Analyzing GRB 080916C

[Alt text](https://astrobites.org/wp-content/uploads/2014/10/NASAGRBwhoa-1024x576.jpg ” (NASA/Swift/Cruz deWilde)”) (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.

%%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

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.

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

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.

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"]

model

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)


Linked parameters (0):

(none)

Independent variables:

(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.

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

Let’s first examine the catalof 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.

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)

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.

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.

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)

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.

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

Run the sampler

We let MultiNest condition the model on the data

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

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
GRB080916009...K	(1.469 +/- 0.019) x 10^-2	1 / (cm2 keV s)
GRB080916009...alpha	-1.069 +/- 0.019
GRB080916009...break_energy	(2.33 -0.34 +0.33) x 10^2	keV
GRB080916009...break_scale	(2.3 -0.8 +0.9) x 10^-1
GRB080916009...beta	-2.18 +/- 0.13

Values of -log(posterior) at the minimum:

	-log(posterior)
b0	-1051.716603
n3	-1023.620676
n4	-1014.293729
total	-3089.631009

Values of statistical measures:

	statistical measures
AIC	6189.432472
BIC	6208.664683
DIC	6177.108362
PDIC	4.764015
log(Z)	-1347.632852

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

 no. of live points =  400
 dimensionality =    5
 *****************************************************
 ln(ev)=  -3103.0393167076077      +/-  0.22210673255198010
 Total Likelihood Evaluations:        23259
 Sampling finished. Exiting MultiNest

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

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

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

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)"
)

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")

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.

n3 = time_series["n3"]

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

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.

fig = n3.view_lightcurve(use_binner=True)

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")

Now our light curve looks much more acceptable.

fig = n3.view_lightcurve(use_binner=True)

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.

time_resolved_plugins = {}

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

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.

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 = (0, 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.

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)

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(3.5 +/- 0.5) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-5.4 +/- 1.2) x 10^-1
grb.spectrum.main.Band.xp	(3.3 +/- 0.6) x 10^2	keV
grb.spectrum.main.Band.beta	-2.29 +/- 0.26

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval0	-280.514899
n3_interval0	-244.982566
n4_interval0	-261.481374
total	-786.978839

Values of statistical measures:

	statistical measures
AIC	1582.070992
BIC	1597.479810
DIC	1560.468829
PDIC	2.383562
log(Z)	-342.961458

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(4.97 +/- 0.15) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-7.39 +/- 0.23) x 10^-1
grb.spectrum.main.Band.xp	(3.89 +/- 0.19) x 10^2	keV
grb.spectrum.main.Band.beta	-1.7901 -0.0032 +0.004

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval1	-686.565534
n3_interval1	-636.802834
n4_interval1	-636.277864
total	-1959.646233

Values of statistical measures:

	statistical measures
AIC	3927.405780
BIC	3942.814597
DIC	3903.677712
PDIC	1.708169
log(Z)	-856.105286

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(3.8 -1.1 +1.3) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-9.8 -2.2 +2.8) x 10^-1
grb.spectrum.main.Band.xp	(3.1 -1.5 +1.4) x 10^2	keV
grb.spectrum.main.Band.beta	-1.91 -0.24 +0.26

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval2	-320.188088
n3_interval2	-292.531940
n4_interval2	-305.748422
total	-918.468450

Values of statistical measures:

	statistical measures
AIC	1845.050214
BIC	1860.459031
DIC	1806.554293
PDIC	-16.252081
log(Z)	-401.924468

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.93 -0.4 +0.35) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-9.1 +/- 0.9) x 10^-1
grb.spectrum.main.Band.xp	(3.5 +/- 0.7) x 10^2	keV
grb.spectrum.main.Band.beta	-2.41 +/- 0.33

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval3	-291.976829
n3_interval3	-237.259936
n4_interval3	-257.311201
total	-786.547967

Values of statistical measures:

	statistical measures
AIC	1581.209248
BIC	1596.618065
DIC	1559.992387
PDIC	3.263738
log(Z)	-342.440293

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.07 +/- 0.11) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-9.7 +/- 0.4) x 10^-1
grb.spectrum.main.Band.xp	(4.0 +/- 0.4) x 10^2	keV
grb.spectrum.main.Band.beta	-2.08 -0.12 +0.13

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval4	-773.524940
n3_interval4	-751.147552
n4_interval4	-741.415460
total	-2266.087951

Values of statistical measures:

	statistical measures
AIC	4540.289217
BIC	4555.698035
DIC	4520.493624
PDIC	3.565624
log(Z)	-986.926335

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.80 +/- 0.19) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-9.0 +/- 0.5) x 10^-1
grb.spectrum.main.Band.xp	(4.2 +/- 0.5) x 10^2	keV
grb.spectrum.main.Band.beta	-2.31 -0.21 +0.22

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval5	-531.985640
n3_interval5	-517.599918
n4_interval5	-522.683209
total	-1572.268767

Values of statistical measures:

	statistical measures
AIC	3152.650849
BIC	3168.059666
DIC	3130.838349
PDIC	3.298274
log(Z)	-684.671678

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.03 -0.11 +0.12) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-9.8 +/- 0.4) x 10^-1
grb.spectrum.main.Band.xp	(4.1 -0.5 +0.4) x 10^2	keV
grb.spectrum.main.Band.beta	-2.45 -0.27 +0.23

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval6	-607.032257
n3_interval6	-577.843573
n4_interval6	-570.861615
total	-1755.737445

Values of statistical measures:

	statistical measures
AIC	3519.588205
BIC	3534.997023
DIC	3496.685280
PDIC	2.966691
log(Z)	-764.608756

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(1.66 -0.10 +0.08) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	-1.05 +/- 0.04
grb.spectrum.main.Band.xp	(4.4 -0.5 +0.6) x 10^2	keV
grb.spectrum.main.Band.beta	-2.69 -0.26 +0.27

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval7	-659.178354
n3_interval7	-635.249795
n4_interval7	-644.256175
total	-1938.684324

Values of statistical measures:

	statistical measures
AIC	3885.481963
BIC	3900.890781
DIC	3862.953654
PDIC	2.820806
log(Z)	-844.414212

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(1.57 +/- 0.13) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-8.2 +/- 0.6) x 10^-1
grb.spectrum.main.Band.xp	(3.6 +/- 0.4) x 10^2	keV
grb.spectrum.main.Band.beta	-2.52 +/- 0.28

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval8	-696.948507
n3_interval8	-693.476716
n4_interval8	-661.296634
total	-2051.721858

Values of statistical measures:

	statistical measures
AIC	4111.557030
BIC	4126.965847
DIC	4090.708365
PDIC	3.368614
log(Z)	-892.992544

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(1.1 -0.5 +0.4) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-9.0 -1.6 +2.0) x 10^-1
grb.spectrum.main.Band.xp	(1.4 -0.5 +0.4) x 10^2	keV
grb.spectrum.main.Band.beta	-2.09 -0.19 +0.20

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval9	-647.173069
n3_interval9	-614.880413
n4_interval9	-613.557902
total	-1875.611384

Values of statistical measures:

	statistical measures
AIC	3759.336082
BIC	3774.744899
DIC	3728.119480
PDIC	-16.882434
log(Z)	-816.907631

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.2 +/- 0.5) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-7.2 +/- 1.5) x 10^-1
grb.spectrum.main.Band.xp	(2.2 +/- 0.5) x 10^2	keV
grb.spectrum.main.Band.beta	-2.21 -0.30 +0.29

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval10	-457.165894
n3_interval10	-433.964047
n4_interval10	-429.072503
total	-1320.202444

Values of statistical measures:

	statistical measures
AIC	2648.518203
BIC	2663.927020
DIC	2629.656209
PDIC	0.828922
log(Z)	-575.206648

  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(3.3 -1.2 +1.3) x 10^-2	1 / (cm2 keV s)
grb.spectrum.main.Band.alpha	(-4.2 -2.2 +2.3) x 10^-1
grb.spectrum.main.Band.xp	(1.27 -0.27 +0.25) x 10^2	keV
grb.spectrum.main.Band.beta	-2.27 -0.34 +0.31

Values of -log(posterior) at the minimum:

	-log(posterior)
b0_interval11	-289.586764
n3_interval11	-269.000188
n4_interval11	-252.483847
total	-811.070799

Values of statistical measures:

	statistical measures
AIC	1630.254913
BIC	1645.663730
DIC	1613.559691
PDIC	1.251006
log(Z)	-353.417466

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

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -789.69794036378914      +/-  0.17645119210319724
 Total Likelihood Evaluations:        16069
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1971.2552686904098      +/-  0.22569033590904114
 Total Likelihood Evaluations:        23331
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -925.46528869750796      +/-  0.19253458682235039
 Total Likelihood Evaluations:        18232
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -788.49791401847779      +/-  0.17340511236201572
 Total Likelihood Evaluations:        18821
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -2272.4818675898682      +/-  0.19389714827720803
 Total Likelihood Evaluations:        23608
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1576.5147993555313      +/-  0.18927536919964608
 Total Likelihood Evaluations:        20224
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1760.5767240175117      +/-  0.19379332344256356
 Total Likelihood Evaluations:        20068
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1944.3355763672516      +/-  0.19795749358915810
 Total Likelihood Evaluations:        18633
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -2056.1913197791114      +/-  0.18605099739858069
 Total Likelihood Evaluations:        20347
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1880.9993326851566      +/-  0.14757785997145398
 Total Likelihood Evaluations:        12896
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1324.4622529535122      +/-  0.16647594199336932
 Total Likelihood Evaluations:        14988
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -813.77378807530204      +/-  0.14721022162359965
 Total Likelihood Evaluations:        12347
 Sampling finished. Exiting MultiNest

Examine the fits

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

for a in analysis:
    a.restore_median_fit()
    _ = display_spectrum_model_counts(a, min_rate=[20, 20, -99], step=False)

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

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),
)

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.