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")

22:49:58 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)

22:50:45 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:356

22:50:50 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

         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

22:50:54 INFO      Now using 120 bins                                                          SpectrumLike.py:1706

22:50:55 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:356

22:51:01 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

22:51:02 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:356

22:51:07 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

22:51:08 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

         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)=  -3100.8309923711349      +/-  0.22190045986099322
 Total Likelihood Evaluations:        28224
 Sampling finished. Exiting MultiNest

22:51:19 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	(1.459 -0.009 +0.028) x 10^-2	1 / (keV s cm2)
GRB080916009...alpha	-1.101 +0.009 +0.05
GRB080916009...break_energy	(1.95 +0.08 +0.7) x 10^2	keV
GRB080916009...break_scale	(0.0 +1.8 +4) x 10^-1
GRB080916009...beta	-1.96 -0.26 -0.05

Values of -log(posterior) at the minimum:

	-log(posterior)
n3	-1018.996191
n4	-1010.939096
b0	-1051.216238
total	-3081.151526

Values of statistical measures:

	statistical measures
AIC	6172.473506
BIC	6191.705717
DIC	6180.437152
PDIC	4.485660
log(Z)	-1346.673789

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)

22:52:22 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

22:52:23 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

22:52:24 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)=  -788.75865615801922      +/-  0.17801604765842316
 Total Likelihood Evaluations:        16097
 Sampling finished. Exiting MultiNest

22:52:31 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 +/- 0.7) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-5.5 -1.7 +1.2) x 10^-1
grb.spectrum.main.Band.xp	(3.1 -0.5 +1.0) x 10^2	keV
grb.spectrum.main.Band.beta	-2.00 -0.33 +0.09

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval0	-250.182966
n4_interval0	-267.919443
b0_interval0	-285.692405
total	-803.794814

Values of statistical measures:

	statistical measures
AIC	1615.702942
BIC	1631.111760
DIC	1570.084672
PDIC	1.957711
log(Z)	-342.553532

         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)=  -1964.7692795441142      +/-  0.23027764727715330
 Total Likelihood Evaluations:        22639
 Sampling finished. Exiting MultiNest

22:52:39 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	(4.378 -0.021 +0.10) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-8.56 -0.04 +0.11) x 10^-1
grb.spectrum.main.Band.xp	(5.052 -0.34 +0.020) x 10^2	keV
grb.spectrum.main.Band.beta	-1.8364 -0.0013 +0.017

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval1	-645.323478
n4_interval1	-644.605440
b0_interval1	-685.576682
total	-1975.505600

Values of statistical measures:

	statistical measures
AIC	3959.124514
BIC	3974.533331
DIC	3903.934422
PDIC	1.476412
log(Z)	-853.288456

         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)=  -906.48818485555364      +/-  0.19570082692025059
 Total Likelihood Evaluations:        20155
 Sampling finished. Exiting MultiNest

22:52:47 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	(2.54 -0.19 +0.17) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.05 -0.06 +0.05
grb.spectrum.main.Band.xp	(5.9 -1.1 +1.8) x 10^2	keV
grb.spectrum.main.Band.beta	-1.90 -0.14 +0.11

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval2	-289.080367
n4_interval2	-312.453827
b0_interval2	-324.454230
total	-925.988424

Values of statistical measures:

	statistical measures
AIC	1860.090162
BIC	1875.498979
DIC	1803.802492
PDIC	2.280009
log(Z)	-393.682817

         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)=  -789.15406999997276      +/-  0.17844576394926071
 Total Likelihood Evaluations:        17832
 Sampling finished. Exiting MultiNest

22:52:54 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	(2.86 -0.21 +0.5) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.3 -0.7 +1.0) x 10^-1
grb.spectrum.main.Band.xp	(3.4 -0.8 +0.4) x 10^2	keV
grb.spectrum.main.Band.beta	-2.209 -0.028 +0.27

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval3	-242.517672
n4_interval3	-262.523454
b0_interval3	-298.454500
total	-803.495626

Values of statistical measures:

	statistical measures
AIC	1615.104566
BIC	1630.513384
DIC	1570.340736
PDIC	2.858816
log(Z)	-342.725258

         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)=  -2272.8068233084396      +/-  0.20356421532686364
 Total Likelihood Evaluations:        20600
 Sampling finished. Exiting MultiNest

22:53:02 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	(2.07 -0.13 +0.06) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.74 -0.5 +0.22) x 10^-1
grb.spectrum.main.Band.xp	(4.04 -0.20 +0.7) x 10^2	keV
grb.spectrum.main.Band.beta	-2.073 -0.20 -0.025

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval4	-757.795432
n4_interval4	-746.396849
b0_interval4	-779.129277
total	-2283.321558

Values of statistical measures:

	statistical measures
AIC	4574.756430
BIC	4590.165247
DIC	4529.429624
PDIC	2.394159
log(Z)	-987.067462

         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)=  -1572.9476421423926      +/-  0.19061159817923440
 Total Likelihood Evaluations:        20592
 Sampling finished. Exiting MultiNest

22:53:09 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	(2.81 -0.23 +0.15) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.1 -0.6 +0.4) x 10^-1
grb.spectrum.main.Band.xp	(4.2 -0.4 +0.8) x 10^2	keV
grb.spectrum.main.Band.beta	-2.15 -0.28 +0.11

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval5	-523.598617
n4_interval5	-527.659470
b0_interval5	-536.838951
total	-1588.097037

Values of statistical measures:

	statistical measures
AIC	3184.307389
BIC	3199.716206
DIC	3137.064867
PDIC	3.511967
log(Z)	-683.122481

         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)=  -1756.6637015507461      +/-  0.19487469000716975
 Total Likelihood Evaluations:        20446
 Sampling finished. Exiting MultiNest

22:53:17 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	(1.97 -0.05 +0.19) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.006 -0.023 +0.06
grb.spectrum.main.Band.xp	(4.33 -0.8 +0.23) x 10^2	keV
grb.spectrum.main.Band.beta	-2.243 -0.029 +0.20

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval6	-584.146763
n4_interval6	-576.571670
b0_interval6	-609.435955
total	-1770.154388

Values of statistical measures:

	statistical measures
AIC	3548.422091
BIC	3563.830908
DIC	3500.482623
PDIC	2.806300
log(Z)	-762.909352

         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)=  -1947.2046661354143      +/-  0.21294112090746764
 Total Likelihood Evaluations:        19409
 Sampling finished. Exiting MultiNest

22:53:25 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	(1.65 -0.05 +0.13) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.077 -0.010 +0.04
grb.spectrum.main.Band.xp	(3.99 -0.5 +0.16) x 10^2	keV
grb.spectrum.main.Band.beta	-1.965 -0.029 +0.09

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval7	-641.432423
n4_interval7	-649.899341
b0_interval7	-663.639986
total	-1954.971750

Values of statistical measures:

	statistical measures
AIC	3918.056815
BIC	3933.465633
DIC	3874.803264
PDIC	4.340113
log(Z)	-845.660242

         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)=  -2054.1748477933575      +/-  0.18970018581072504
 Total Likelihood Evaluations:        18516
 Sampling finished. Exiting MultiNest

22:53:32 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	(1.51 -0.10 +0.14) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-8.6 -0.5 +0.7) x 10^-1
grb.spectrum.main.Band.xp	(3.9 -0.5 +0.6) x 10^2	keV
grb.spectrum.main.Band.beta	-2.33 -0.33 +0.16

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval8	-698.394736
n4_interval8	-666.454936
b0_interval8	-702.332694
total	-2067.182366

Values of statistical measures:

	statistical measures
AIC	4142.478047
BIC	4157.886864
DIC	4097.469952
PDIC	2.979969
log(Z)	-892.116801

         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)=  -1878.4861074875325      +/-  0.14311557580351283
 Total Likelihood Evaluations:        13367
 Sampling finished. Exiting MultiNest

22:53:36 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	(1.01 -0.22 +1.5) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.0 -1.5 +4) x 10^-1
grb.spectrum.main.Band.xp	(1.24 -0.6 +0.33) x 10^2	keV
grb.spectrum.main.Band.beta	-1.88 -0.29 +0.17

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval9	-617.155794
n4_interval9	-616.323117
b0_interval9	-648.628975
total	-1882.107886

Values of statistical measures:

	statistical measures
AIC	3772.329086
BIC	3787.737904
DIC	3674.743198
PDIC	-72.343847
log(Z)	-815.816151

         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)=  -1322.6338839046537      +/-  0.16866567840250193
 Total Likelihood Evaluations:        15503
 Sampling finished. Exiting MultiNest

22:53:42 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	(2.04 -0.14 +0.8) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-7.5 -0.5 +2.0) x 10^-1
grb.spectrum.main.Band.xp	(2.14 -0.5 +0.24) x 10^2	keV
grb.spectrum.main.Band.beta	-1.93 -0.12 +0.13

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval10	-437.702131
n4_interval10	-433.090709
b0_interval10	-460.758707
total	-1331.551547

Values of statistical measures:

	statistical measures
AIC	2671.216409
BIC	2686.625227
DIC	2634.015089
PDIC	0.802792
log(Z)	-574.412597

         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)=  -813.09765170744333      +/-  0.15478106131222949
 Total Likelihood Evaluations:        11824
 Sampling finished. Exiting MultiNest

22:53:47 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	(2.7 -0.6 +0.9) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-5.3 -1.6 +2.1) x 10^-1
grb.spectrum.main.Band.xp	(1.28 -0.19 +0.29) x 10^2	keV
grb.spectrum.main.Band.beta	-2.10 -0.4 +0.07

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval11	-272.360340
n4_interval11	-255.723884
b0_interval11	-292.217532
total	-820.301756

Values of statistical measures:

	statistical measures
AIC	1648.716826
BIC	1664.125644
DIC	1617.409207
PDIC	0.600805
log(Z)	-353.123823

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

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

22:53:48 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

22:53:49 INFO      fit restored to median of posterior                                          sampler_base.py:157

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

22:53:50 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

22:53:51 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.