Fermi-LAT via FermiPyLike

In this Example we show how to use the fermipy plugin in threeML. We perform a Binned likelihood analysis and a Bayesian analysis of the Crab, optimizing the parameters of the Crab Pulsar (PSR J0534+2200) keeping fixed the parameters of the Crab Nebula. In the model, the nebula is described by two sources, one representing the synchrotron spectrum, the othet the Inverse Compton emission. In this example we show how to download Fermi-LAT data, how to build a model starting from the 4FGL, how to free and fix parameters of the sources in the model, and how to perform a spectral analysis using the fermipy plugin.

[1]:
import warnings

warnings.simplefilter("ignore")
import numpy as np

np.seterr(all="ignore")
import shutil
from IPython.display import Image, display
import glob
from pathlib import Path
import matplotlib as mpl
from matplotlib import pyplot as plt
from astropy.io import fits as pyfits
import scipy as sp
[2]:
%%capture
from threeML import *
[3]:
from jupyterthemes import jtplot

%matplotlib inline
jtplot.style(context="talk", fscale=1, ticks=True, grid=False)
set_threeML_style()
silence_warnings()

The Fermi 4FGL catalog

Let’s interrogate the 4FGL to get the sources in a radius of 20.0 deg around the Crab

[4]:
lat_catalog = FermiLATSourceCatalog()

ra, dec, table = lat_catalog.search_around_source("Crab", radius=20.0)

table
Trying https://heasarc.gsfc.nasa.gov/cgi-bin/vo/cone/coneGet.pl?table=fermilpsc&
[4]:
Table length=172
namesource_typeshort_source_typeradecassoc_nametevcat_assocSearch_Offset
degdeg
objectstr32objectfloat64float64objectobjectfloat64
4FGL J0534.5+2200pulsar, identified by pulsationsPSR83.636722.0149PSR J0534+2200Crab pulsar0.2093
4FGL J0534.5+2201spulsar wind nebulaPWN83.633122.0199Crab NebulaCrab0.3945
4FGL J0534.5+2201ipulsar wind nebulaPWN83.633022.0200Crab NebulaCrab0.4008
4FGL J0526.3+2246active galaxy of uncertain typebcu81.590822.7778NVSS J052622+224801122.2389
4FGL J0544.4+2238unknown86.109322.6418142.4970
4FGL J0521.7+2112BL Lac type of blazarbll80.444521.2131TXS 0518+211VER J0521+211184.2394
4FGL J0528.3+1817unknownunk82.094618.29431RXS J052829.6+181657239.3750
4FGL J0519.7+1939unknown79.945719.6646250.2177
4FGL J0536.2+1733BL Lac type of blazarbll84.071917.5534TXS 0533+175268.7380
........................
4FGL J0639.6+3503active galaxy of uncertain typebcu99.902435.0586B2 0635+351157.9723
4FGL J0552.8+0313active galaxy of uncertain typebcu88.21803.2322PKS 0550+0321158.0759
4FGL J0431.0+3529cunknown67.765035.49491159.2470
4FGL J0653.6+1636active galaxy of uncertain typebcu103.410516.61062MASX J06533986+16364321164.6964
4FGL J0658.7+2318unknown104.680823.30271166.9279
4FGL J0555.1+0304active galaxy of uncertain typebcu88.77763.0710GB6 J0555+03041175.3905
4FGL J0658.2+2709active galaxy of uncertain typebcu104.573527.1501B2 0655+27A1181.7022
4FGL J0642.4+1048unknown100.608110.81351183.9156
4FGL J0506.9+0323BL Lac type of blazarbll76.73143.3917NVSS J050650+0324011187.3856
4FGL J0409.2+2542unknown62.314425.70221189.1777

This gets a 3ML model (a Model instance) from the table above, where every source in the 4FGL becomes a Source instance. Note that by default all parameters of all sources are fixed.

[5]:
model = lat_catalog.get_model()

Let’s free all the normalizations within 3 deg from the center.

[6]:
model.free_point_sources_within_radius(3.0, normalization_only=True)

model.display()
Model summary:

N
Point sources 172
Extended sources 0
Particle sources 0


Free parameters (5):

value min_value max_value unit
PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.K 0.0 0.0 0.0 keV-1 s-1 cm-2
Crab_synch.spectrum.main.Log_parabola.K 0.0 0.0 0.0 keV-1 s-1 cm-2
Crab_IC.spectrum.main.Log_parabola.K 0.0 0.0 0.0 keV-1 s-1 cm-2
NVSS_J052622p224801.spectrum.main.Powerlaw.K 0.0 0.0 0.0 keV-1 s-1 cm-2
_4FGL_J0544d4p2238.spectrum.main.Powerlaw.K 0.0 0.0 0.0 keV-1 s-1 cm-2


Fixed parameters (957):
(abridged. Use complete=True to see all fixed parameters)


Properties (0):

(none)


Linked parameters (0):

(none)

Independent variables:

(none)

Linked functions (0):

(none)

but then let’s fix the sync and the IC components of the Crab nebula (cannot fit them with just one month of data) (these two methods are equivalent)

[7]:
model["Crab_IC.spectrum.main.Log_parabola.K"].fix = True
model.Crab_synch.spectrum.main.Log_parabola.K.fix = True

However, let’s free the index of the Crab Pulsar

[8]:
model.PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.index.free = True

model.display()
Model summary:

N
Point sources 172
Extended sources 0
Particle sources 0


Free parameters (4):

value min_value max_value unit
PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.K 0.0 0.0 0.0 keV-1 s-1 cm-2
PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.index -1.932218 -10.0 10.0
NVSS_J052622p224801.spectrum.main.Powerlaw.K 0.0 0.0 0.0 keV-1 s-1 cm-2
_4FGL_J0544d4p2238.spectrum.main.Powerlaw.K 0.0 0.0 0.0 keV-1 s-1 cm-2


Fixed parameters (958):
(abridged. Use complete=True to see all fixed parameters)


Properties (0):

(none)


Linked parameters (0):

(none)

Independent variables:

(none)

Linked functions (0):

(none)
[9]:
# Download data from Jan 01 2010 to February 1 2010

tstart = "2010-01-01 00:00:00"
tstop = "2010-02-01 00:00:00"

# Note that this will understand if you already download these files, and will
# not do it twice unless you change your selection or the outdir

evfile, scfile = download_LAT_data(
    ra,
    dec,
    20.0,
    tstart,
    tstop,
    time_type="Gregorian",
    destination_directory="Crab_data",
)

Configuration for Fermipy

3ML provides and intreface into Fermipy via the FermipyLike plugin. We can use it to generate basic configuration files.

Note

Currently, the FermipyLike plugin does not provide an interface to handle extended sources. This will change

[10]:
config = FermipyLike.get_basic_config(
    evfile=evfile,
    scfile=scfile,
    ra=ra,
    dec=dec,
    fermipy_verbosity=1,
    fermitools_chatter=0,
)

# See what we just got

config.display()
binning:
  binsperdec: 8
  binsz: 0.1
  roiwidth: 10.0
data:
  evfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L1d06e9b939821424b65e80e4641be3c8_FT1.fits
  scfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L220525162538E5ECAA3846_SC00.fits
logging:
  chatter: 0
  verbosity: 1
selection:
  dec: 22.013328116621064
  emax: 100000.0
  emin: 100.0
  evclass: 128
  evtype: 3
  filter: DATA_QUAL>0 && LAT_CONFIG==1
  ra: 83.63334095460648
  tmax: 286675202.0
  tmin: 283996802.0
  zmax: 100.0

binning:
  binsperdec: 8
  binsz: 0.1
  roiwidth: 10.0
data:
  evfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L1d06e9b939821424b65e80e4641be3c8_FT1.fits
  scfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L220525162538E5ECAA3846_SC00.fits
logging:
  chatter: 0
  verbosity: 1
selection:
  dec: 22.013328116621064
  emax: 100000.0
  emin: 100.0
  evclass: 128
  evtype: 3
  filter: DATA_QUAL>0 && LAT_CONFIG==1
  ra: 83.63334095460648
  tmax: 286675202.0
  tmin: 283996802.0
  zmax: 100.0

You can of course modify the configuration as a dictionary

[11]:
config["selection"]["emax"] = 300000.0

and even add sections

[12]:
config["gtlike"] = {"edisp": False}

config.display()
binning:
  binsperdec: 8
  binsz: 0.1
  roiwidth: 10.0
data:
  evfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L1d06e9b939821424b65e80e4641be3c8_FT1.fits
  scfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L220525162538E5ECAA3846_SC00.fits
gtlike:
  edisp: false
logging:
  chatter: 0
  verbosity: 1
selection:
  dec: 22.013328116621064
  emax: 300000.0
  emin: 100.0
  evclass: 128
  evtype: 3
  filter: DATA_QUAL>0 && LAT_CONFIG==1
  ra: 83.63334095460648
  tmax: 286675202.0
  tmin: 283996802.0
  zmax: 100.0

binning:
  binsperdec: 8
  binsz: 0.1
  roiwidth: 10.0
data:
  evfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L1d06e9b939821424b65e80e4641be3c8_FT1.fits
  scfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L220525162538E5ECAA3846_SC00.fits
gtlike:
  edisp: false
logging:
  chatter: 0
  verbosity: 1
selection:
  dec: 22.013328116621064
  emax: 300000.0
  emin: 100.0
  evclass: 128
  evtype: 3
  filter: DATA_QUAL>0 && LAT_CONFIG==1
  ra: 83.63334095460648
  tmax: 286675202.0
  tmin: 283996802.0
  zmax: 100.0

FermipyLike

Let’s create an instance of the plugin/ Note that here no processing is made, because fermipy still doesn’t know about the model you want to use.

[13]:
LAT = FermipyLike("LAT", config)

The plugin modifies the configuration as needed to get the output files in a unique place, which will stay the same as long as your selection does not change.

[14]:
config.display()
binning:
  binsperdec: 8
  binsz: 0.1
  roiwidth: 10.0
data:
  evfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L1d06e9b939821424b65e80e4641be3c8_FT1.fits
  scfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L220525162538E5ECAA3846_SC00.fits
fileio:
  outdir: __be5bc8387aba6146427a0bf98ddc46d1
gtlike:
  edisp: false
logging:
  chatter: 0
  verbosity: 1
selection:
  dec: 22.013328116621064
  emax: 300000.0
  emin: 100.0
  evclass: 128
  evtype: 3
  filter: DATA_QUAL>0 && LAT_CONFIG==1
  ra: 83.63334095460648
  tmax: 286675202.0
  tmin: 283996802.0
  zmax: 100.0

binning:
  binsperdec: 8
  binsz: 0.1
  roiwidth: 10.0
data:
  evfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L1d06e9b939821424b65e80e4641be3c8_FT1.fits
  scfile: /Users/runner/work/threeML/threeML/docs/md_docs/slow_execute/Crab_data/L220525162538E5ECAA3846_SC00.fits
fileio:
  outdir: __be5bc8387aba6146427a0bf98ddc46d1
gtlike:
  edisp: false
logging:
  chatter: 0
  verbosity: 1
selection:
  dec: 22.013328116621064
  emax: 300000.0
  emin: 100.0
  evclass: 128
  evtype: 3
  filter: DATA_QUAL>0 && LAT_CONFIG==1
  ra: 83.63334095460648
  tmax: 286675202.0
  tmin: 283996802.0
  zmax: 100.0

Here is where the fermipy processing happens (the .setup method)

[15]:
fermipy_output_directory = Path(config["fileio"]["outdir"])
print("Fermipy Output directoty: %s" % fermipy_output_directory)

# This remove the output directory, to start a fresh analysis...

if fermipy_output_directory.exists():
    shutil.rmtree(fermipy_output_directory)

# Here is where the fermipy processing happens (the .setup method)

data = DataList(LAT)

jl = JointLikelihood(model, data)
Fermipy Output directoty: __be5bc8387aba6146427a0bf98ddc46d1

Found Galactic template for IRF. P8R3_SOURCE_V3: /usr/local/miniconda/envs/test_env/share/fermitools/refdata/fermi/galdiffuse/gll_iem_v07.fits

Cutting the template around the ROI:


Found Isotropic template for irf P8R3_SOURCE_V3: /usr/local/miniconda/envs/test_env/share/fermitools/refdata/fermi/galdiffuse/iso_P8R3_SOURCE_V3_v1.txt
WARNING: Point source PKS_0459p060 lies 17.6413 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4C_p06d21 lies 17.7192 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _1ES_0647p250 lies 17.7261 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source NVSS_J065035p205556 lies 17.7265 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0551d7p0446 lies 17.7346 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0454p3724 lies 17.751 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PSR_J0631p1036 lies 17.7813 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0648p1749 lies 17.8402 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0537d6p0400 lies 18.0266 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0442d8p3609 lies 18.0663 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0424d8p3117 lies 18.0883 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0450d7p0715 lies 18.1415 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0552p39A lies 18.3309 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _1RXS_J064814d1p160708 lies 18.3607 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source MG2_J065230p1934 lies 18.3948 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PKS_0502p049 lies 18.4197 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source RX_J0648d7p1516 lies 18.8078 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0620p3806 lies 18.8402 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PSR_J0622p3749 lies 18.8564 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0641d4p3349 lies 18.8672 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0654p24 lies 18.868 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _1RXS_J065331d8p181448 lies 18.8952 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0412d6p2253c lies 18.932 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0653p2816 lies 18.95 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PMN_J0444p0717 lies 18.9984 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source MG1_J050533p0415 lies 19.0804 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0559d3p0352 lies 19.1138 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source TXS_0431p092 lies 19.1185 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B3_0509p406 lies 19.2344 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0635p35 lies 19.2995 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PKS_0550p032 lies 19.3013 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0431d0p3529c lies 19.3208 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _2MASX_J06533986p1636432 lies 19.4116 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0658d7p2318 lies 19.4488 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0555p0304 lies 19.5898 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0655p27A lies 19.695 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0642d4p1048 lies 19.7319 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source NVSS_J050650p032401 lies 19.7898 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0409d2p2542 lies 19.8196 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107

Performing the fit

[16]:
jl.set_minimizer("ROOT")

res = jl.fit()
Best fit values:

result unit
parameter
PSR_J0534p2200...K (1.07 +/- 0.04) x 10^-13 1 / (cm2 keV s)
PSR_J0534p2200...index -1.908 +/- 0.026
NVSS_J052622p224801.spectrum.main.Powerlaw.K (4.0 -2.8 +9) x 10^-17 1 / (cm2 keV s)
_4FGL_J0544d4p2238.spectrum.main.Powerlaw.K (6.3 -3.1 +6) x 10^-16 1 / (cm2 keV s)

Correlation matrix:

1.000.80-0.040.02
0.801.000.090.23
-0.040.091.000.03
0.020.230.031.00

Values of -log(likelihood) at the minimum:

-log(likelihood)
LAT 118124.597695
total 118124.597695

Values of statistical measures:

statistical measures
AIC 236257.195534
BIC 236299.365570

Now let’s compute the errors on the best fit parameters

[17]:
res = jl.get_errors()
result unit
parameter
PSR_J0534p2200...K (1.07 +/- 0.04) x 10^-13 1 / (cm2 keV s)
PSR_J0534p2200...index -1.908 +/- 0.026
NVSS_J052622p224801.spectrum.main.Powerlaw.K (4 -4 +5) x 10^-17 1 / (cm2 keV s)
_4FGL_J0544d4p2238.spectrum.main.Powerlaw.K (6 +/- 4) x 10^-16 1 / (cm2 keV s)

We might also want to look at the profile of the likelihood for each parameter.

[18]:
res = jl.get_contours(
    model.PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.index, -2.0, -1.6, 30
)
[19]:
res[-1]
[19]:
../_images/notebooks_Fermipy_LAT_36_0.png

Or we might want to produce a contour plot

[20]:
res = jl.get_contours(
    "PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.K",
    0.7e-13,
    1.3e-13,
    20,
    "PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.index",
    -2.0,
    -1.6,
    20,
)
[21]:
res[-1]
[21]:
../_images/notebooks_Fermipy_LAT_39_0.png

Pro-trick: We can also axcess the GTAnalysis object of fermipy:

[22]:
# res = jl.fit()
# LAT.gta.write_roi('test',make_plots=True)

All the plots are saved in the output directory as png files:

[23]:
# pngs=Path(f"{fermipy_output_directory}").glob("*png")
# for png in pngs:
#    print(png)
#    my_image=Image(str(png))
#    display(my_image)

We can also plot the resulting model:

[24]:
energies = sp.logspace(1, 8, 100) * u.MeV
fig, ax = plt.subplots()
# we only want to visualize the relevant sources...
src_to_plot = ["Crab", "PSR_J0534p2200"]
# Now loop over all point sources and plot them
for source_name, point_source in model.point_sources.items():
    for src in src_to_plot:
        if src in source_name:
            # Plot the sum of all components for this source

            ax.loglog(energies, point_source(energies), label=source_name)
            # If there is more than one component, plot them also separately

            if len(point_source.components) > 1:

                for component_name, component in point_source.components.items():
                    ax.loglog(
                        energies,
                        component.shape(energies),
                        "--",
                        label=f"{component_name} of {source_name}",
                    )

# Add a legend
ax.legend(loc=0, frameon=False)

ax.set_xlabel("Energy (MeV)")
ax.set_ylabel(r"Flux (ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$")
ax.set_ylim([1e-20, 1e-3])

# show the plot
fig
[24]:
../_images/notebooks_Fermipy_LAT_45_0.png

We can also do a bayesian analysis.

This will set priors based on the current defined min-max (log-uniform or uniform)

[25]:

for param in model.free_parameters.values(): if param.has_transformation(): param.set_uninformative_prior(Log_uniform_prior) else: param.set_uninformative_prior(Uniform_prior)
[26]:
# It's better to remove the output directory,...
shutil.rmtree(fermipy_output_directory)

bayes = BayesianAnalysis(model, data)

Found Galactic template for IRF. P8R3_SOURCE_V3: /usr/local/miniconda/envs/test_env/share/fermitools/refdata/fermi/galdiffuse/gll_iem_v07.fits

Cutting the template around the ROI:


Found Isotropic template for irf P8R3_SOURCE_V3: /usr/local/miniconda/envs/test_env/share/fermitools/refdata/fermi/galdiffuse/iso_P8R3_SOURCE_V3_v1.txt
WARNING: Point source PKS_0459p060 lies 17.6413 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4C_p06d21 lies 17.7192 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _1ES_0647p250 lies 17.7261 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source NVSS_J065035p205556 lies 17.7265 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0551d7p0446 lies 17.7346 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0454p3724 lies 17.751 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PSR_J0631p1036 lies 17.7813 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0648p1749 lies 17.8402 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0537d6p0400 lies 18.0266 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0442d8p3609 lies 18.0663 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0424d8p3117 lies 18.0883 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0450d7p0715 lies 18.1415 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0552p39A lies 18.3309 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _1RXS_J064814d1p160708 lies 18.3607 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source MG2_J065230p1934 lies 18.3948 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PKS_0502p049 lies 18.4197 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source RX_J0648d7p1516 lies 18.8078 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0620p3806 lies 18.8402 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PSR_J0622p3749 lies 18.8564 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0641d4p3349 lies 18.8672 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0654p24 lies 18.868 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _1RXS_J065331d8p181448 lies 18.8952 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0412d6p2253c lies 18.932 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0653p2816 lies 18.95 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PMN_J0444p0717 lies 18.9984 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source MG1_J050533p0415 lies 19.0804 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0559d3p0352 lies 19.1138 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source TXS_0431p092 lies 19.1185 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B3_0509p406 lies 19.2344 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0635p35 lies 19.2995 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source PKS_0550p032 lies 19.3013 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0431d0p3529c lies 19.3208 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _2MASX_J06533986p1636432 lies 19.4116 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0658d7p2318 lies 19.4488 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source GB6_J0555p0304 lies 19.5898 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source B2_0655p27A lies 19.695 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0642d4p1048 lies 19.7319 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source NVSS_J050650p032401 lies 19.7898 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
WARNING: Point source _4FGL_J0409d2p2542 lies 19.8196 degrees from the ROI center at RA, Dec = 83.6333, 22.0133 7.57107
[27]:
bayes.set_sampler("emcee")

n_walkers = 10
burn_in = 10
n_samples = 500

bayes.sampler.setup(n_iterations=n_samples, n_burn_in=burn_in, n_walkers=n_walkers)

res = bayes.sample()
Maximum a posteriori probability (MAP) point:

result unit
parameter
PSR_J0534p2200...K (1.07 +/- 0.04) x 10^-13 1 / (cm2 keV s)
PSR_J0534p2200...index -1.908 -0.023 +0.024
NVSS_J052622p224801.spectrum.main.Powerlaw.K (3.1 -3.1 +3.5) x 10^-17 1 / (cm2 keV s)
_4FGL_J0544d4p2238.spectrum.main.Powerlaw.K (5 -4 +5) x 10^-16 1 / (cm2 keV s)

Values of -log(posterior) at the minimum:

-log(posterior)
LAT -118044.97048
total -118044.97048

Values of statistical measures:

statistical measures
AIC 236097.941103
BIC 236140.111140
DIC 236101.430436
PDIC -0.682621

You can access to the parameter range like this (HPD):

[28]:
this_K = bayes.results.get_variates(
    "PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.K"
)
this_idx = bayes.results.get_variates(
    "PSR_J0534p2200.spectrum.main.Super_cutoff_powerlaw.index"
)

print("Highest_posterior_density_intervals :")
print(
    "K (68%%):     %10.2e,%10.2e" % this_K.highest_posterior_density_interval(cl=0.68)
)
print(
    "K (95%%):     %10.2e,%10.2e" % this_K.highest_posterior_density_interval(cl=0.95)
)
print(
    "Index (68%%): %10.2e,%10.2e" % this_idx.highest_posterior_density_interval(cl=0.68)
)
print(
    "Index (95%%): %10.2e,%10.2e" % this_idx.highest_posterior_density_interval(cl=0.95)
)
Highest_posterior_density_intervals :
K (68%):       1.02e-13,  1.10e-13
K (95%):       9.96e-14,  1.15e-13
Index (68%):  -1.93e+00, -1.89e+00
Index (95%):  -1.95e+00, -1.85e+00
[29]:
corner_figure = bayes.results.corner_plot()
corner_figure
[29]:
../_images/notebooks_Fermipy_LAT_52_0.png
[ ]: