Analyzing GRB 080916C with Fermi-GBM

Alt text (NASA/Swift/Cruz deWilde)

To demonstrate the capabilities and features of 3ML in, we will go through a time-integrated and time-resolved analysis. This example serves as a standard way to analyze Fermi-GBM data with 3ML as well as a template for how you can design your instrument’s analysis pipeline with 3ML if you have similar data.

3ML provides utilities to reduce time series data to plugins in a correct and statistically justified way (e.g., background fitting of Poisson data is done with a Poisson likelihood). The approach is generic and can be extended. For more details, see the time series documentation.

[1]:

import warnings

warnings.simplefilter("ignore")

[2]:

%%capture
import matplotlib.pyplot as plt
import numpy as np

np.seterr(all="ignore")


from threeML import *
from threeML.io.package_data import get_path_of_data_file

[3]:

silence_warnings()
%matplotlib inline
from jupyterthemes import jtplot

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

Examining the catalog

As with Swift and Fermi-LAT, 3ML provides a simple interface to the on-line Fermi-GBM catalog. Let’s get the information for GRB 080916C.

[4]:

gbm_catalog = FermiGBMBurstCatalog()
gbm_catalog.query_sources("GRB080916009")

00:49:36 INFO      The cache for fermigbrst does not yet exist. We will try to    get_heasarc_table_as_pandas.py:63
                  build it

         INFO      Building cache for fermigbrst                                 get_heasarc_table_as_pandas.py:103

[4]:

Table length=1

name	ra	dec	trigger_time	t90
object	float64	float64	float64	float64
GRB080916009	119.800	-56.600	54725.0088613	62.977

To aid in quickly replicating the catalog analysis, and thanks to the tireless efforts of the Fermi-GBM team, we have added the ability to extract the analysis parameters from the catalog as well as build an astromodels model with the best fit parameters baked in. Using this information, one can quickly run through the catalog an replicate the entire analysis with a script. Let’s give it a try.

[5]:

grb_info = gbm_catalog.get_detector_information()["GRB080916009"]

gbm_detectors = grb_info["detectors"]
source_interval = grb_info["source"]["fluence"]
background_interval = grb_info["background"]["full"]
best_fit_model = grb_info["best fit model"]["fluence"]
model = gbm_catalog.get_model(best_fit_model, "fluence")["GRB080916009"]

[6]:

model

[6]:

Model summary:

	N
Point sources	1
Extended sources	0
Particle sources	0

Free parameters (5):

	value	min_value	max_value	unit
GRB080916009.spectrum.main.SmoothlyBrokenPowerLaw.K	0.012255	0.0	None	keV-1 s-1 cm-2
GRB080916009.spectrum.main.SmoothlyBrokenPowerLaw.alpha	-1.130424	-1.5	2.0
GRB080916009.spectrum.main.SmoothlyBrokenPowerLaw.break_energy	309.2031	10.0	None	keV
GRB080916009.spectrum.main.SmoothlyBrokenPowerLaw.break_scale	0.3	0.0	10.0
GRB080916009.spectrum.main.SmoothlyBrokenPowerLaw.beta	-2.096931	-5.0	-1.6

Fixed parameters (3):
(abridged. Use complete=True to see all fixed parameters)

Properties (0):

(none)

Linked parameters (0):

(none)

Independent variables:

(none)

Linked functions (0):

(none)

Downloading the data

We provide a simple interface to download the Fermi-GBM data. Using the information from the catalog that we have extracted, we can download just the data from the detectors that were used for the catalog analysis. This will download the CSPEC, TTE and instrument response files from the on-line database.

[7]:

dload = download_GBM_trigger_data("bn080916009", detectors=gbm_detectors)

Let’s first examine the catalog fluence fit. Using the TimeSeriesBuilder, we can fit the background, set the source interval, and create a 3ML plugin for the analysis. We will loop through the detectors, set their appropriate channel selections, and ensure there are enough counts in each bin to make the PGStat profile likelihood valid.

First we use the CSPEC data to fit the background using the background selections. We use CSPEC because it has a longer duration for fitting the background.
The background is saved to an HDF5 file that stores the polynomial coefficients and selections which we can read in to the TTE file later.
The light curve is plotted.
The source selection from the catalog is set and DispersionSpectrumLike plugin is created.
The plugin has the standard GBM channel selections for spectral analysis set.

[8]:

fluence_plugins = []
time_series = {}
for det in gbm_detectors:

    ts_cspec = TimeSeriesBuilder.from_gbm_cspec_or_ctime(
        det, cspec_or_ctime_file=dload[det]["cspec"], rsp_file=dload[det]["rsp"]
    )

    ts_cspec.set_background_interval(*background_interval.split(","))
    ts_cspec.save_background(f"{det}_bkg.h5", overwrite=True)

    ts_tte = TimeSeriesBuilder.from_gbm_tte(
        det,
        tte_file=dload[det]["tte"],
        rsp_file=dload[det]["rsp"],
        restore_background=f"{det}_bkg.h5",
    )

    time_series[det] = ts_tte

    ts_tte.set_active_time_interval(source_interval)

    ts_tte.view_lightcurve(-40, 100)

    fluence_plugin = ts_tte.to_spectrumlike()

    if det.startswith("b"):

        fluence_plugin.set_active_measurements("250-30000")

    else:

        fluence_plugin.set_active_measurements("9-900")

    fluence_plugin.rebin_on_background(1.0)

    fluence_plugins.append(fluence_plugin)

00:50:19 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:356

00:50:24 INFO      None 0-order polynomial fit with the mle method                               time_series.py:426

         INFO      Saved fitted background to n3_bkg.h5                                          time_series.py:972

         INFO      Saved background to n3_bkg.h5                                         time_series_builder.py:430

00:50:25 INFO      Successfully restored fit from n3_bkg.h5                              time_series_builder.py:166

         INFO      Interval set to 1.28-64.257 for n3                                    time_series_builder.py:274

         INFO      Auto-probed noise models:                                                    SpectrumLike.py:486

         INFO      - observation: poisson                                                       SpectrumLike.py:487

         INFO      - background: gaussian                                                       SpectrumLike.py:488

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

00:50:29 INFO      Now using 120 bins                                                          SpectrumLike.py:1706

00:50:30 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:356

00:50:36 INFO      None 1-order polynomial fit with the mle method                               time_series.py:426

         INFO      Saved fitted background to n4_bkg.h5                                          time_series.py:972

         INFO      Saved background to n4_bkg.h5                                         time_series_builder.py:430

         INFO      Successfully restored fit from n4_bkg.h5                              time_series_builder.py:166

         INFO      Interval set to 1.28-64.257 for n4                                    time_series_builder.py:274

         INFO      Auto-probed noise models:                                                    SpectrumLike.py:486

         INFO      - observation: poisson                                                       SpectrumLike.py:487

         INFO      - background: gaussian                                                       SpectrumLike.py:488

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

00:50:37 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:356

00:50:43 INFO      None 1-order polynomial fit with the mle method                               time_series.py:426

         INFO      Saved fitted background to b0_bkg.h5                                          time_series.py:972

         INFO      Saved background to b0_bkg.h5                                         time_series_builder.py:430

         INFO      Successfully restored fit from b0_bkg.h5                              time_series_builder.py:166

         INFO      Interval set to 1.28-64.257 for b0                                    time_series_builder.py:274

00:50:44 INFO      Auto-probed noise models:                                                    SpectrumLike.py:486

         INFO      - observation: poisson                                                       SpectrumLike.py:487

         INFO      - background: gaussian                                                       SpectrumLike.py:488

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

../_images/notebooks_grb080916C_12_42.png

../_images/notebooks_grb080916C_12_43.png

../_images/notebooks_grb080916C_12_44.png

Setting up the fit

Let’s see if we can reproduce the results from the catalog.

Set priors for the model

We will fit the spectrum using Bayesian analysis, so we must set priors on the model parameters.

[9]:

model.GRB080916009.spectrum.main.shape.alpha.prior = Truncated_gaussian(
    lower_bound=-1.5, upper_bound=1, mu=-1, sigma=0.5
)
model.GRB080916009.spectrum.main.shape.beta.prior = Truncated_gaussian(
    lower_bound=-5, upper_bound=-1.6, mu=-2.25, sigma=0.5
)
model.GRB080916009.spectrum.main.shape.break_energy.prior = Log_normal(mu=2, sigma=1)
model.GRB080916009.spectrum.main.shape.break_energy.bounds = (None, None)
model.GRB080916009.spectrum.main.shape.K.prior = Log_uniform_prior(
    lower_bound=1e-3, upper_bound=1e1
)
model.GRB080916009.spectrum.main.shape.break_scale.prior = Log_uniform_prior(
    lower_bound=1e-4, upper_bound=10
)

Clone the model and setup the Bayesian analysis class

Next, we clone the model we built from the catalog so that we can look at the results later and fit the cloned model. We pass this model and the DataList of the plugins to a BayesianAnalysis class and set the sampler to MultiNest.

[10]:

new_model = clone_model(model)

bayes = BayesianAnalysis(new_model, DataList(*fluence_plugins))

# share spectrum gives a linear speed up when
# spectrumlike plugins have the same RSP input energies
bayes.set_sampler("multinest", share_spectrum=True)

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

Examine at the catalog fitted model

We can quickly examine how well the catalog fit matches the data. There appears to be a discrepancy between the data and the model! Let’s refit to see if we can fix it.

[11]:

fig = display_spectrum_model_counts(bayes, min_rate=20, step=False)

../_images/notebooks_grb080916C_18_0.png

Run the sampler

We let MultiNest condition the model on the data

[12]:

bayes.sampler.setup(n_live_points=400)
bayes.sample()

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  400
 dimensionality =    5
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -3101.0933856490624      +/-  0.22327937858070207
 Total Likelihood Evaluations:        26476
 Sampling finished. Exiting MultiNest

00:50:57 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
GRB080916009...K	(1.4503 -0.0004 +0.04) x 10^-2	1 / (keV s cm2)
GRB080916009...alpha	-1.101 +0.008 +0.05
GRB080916009...break_energy	(1.91 +0.10 +0.7) x 10^2	keV
GRB080916009...break_scale	(0.0 +1.6 +3.5) x 10^-1
GRB080916009...beta	-1.96 -0.25 -0.05

Values of -log(posterior) at the minimum:

	-log(posterior)
n3	-1018.844389
n4	-1011.337064
b0	-1050.185200
total	-3080.366653

Values of statistical measures:

	statistical measures
AIC	6170.903760
BIC	6190.135971
DIC	6180.233910
PDIC	4.466351
log(Z)	-1346.787745

Now our model seems to match much better with the data!

[13]:

bayes.restore_median_fit()
fig = display_spectrum_model_counts(bayes, min_rate=20)

         INFO      fit restored to median of posterior                                          sampler_base.py:157

../_images/notebooks_grb080916C_22_1.png

But how different are we from the catalog model? Let’s plot our fit along with the catalog model. Luckily, 3ML can handle all the units for is

[14]:

conversion = u.Unit("keV2/(cm2 s keV)").to("erg2/(cm2 s keV)")
energy_grid = np.logspace(1, 4, 100) * u.keV
vFv = (energy_grid**2 * model.get_point_source_fluxes(0, energy_grid)).to(
    "erg2/(cm2 s keV)"
)

[15]:

fig = plot_spectra(bayes.results, flux_unit="erg2/(cm2 s keV)")
ax = fig.get_axes()[0]
_ = ax.loglog(energy_grid, vFv, color="blue", label="catalog model")

../_images/notebooks_grb080916C_25_2.png

Time Resolved Analysis

Now that we have examined fluence fit, we can move to performing a time-resolved analysis.

Selecting a temporal binning

We first get the brightest NaI detector and create time bins via the Bayesian blocks algorithm. We can use the fitted background to make sure that our intervals are chosen in an unbiased way.

[16]:

n3 = time_series["n3"]

[17]:

n3.create_time_bins(0, 60, method="bayesblocks", use_background=True, p0=0.2)

00:51:51 INFO      Created 15 bins via bayesblocks                                       time_series_builder.py:632

Sometimes, glitches in the GBM data cause spikes in the data that the Bayesian blocks algorithm detects as fast changes in the count rate. We will have to remove those intervals manually.

Note: In the future, 3ML will provide an automated method to remove these unwanted spikes.

[18]:

fig = n3.view_lightcurve(use_binner=True)

../_images/notebooks_grb080916C_30_0.png

[19]:

bad_bins = []
for i, w in enumerate(n3.bins.widths):

    if w < 5e-2:
        bad_bins.append(i)

edges = [n3.bins.starts[0]]

for i, b in enumerate(n3.bins):

    if i not in bad_bins:
        edges.append(b.stop)

starts = edges[:-1]
stops = edges[1:]

n3.create_time_bins(starts, stops, method="custom")

         INFO      Created 12 bins via custom                                            time_series_builder.py:632

Now our light curve looks much more acceptable.

[20]:

fig = n3.view_lightcurve(use_binner=True)

../_images/notebooks_grb080916C_33_0.png

The time series objects can read time bins from each other, so we will map these time bins onto the other detectors’ time series and create a list of time plugins for each detector and each time bin created above.

[21]:

time_resolved_plugins = {}

for k, v in time_series.items():
    v.read_bins(n3)
    time_resolved_plugins[k] = v.to_spectrumlike(from_bins=True)

         INFO      Created 12 bins via custom                                            time_series_builder.py:632

00:51:52 INFO      Interval set to 1.28-64.257 for n3                                    time_series_builder.py:274

         INFO      Created 12 bins via custom                                            time_series_builder.py:632

         INFO      Interval set to 1.28-64.257 for n4                                    time_series_builder.py:274

         INFO      Created 12 bins via custom                                            time_series_builder.py:632

00:51:53 INFO      Interval set to 1.28-64.257 for b0                                    time_series_builder.py:274

Setting up the model

For the time-resolved analysis, we will fit the classic Band function to the data. We will set some principled priors.

[22]:

band = Band()
band.alpha.prior = Truncated_gaussian(lower_bound=-1.5, upper_bound=1, mu=-1, sigma=0.5)
band.beta.prior = Truncated_gaussian(lower_bound=-5, upper_bound=-1.6, mu=-2, sigma=0.5)
band.xp.prior = Log_normal(mu=2, sigma=1)
band.xp.bounds = (None, None)
band.K.prior = Log_uniform_prior(lower_bound=1e-10, upper_bound=1e3)
ps = PointSource("grb", 0, 0, spectral_shape=band)
band_model = Model(ps)

Perform the fits

One way to perform Bayesian spectral fits to all the intervals is to loop through each one. There can are many ways to do this, so find an analysis pattern that works for you.

[23]:

models = []
results = []
analysis = []
for interval in range(12):

    # clone the model above so that we have a separate model
    # for each fit

    this_model = clone_model(band_model)

    # for each detector set up the plugin
    # for this time interval

    this_data_list = []
    for k, v in time_resolved_plugins.items():

        pi = v[interval]

        if k.startswith("b"):
            pi.set_active_measurements("250-30000")
        else:
            pi.set_active_measurements("9-900")

        pi.rebin_on_background(1.0)

        this_data_list.append(pi)

    # create a data list

    dlist = DataList(*this_data_list)

    # set up the sampler and fit

    bayes = BayesianAnalysis(this_model, dlist)

    # get some speed with share spectrum
    bayes.set_sampler("multinest", share_spectrum=True)
    bayes.sampler.setup(n_live_points=500)
    bayes.sample()

    # at this stage we coudl also
    # save the analysis result to
    # disk but we will simply hold
    # onto them in memory

    analysis.append(bayes)

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 107 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt
 ln(ev)=  -789.13455524420340      +/-  0.18070791032123290
 Total Likelihood Evaluations:        15762
 Sampling finished. Exiting MultiNest

00:52:02 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(3.6 +/- 0.5) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-5.4 -1.1 +1.0) x 10^-1
grb.spectrum.main.Band.xp	(3.02 -0.28 +0.7) x 10^2	keV
grb.spectrum.main.Band.beta	-2.06 -0.29 +0.08

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval0	-249.970667
n4_interval0	-267.971373
b0_interval0	-285.653659
total	-803.595700

Values of statistical measures:

	statistical measures
AIC	1615.304714
BIC	1630.713531
DIC	1570.371598
PDIC	2.630311
log(Z)	-342.716783

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt
 ln(ev)=  -1943.3721219159743      +/-  0.21340577020317913
 Total Likelihood Evaluations:        23864
 Sampling finished. Exiting MultiNest

00:52:14 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(4.14 -0.13 +0.11) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-8.53 -0.28 +0.21) x 10^-1
grb.spectrum.main.Band.xp	(6.2 -0.4 +0.6) x 10^2	keV
grb.spectrum.main.Band.beta	-2.15 -0.09 +0.07

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval1	-641.746039
n4_interval1	-645.317987
b0_interval1	-674.026681
total	-1961.090707

Values of statistical measures:

	statistical measures
AIC	3930.294729
BIC	3945.703546
DIC	3872.984663
PDIC	3.503634
log(Z)	-843.995789

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 115 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -905.70294620606569      +/-  0.19039934034954512
 Total Likelihood Evaluations:        21312
 Sampling finished. Exiting MultiNest

00:52:26 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.57 -0.22 +0.21) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.03 -0.08 +0.05
grb.spectrum.main.Band.xp	(5.6 -1.1 +2.4) x 10^2	keV
grb.spectrum.main.Band.beta	-1.87 -0.31 +0.09

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval2	-288.813177
n4_interval2	-312.443753
b0_interval2	-324.302068
total	-925.558998

Values of statistical measures:

	statistical measures
AIC	1859.231311
BIC	1874.640129
DIC	1804.904202
PDIC	2.665517
log(Z)	-393.341792

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 109 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -794.95346221209729      +/-  0.17827155957688359
 Total Likelihood Evaluations:        16199
 Sampling finished. Exiting MultiNest

00:52:37 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.8 +1.2 +2.6) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.03 +0.22 +0.5
grb.spectrum.main.Band.xp	(3.1 -1.5 -1.0) x 10^2	keV
grb.spectrum.main.Band.beta	-1.94 +0.10 +0.22

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval3	-243.648163
n4_interval3	-261.928504
b0_interval3	-299.037943
total	-804.614611

Values of statistical measures:

	statistical measures
AIC	1617.342536
BIC	1632.751353
DIC	1577.600137
PDIC	2.276789
log(Z)	-345.243902

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -2270.6478183685745      +/-  0.19756741094050750
 Total Likelihood Evaluations:        20433
 Sampling finished. Exiting MultiNest

00:52:48 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.06 -0.12 +0.10) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.80 -0.4 +0.32) x 10^-1
grb.spectrum.main.Band.xp	(3.96 -0.34 +0.7) x 10^2	keV
grb.spectrum.main.Band.beta	-1.96 -0.15 +0.05

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval4	-757.072768
n4_interval4	-746.725000
b0_interval4	-778.465011
total	-2282.262779

Values of statistical measures:

	statistical measures
AIC	4572.638873
BIC	4588.047691
DIC	4528.293709
PDIC	3.610763
log(Z)	-986.129818

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -1575.0822179164393      +/-  0.19737815915919171
 Total Likelihood Evaluations:        20571
 Sampling finished. Exiting MultiNest

00:53:00 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.83 -0.09 +0.20) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.15 -0.35 +0.5) x 10^-1
grb.spectrum.main.Band.xp	(4.00 -0.4 +0.12) x 10^2	keV
grb.spectrum.main.Band.beta	-2.03 +0.04 +0.13

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval5	-523.013937
n4_interval5	-528.172114
b0_interval5	-536.641091
total	-1587.827142

Values of statistical measures:

	statistical measures
AIC	3183.767598
BIC	3199.176415
DIC	3135.257681
PDIC	1.726562
log(Z)	-684.049516

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -1757.9621740617522      +/-  0.19772435429826762
 Total Likelihood Evaluations:        19031
 Sampling finished. Exiting MultiNest

00:53:10 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.01 -0.07 +0.21) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-10.0 -0.26 +0.7) x 10^-1
grb.spectrum.main.Band.xp	(4.09 -0.7 +0.31) x 10^2	keV
grb.spectrum.main.Band.beta	-2.107 +0.007 +0.11

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval6	-583.993411
n4_interval6	-576.271683
b0_interval6	-609.564291
total	-1769.829385

Values of statistical measures:

	statistical measures
AIC	3547.772085
BIC	3563.180902
DIC	3501.190979
PDIC	2.571383
log(Z)	-763.473272

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -1939.1757014826214      +/-  0.19071254397103671
 Total Likelihood Evaluations:        22031
 Sampling finished. Exiting MultiNest

00:53:20 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

00:53:21 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(1.67 -0.11 +0.10) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.05 +/- 0.05
grb.spectrum.main.Band.xp	(4.3 -0.5 +0.8) x 10^2	keV
grb.spectrum.main.Band.beta	-2.21 -0.5 +0.09

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval7	-640.899824
n4_interval7	-650.209300
b0_interval7	-662.364764
total	-1953.473887

Values of statistical measures:

	statistical measures
AIC	3915.061089
BIC	3930.469907
DIC	3868.883898
PDIC	3.358113
log(Z)	-842.173307

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt
 ln(ev)=  -2056.7935521304630      +/-  0.19788057276274881
 Total Likelihood Evaluations:        18340
 Sampling finished. Exiting MultiNest

00:53:32 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(1.59 -0.09 +0.17) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-8.3 -0.6 +0.8) x 10^-1
grb.spectrum.main.Band.xp	(3.5 -0.5 +0.4) x 10^2	keV
grb.spectrum.main.Band.beta	-2.26 -0.4 +0.07

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval8	-698.624604
n4_interval8	-665.823339
b0_interval8	-701.858200
total	-2066.306143

Values of statistical measures:

	statistical measures
AIC	4140.725600
BIC	4156.134418
DIC	4098.588302
PDIC	3.113037
log(Z)	-893.254090

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt
 ln(ev)=  -1878.9272283227331      +/-  0.14588581632491374
 Total Likelihood Evaluations:        12764
 Sampling finished. Exiting MultiNest

00:53:38 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(1.02 -0.17 +1.7) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.0 -0.9 +4) x 10^-1
grb.spectrum.main.Band.xp	(1.18 -0.5 +0.34) x 10^2	keV
grb.spectrum.main.Band.beta	-1.85 -0.4 +0.13

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval9	-616.993126
n4_interval9	-616.281454
b0_interval9	-648.496684
total	-1881.771263

Values of statistical measures:

	statistical measures
AIC	3771.655841
BIC	3787.064659
DIC	3708.610153
PDIC	-38.559908
log(Z)	-816.007727

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt
 ln(ev)=  -1322.5334812878282      +/-  0.16889554438679191
 Total Likelihood Evaluations:        15218
 Sampling finished. Exiting MultiNest

00:53:47 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.2 -0.4 +0.6) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-7.1 -1.1 +1.5) x 10^-1
grb.spectrum.main.Band.xp	(2.1 -0.4 +0.5) x 10^2	keV
grb.spectrum.main.Band.beta	-1.93 -0.30 +0.08

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval10	-437.745002
n4_interval10	-432.972402
b0_interval10	-460.773845
total	-1331.491249

Values of statistical measures:

	statistical measures
AIC	2671.095813
BIC	2686.504630
DIC	2633.874034
PDIC	0.462129
log(Z)	-574.368993

         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1237

         INFO      Now using 120 bins                                                          SpectrumLike.py:1706

         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1237

         INFO      Now using 119 bins                                                          SpectrumLike.py:1706

         INFO      sampler set to multinest                                                bayesian_analysis.py:186

 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -812.42944564751065      +/-  0.15001637397101170
 Total Likelihood Evaluations:        11847
 Sampling finished. Exiting MultiNest

00:53:54 INFO      fit restored to maximum of posterior                                         sampler_base.py:168

         INFO      fit restored to maximum of posterior                                         sampler_base.py:168

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.7 -0.6 +2.4) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-5.1 -1.8 +3.2) x 10^-1
grb.spectrum.main.Band.xp	(1.31 -0.32 +0.23) x 10^2	keV
grb.spectrum.main.Band.beta	-2.07 -0.4 +0.15

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval11	-272.349204
n4_interval11	-255.871867
b0_interval11	-292.318748
total	-820.539818

Values of statistical measures:

	statistical measures
AIC	1649.192950
BIC	1664.601767
DIC	1616.519926
PDIC	-0.462507
log(Z)	-352.833625

Examine the fits

Now we can look at the fits in count space to make sure they are ok.

[24]:

for a in analysis:
    a.restore_median_fit()
    _ = display_spectrum_model_counts(a, min_rate=[20, 20, 20], step=False)

         INFO      fit restored to median of posterior                                          sampler_base.py:157

00:53:55 INFO      fit restored to median of posterior                                          sampler_base.py:157

         INFO      fit restored to median of posterior                                          sampler_base.py:157

         INFO      fit restored to median of posterior                                          sampler_base.py:157

00:53:56 INFO      fit restored to median of posterior                                          sampler_base.py:157

         INFO      fit restored to median of posterior                                          sampler_base.py:157

00:53:57 INFO      fit restored to median of posterior                                          sampler_base.py:157

         INFO      fit restored to median of posterior                                          sampler_base.py:157

         INFO      fit restored to median of posterior                                          sampler_base.py:157

00:53:58 INFO      fit restored to median of posterior                                          sampler_base.py:157

         INFO      fit restored to median of posterior                                          sampler_base.py:157

         INFO      fit restored to median of posterior                                          sampler_base.py:157

../_images/notebooks_grb080916C_41_12.png

../_images/notebooks_grb080916C_41_13.png

../_images/notebooks_grb080916C_41_14.png

../_images/notebooks_grb080916C_41_15.png

../_images/notebooks_grb080916C_41_16.png

../_images/notebooks_grb080916C_41_17.png

../_images/notebooks_grb080916C_41_18.png

../_images/notebooks_grb080916C_41_19.png

../_images/notebooks_grb080916C_41_20.png

../_images/notebooks_grb080916C_41_21.png

../_images/notebooks_grb080916C_41_22.png

../_images/notebooks_grb080916C_41_23.png

Finally, we can plot the models together to see how the spectra evolve with time.

[25]:

fig = plot_spectra(
    *[a.results for a in analysis[::1]],
    flux_unit="erg2/(cm2 s keV)",
    fit_cmap="viridis",
    contour_cmap="viridis",
    contour_style_kwargs=dict(alpha=0.1),
)

../_images/notebooks_grb080916C_43_13.png

This example can serve as a template for performing analysis on GBM data. However, as 3ML provides an abstract interface and modular building blocks, similar analysis pipelines can be built for any time series data.