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

Flux point fitting

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
import matplotlib.pyplot as plt
[2]:
from astropy import units as u
from gammapy.modeling.models import (
    PowerLawSpectralModel,
    ExpCutoffPowerLawSpectralModel,
    LogParabolaSpectralModel,
    SkyModel,
)
from gammapy.estimators import FluxPoints
from gammapy.datasets import FluxPointsDataset, Datasets
from gammapy.catalog import CATALOG_REGISTRY
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 = CATALOG_REGISTRY.get_cls("3fgl")()
catalog_3fhl = CATALOG_REGISTRY.get_cls("3fhl")()
catalog_gammacat = CATALOG_REGISTRY.get_cls("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]:
dataset_gammacat = FluxPointsDataset(data=source_gammacat.flux_points, name="gammacat")
dataset_gammacat.data.to_table(sed_type="dnde", formatted=True)
[5]:
Table length=6
e_refe_mine_maxdndednde_errpdnde_errn
TeVTeVTeV1 / (cm2 s TeV)1 / (cm2 s TeV)1 / (cm2 s TeV)
float64float64float64float64float64float64
0.8610.6391.1592.291e-128.955e-138.705e-13
1.5621.1592.0776.982e-132.304e-132.204e-13
2.7642.0773.6771.691e-137.188e-146.759e-14
4.8923.6776.9907.729e-142.607e-142.401e-14
9.9896.99016.4351.033e-145.642e-155.063e-15
27.04016.43544.4907.450e-167.260e-165.721e-16
[6]:
dataset_3fgl = FluxPointsDataset(data=source_fermi_3fgl.flux_points, name="3fgl")
dataset_3fgl.data.to_table(sed_type="dnde", formatted=True)
[6]:
Table length=5
e_refe_mine_maxdndednde_errpdnde_errndnde_ulsqrt_ts
MeVMeVMeV1 / (cm2 MeV s)1 / (cm2 MeV s)1 / (cm2 MeV s)1 / (cm2 MeV s)
float64float64float64float64float64float64float64float32
173.205100.000300.0001.798e-105.566e-115.710e-11nan2.843
547.723300.0001000.0002.171e-131.689e-12nan3.595e-120.000
1732.0511000.0003000.0002.528e-131.058e-139.991e-14nan2.661
5477.2263000.00010000.0002.654e-148.936e-157.932e-15nan4.265
31622.77710000.000100000.0001.274e-154.237e-163.658e-16nan5.774
[7]:
dataset_3fhl = FluxPointsDataset(data=source_fermi_3fhl.flux_points, name="3fhl")
dataset_3fhl.data.to_table(sed_type="dnde", formatted=True)
[7]:
Table length=5
e_refe_mine_maxdndednde_errpdnde_errndnde_ulsqrt_ts
GeVGeVGeV1 / (cm2 GeV s)1 / (cm2 GeV s)1 / (cm2 GeV s)1 / (cm2 GeV s)
float64float64float64float64float64float64float64float32
14.14210.00020.0009.288e-122.343e-122.128e-12nan5.660
31.62320.00050.0002.777e-126.572e-135.818e-13nan6.940
86.60350.000150.0002.335e-131.055e-138.554e-14nan3.835
273.861150.000500.0006.411e-142.697e-142.133e-14nan5.697
1000.000500.0002000.0009.188e-214.034e-15nan8.068e-150.000

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, name="j1507-pl")

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

[9]:
datasets = Datasets([dataset_gammacat, dataset_3fgl, dataset_3fhl])
datasets.models = model
print(datasets)
Datasets
--------

Dataset 0:

  Type       : FluxPointsDataset
  Name       : gammacat
  Instrument :
  Models     : ['j1507-pl']

Dataset 1:

  Type       : FluxPointsDataset
  Name       : 3fgl
  Instrument :
  Models     : ['j1507-pl']

Dataset 2:

  Type       : FluxPointsDataset
  Name       : 3fhl
  Instrument :
  Models     : ['j1507-pl']


[10]:
fitter = Fit()
result_pwl = fitter.run(datasets=datasets)

And print the result:

[11]:
print(result_pwl["optimize_result"])
OptimizeResult

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

[12]:
print(model)
SkyModel

  Name                      : j1507-pl
  Datasets names            : None
  Spectral model type       : PowerLawSpectralModel
  Spatial  model type       :
  Temporal model type       :
  Parameters:
    index                   :   1.985
    amplitude               :   1.28e-12  1 / (cm2 s TeV)
    reference    (frozen)   :   1.000  TeV


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

[13]:
ax = plt.subplot()

for d in datasets:
    d.data.plot(label=d.name, sed_type="e2dnde", yunits=u.Unit("TeV cm-2 s-1"))

pwl.plot(energy_bounds=[1e-4, 1e2] * u.TeV, ax=ax, color="k", sed_type="e2dnde")
pwl.plot_error(energy_bounds=[1e-4, 1e2] * u.TeV, ax=ax, sed_type="e2dnde")
ax.set_ylim(1e-13, 1e-11);
ax.legend()
[13]:
<matplotlib.legend.Legend at 0x7fdcd7692ef0>
../../../_images/tutorials_analysis_1D_sed_fitting_21_1.png

Exponential Cut-Off Powerlaw Fit

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

[14]:
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, name="j1507-ecpl")

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

[15]:
datasets.models = model
result_ecpl = fitter.run(datasets=datasets)
print(model)
SkyModel

  Name                      : j1507-ecpl
  Datasets names            : None
  Spectral model type       : ExpCutoffPowerLawSpectralModel
  Spatial  model type       :
  Temporal model type       :
  Parameters:
    index                   :   1.894
    amplitude               :   1.96e-12  1 / (cm2 s TeV)
    reference    (frozen)   :   1.000  TeV
    lambda_                 :   0.078  1 / TeV
    alpha        (frozen)   :   1.000


We plot the data and best fit model:

[16]:
ax = plt.subplot()

for d in datasets:
    d.data.plot(sed_type="e2dnde", label=d.name, yunits=u.Unit("TeV cm-2 s-1"))

ecpl.plot(energy_bounds=[1e-4, 1e2] * u.TeV, ax=ax, sed_type="e2dnde", color="k")
ecpl.plot_error(energy_bounds=[1e-4, 1e2] * u.TeV, ax=ax, sed_type="e2dnde")
ax.set_ylim(1e-13, 1e-11);
ax.legend()
[16]:
<matplotlib.legend.Legend at 0x7fdcd7006908>
../../../_images/tutorials_analysis_1D_sed_fitting_27_1.png

Log-Parabola Fit

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

[17]:
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, name="j1507-lp")
[18]:
datasets.models = model
result_log_parabola = fitter.run(datasets=datasets)
print(model)
SkyModel

  Name                      : j1507-lp
  Datasets names            : None
  Spectral model type       : LogParabolaSpectralModel
  Spatial  model type       :
  Temporal model type       :
  Parameters:
    amplitude               :   1.88e-12  1 / (cm2 s TeV)
    reference    (frozen)   :   1.000  TeV
    alpha                   :   2.144
    beta                    :   0.049


[19]:
ax = plt.subplot()

for d in datasets:
    d.data.plot(sed_type="e2dnde", label=d.name, yunits=u.Unit("TeV cm-2 s-1"))

log_parabola.plot(energy_bounds=[1e-4, 1e2] * u.TeV, ax=ax, sed_type="e2dnde", color="k")
log_parabola.plot_error(energy_bounds=[1e-4, 1e2] * u.TeV, ax=ax, sed_type="e2dnde")
ax.set_ylim(1e-13, 1e-11);
ax.legend()
[19]:
<matplotlib.legend.Legend at 0x7fdcd6e58a58>
../../../_images/tutorials_analysis_1D_sed_fitting_31_1.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

[ ]: