 TeV",
    lambda_="0.1 TeV-1",
)
model = SkyModel(spectral_model=ecpl)

We run the fitter again by passing the flux points and the model instance:

[14]:

dataset_ecpl = FluxPointsDataset(model, flux_points)
fitter = Fit([dataset_ecpl])
result_ecpl = fitter.run()
print(ecpl)

ExpCutoffPowerLawSpectralModel

   name     value   error      unit      min max frozen
--------- --------- ----- -------------- --- --- ------
    index 1.894e+00   nan                nan nan  False
amplitude 1.962e-12   nan cm-2 s-1 TeV-1 nan nan  False
reference 1.000e+00   nan            TeV nan nan   True
  lambda_ 7.766e-02   nan          TeV-1 nan nan  False
    alpha 1.000e+00   nan                nan nan   True

We plot the data and best fit model:

[15]:

ax = flux_points.plot(energy_power=2)
ecpl.plot(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)

# assign covariance for plotting
ecpl.parameters.covariance = result_ecpl.parameters.covariance

ecpl.plot_error(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)
ax.set_ylim(1e-13, 1e-11)

[15]:

(1e-13, 1e-11)
../_images/notebooks_sed_fitting_28_1.png

Log-Parabola Fit

Finally we try to fit a gammapy.modeling.models.LogParabolaSpectralModel model:

[16]:

log_parabola = LogParabolaSpectralModel(
    alpha=2, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV", beta=0.1
)
model = SkyModel(spectral_model=log_parabola)

[17]:

dataset_log_parabola = FluxPointsDataset(model, flux_points)
fitter = Fit([dataset_log_parabola])
result_log_parabola = fitter.run()
print(log_parabola)

LogParabolaSpectralModel

   name     value   error      unit      min max frozen
--------- --------- ----- -------------- --- --- ------
amplitude 1.877e-12   nan cm-2 s-1 TeV-1 nan nan  False
reference 1.000e+00   nan            TeV nan nan   True
    alpha 2.144e+00   nan                nan nan  False
     beta 4.936e-02   nan                nan nan  False

[18]:

ax = flux_points.plot(energy_power=2)
log_parabola.plot(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)

# assign covariance for plotting
log_parabola.parameters.covariance = result_log_parabola.parameters.covariance

log_parabola.plot_error(
    energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2
)
ax.set_ylim(1e-13, 1e-11);
../_images/notebooks_sed_fitting_32_0.png

Exercises

What next?

This was an introduction to SED fitting in Gammapy.

  • If you would like to learn how to perform a full Poisson maximum likelihood spectral fit, please check out the spectrum analysis tutorial.

  • To learn how to combine heterogeneous datasets to perform a multi-instrument forward-folding fit see the MWL analysis tutorial

[ ]: