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_tsis_ul
MeVMeVMeV1 / (cm2 MeV s)1 / (cm2 MeV s)1 / (cm2 MeV s)1 / (cm2 MeV s)
float64float64float64float64float64float64float64float32bool
173.205100.000300.0001.798e-105.566e-115.710e-11nan2.843False
547.723300.0001000.0002.171e-131.689e-12nan3.595e-120.000True
1732.0511000.0003000.0002.528e-131.058e-139.991e-14nan2.661False
5477.2263000.00010000.0002.654e-148.936e-157.932e-15nan4.265False
31622.77710000.000100000.0001.274e-154.237e-163.658e-16nan5.774False
[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_tsis_ul
GeVGeVGeV1 / (cm2 GeV s)1 / (cm2 GeV s)1 / (cm2 GeV s)1 / (cm2 GeV s)
float64float64float64float64float64float64float64float32bool
14.14210.00020.0009.288e-122.343e-122.128e-12nan5.660False
31.62320.00050.0002.777e-126.572e-135.818e-13nan6.940False
86.60350.000150.0002.335e-131.055e-138.554e-14nan3.835False
273.861150.000500.0006.411e-142.697e-142.133e-14nan5.697False
1000.000500.0002000.0009.188e-214.034e-15nan8.068e-150.000True

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   +/-    0.03
    amplitude               :   1.28e-12   +/- 1.6e-13 1 / (cm2 s TeV)
    reference    (frozen)   :      1.000       TeV


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

[13]:
ax = plt.subplot()

kwargs = {
    "ax": ax,
    "sed_type": "e2dnde",
    "yunits": u.Unit("TeV cm-2 s-1")
}

for d in datasets:
    d.data.plot(label=d.name, **kwargs)

pwl.plot(energy_bounds=[1e-4, 1e2] * u.TeV, color="k", **kwargs)
pwl.plot_error(energy_bounds=[1e-4, 1e2] * u.TeV, **kwargs)
ax.set_ylim(1e-13, 1e-11);
ax.legend()
[13]:
<matplotlib.legend.Legend at 0x7f562aa5ec18>
../../../_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.to_dict())
{'name': 'j1507-ecpl', 'type': 'SkyModel', 'spectral': {'type': 'ExpCutoffPowerLawSpectralModel', 'parameters': [{'name': 'index', 'value': 1.8943430533300762, 'error': 0.047047318380120814, 'interp': 'lin'}, {'name': 'amplitude', 'value': 1.962110470599695e-12, 'unit': 'cm-2 s-1 TeV-1', 'error': 3.903255254322202e-13, 'interp': 'log', 'scale_method': 'scale10'}, {'name': 'reference', 'value': 1.0, 'unit': 'TeV', 'frozen': True, 'interp': 'lin'}, {'name': 'lambda_', 'value': 0.07765585367928371, 'unit': 'TeV-1', 'error': 0.05255178831722765, 'interp': 'lin'}, {'name': 'alpha', 'value': 1.0, 'frozen': True, 'interp': 'lin'}]}}

We plot the data and best fit model:

[16]:
ax = plt.subplot()

kwargs = {
    "ax": ax,
    "sed_type": "e2dnde",
    "yunits": u.Unit("TeV cm-2 s-1")
}

for d in datasets:
    d.data.plot(label=d.name,**kwargs)

ecpl.plot(energy_bounds=[1e-4, 1e2] * u.TeV, color="k", **kwargs)
ecpl.plot_error(energy_bounds=[1e-4, 1e2] * u.TeV, **kwargs)
ax.set_ylim(1e-13, 1e-11);
ax.legend()
[16]:
<matplotlib.legend.Legend at 0x7f562a3ae940>
../../../_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   +/- 2.8e-13 1 / (cm2 s TeV)
    reference    (frozen)   :      1.000       TeV
    alpha                   :      2.144   +/-    0.07
    beta                    :      0.049   +/-    0.02


[19]:
ax = plt.subplot()

kwargs = {
    "ax": ax,
    "sed_type": "e2dnde",
    "yunits": u.Unit("TeV cm-2 s-1")
}

for d in datasets:
    d.data.plot(label=d.name, **kwargs)

log_parabola.plot(energy_bounds=[1e-4, 1e2] * u.TeV, color="k", **kwargs)
log_parabola.plot_error(energy_bounds=[1e-4, 1e2] * u.TeV, **kwargs)
ax.set_ylim(1e-13, 1e-11);
ax.legend()
[19]:
<matplotlib.legend.Legend at 0x7f562a167240>
../../../_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

[ ]: