Analyzing GRB 080916C

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")
08:45:40 INFO      The cache for fermigbrst does not yet exist. We will try to    get_heasarc_table_as_pandas.py:64
                  build it                                                                                         
                                                                                                                   
         INFO      Building cache for fermigbrst                                 get_heasarc_table_as_pandas.py:112
[4]:
Table length=1
nameradectrigger_timet90
objectfloat64float64float64float64
GRB080916009119.800-56.60054725.008861362.977

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)
08:46:36 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:389
08:46:54 INFO      None 0-order polynomial fit with the mle method                               time_series.py:458
         INFO      Saved fitted background to n3_bkg.h5                                         time_series.py:1064
         INFO      Saved background to n3_bkg.h5                                         time_series_builder.py:471
         INFO      Successfully restored fit from n3_bkg.h5                              time_series_builder.py:171
         INFO      Interval set to 1.28-64.257 for n3                                    time_series_builder.py:290
         INFO      Auto-probed noise models:                                                    SpectrumLike.py:469
         INFO      - observation: poisson                                                       SpectrumLike.py:470
         INFO      - background: gaussian                                                       SpectrumLike.py:471
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
08:46:58 INFO      Now using 120 bins                                                          SpectrumLike.py:1673
08:47:00 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389
08:47:19 INFO      None 1-order polynomial fit with the mle method                               time_series.py:458
         INFO      Saved fitted background to n4_bkg.h5                                         time_series.py:1064
         INFO      Saved background to n4_bkg.h5                                         time_series_builder.py:471
         INFO      Successfully restored fit from n4_bkg.h5                              time_series_builder.py:171
         INFO      Interval set to 1.28-64.257 for n4                                    time_series_builder.py:290
         INFO      Auto-probed noise models:                                                    SpectrumLike.py:469
         INFO      - observation: poisson                                                       SpectrumLike.py:470
         INFO      - background: gaussian                                                       SpectrumLike.py:471
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
08:47:22 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389
08:47:40 INFO      None 1-order polynomial fit with the mle method                               time_series.py:458
         INFO      Saved fitted background to b0_bkg.h5                                         time_series.py:1064
         INFO      Saved background to b0_bkg.h5                                         time_series_builder.py:471
08:47:41 INFO      Successfully restored fit from b0_bkg.h5                              time_series_builder.py:171
         INFO      Interval set to 1.28-64.257 for b0                                    time_series_builder.py:290
         INFO      Auto-probed noise models:                                                    SpectrumLike.py:469
         INFO      - observation: poisson                                                       SpectrumLike.py:470
         INFO      - background: gaussian                                                       SpectrumLike.py:471
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
../_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)
08:47:42 INFO      sampler set to multinest                                                bayesian_analysis.py:197

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
 *****************************************************
 ln(ev)=  -3100.8148981185846      +/-  0.22199760823701009
  analysing data from chains/fit-.txt
 Total Likelihood Evaluations:        26294
 Sampling finished. Exiting MultiNest
08:48:08 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
GRB080916009...K (1.462 -0.011 +0.026) x 10^-2 1 / (cm2 keV s)
GRB080916009...alpha -1.0950 +0.0021 +0.04
GRB080916009...break_energy (1.86 +0.16 +0.8) x 10^2 keV
GRB080916009...break_scale (0.0 +1.8 +3.5) x 10^-1
GRB080916009...beta -1.96 -0.26 -0.05
Values of -log(posterior) at the minimum:

-log(posterior)
b0 -1049.505738
n3 -1019.128163
n4 -1010.348015
total -3078.981917
Values of statistical measures:

statistical measures
AIC 6168.134288
BIC 6187.366498
DIC 6180.527038
PDIC 4.610213
log(Z) -1346.666800

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:164
../_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)
08:50:13 INFO      Created 15 bins via bayesblocks                                       time_series_builder.py:708

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:708

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)
08:50:14 INFO      Created 12 bins via custom                                            time_series_builder.py:708
         INFO      Interval set to 1.28-64.257 for n3                                    time_series_builder.py:290
         INFO      Created 12 bins via custom                                            time_series_builder.py:708
08:50:15 INFO      Interval set to 1.28-64.257 for n4                                    time_series_builder.py:290
         INFO      Created 12 bins via custom                                            time_series_builder.py:708
08:50:16 INFO      Interval set to 1.28-64.257 for b0                                    time_series_builder.py:290

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 = (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.

[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:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 107 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -788.65752046363434      +/-  0.17877090246564475
 Total Likelihood Evaluations:        16588
 Sampling finished. Exiting MultiNest

08:50:32 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (3.6 -0.5 +0.6) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-5.5 -1.1 +1.2) x 10^-1
grb.spectrum.main.Band.xp (3.1 -0.4 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.04 -0.20 +0.10
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval0 -285.684371
n3_interval0 -250.133309
n4_interval0 -267.966734
total -803.784414
Values of statistical measures:

statistical measures
AIC 1615.682142
BIC 1631.090960
DIC 1569.511166
PDIC 2.178541
log(Z) -342.509609
08:50:33 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -1951.2949337804603      +/-  0.23344957676457673
 Total Likelihood Evaluations:        23474
 Sampling finished. Exiting MultiNest

08:50:56 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (4.418 -0.13 +0.029) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.212 -0.15 -0.029) x 10^-1
grb.spectrum.main.Band.xp (5.18 -0.05 +0.31) x 10^2 keV
grb.spectrum.main.Band.beta -1.9907 +0.0011 +0.05
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval1 -676.386133
n3_interval1 -642.263496
n4_interval1 -643.616871
total -1962.266499
Values of statistical measures:

statistical measures
AIC 3932.646313
BIC 3948.055131
DIC 3876.572663
PDIC 1.918371
log(Z) -847.436622
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 115 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -909.37539412786566      +/-  0.20057867534896762
 Total Likelihood Evaluations:        18203
 Sampling finished. Exiting MultiNest
  analysing data from chains/fit-.txt
08:51:14 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.63 -0.09 +0.25) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.05 -0.06 +0.10
grb.spectrum.main.Band.xp (5.4 -1.0 +0.6) x 10^2 keV
grb.spectrum.main.Band.beta -1.89 -0.31 +0.04
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval2 -324.252297
n3_interval2 -289.090744
n4_interval2 -311.983466
total -925.326507
Values of statistical measures:

statistical measures
AIC 1858.766328
BIC 1874.175145
DIC 1806.625536
PDIC 2.877493
log(Z) -394.936716
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 109 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -794.72038480273977      +/-  0.18292072947246141
 Total Likelihood Evaluations:        18948
 Sampling finished. Exiting MultiNest

08:51:34 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (3.20 -0.13 +1.2) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.9 -0.5 +1.7) x 10^-1
grb.spectrum.main.Band.xp (2.575 -0.9 +0.019) x 10^2 keV
grb.spectrum.main.Band.beta -1.82 +0.05 +0.15
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval3 -299.104180
n3_interval3 -241.931446
n4_interval3 -261.770918
total -802.806544
Values of statistical measures:

statistical measures
AIC 1613.726403
BIC 1629.135220
DIC 1574.701064
PDIC 0.757296
log(Z) -345.142678
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -2272.3135710858282      +/-  0.20491477561902821
 Total Likelihood Evaluations:        19758
 Sampling finished. Exiting MultiNest

08:51:53 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.96 -0.08 +0.07) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.014 -0.030 +0.015
grb.spectrum.main.Band.xp (4.5 +/- 0.4) x 10^2 keV
grb.spectrum.main.Band.beta -2.03 -0.17 +0.07
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval4 -779.240508
n3_interval4 -757.022014
n4_interval4 -747.801179
total -2284.063701
Values of statistical measures:

statistical measures
AIC 4576.240717
BIC 4591.649534
DIC 4529.101847
PDIC 3.328601
log(Z) -986.853245
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -1573.1894657511657      +/-  0.19161796488723146
 Total Likelihood Evaluations:        20449
 Sampling finished. Exiting MultiNest

08:52:12 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.80 -0.20 +0.17) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.1 -0.6 +0.4) x 10^-1
grb.spectrum.main.Band.xp (4.2 -0.4 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.15 -0.27 +0.11
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval5 -536.817219
n3_interval5 -523.727965
n4_interval5 -527.538981
total -1588.084165
Values of statistical measures:

statistical measures
AIC 3184.281645
BIC 3199.690462
DIC 3137.036671
PDIC 3.479125
log(Z) -683.227504
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -1754.9770593592723      +/-  0.19116211991846807
 Total Likelihood Evaluations:        21154
 Sampling finished. Exiting MultiNest

08:52:31 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.95 -0.11 +0.13) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.02 -0.04 +0.05
grb.spectrum.main.Band.xp (4.5 -0.6 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.35 -0.4 +0.17
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval6 -609.588855
n3_interval6 -584.287288
n4_interval6 -576.885015
total -1770.761158
Values of statistical measures:

statistical measures
AIC 3549.635629
BIC 3565.044447
DIC 3501.196800
PDIC 3.357205
log(Z) -762.176853
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -1939.5714510775526      +/-  0.19244524725443438
 Total Likelihood Evaluations:        19615
 Sampling finished. Exiting MultiNest

08:52:50 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.66 -0.08 +0.12) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.04 -0.04 +0.05
grb.spectrum.main.Band.xp (4.4 -0.6 +0.5) x 10^2 keV
grb.spectrum.main.Band.beta -2.30 -0.24 +0.21
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval7 -662.346419
n3_interval7 -640.934991
n4_interval7 -650.525199
total -1953.806608
Values of statistical measures:

statistical measures
AIC 3915.726531
BIC 3931.135349
DIC 3868.340461
PDIC 3.185416
log(Z) -842.345178
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -2053.6847048340865      +/-  0.18619585356293153
 Total Likelihood Evaluations:        20183
 Sampling finished. Exiting MultiNest

08:53:08 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.52 -0.12 +0.13) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.6 -0.6 +0.7) x 10^-1
grb.spectrum.main.Band.xp (3.8 -0.4 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.30 -0.4 +0.13
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval8 -702.160603
n3_interval8 -698.348599
n4_interval8 -666.417395
total -2066.926597
Values of statistical measures:

statistical measures
AIC 4141.966508
BIC 4157.375326
DIC 4098.424220
PDIC 3.371964
log(Z) -891.903935
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1878.8218264325071      +/-  0.14538140018730025
 Total Likelihood Evaluations:        12621
 Sampling finished. Exiting MultiNest
  analysing data from chains/fit-.txt
08:53:20 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.01 -0.18 +1.7) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.0 -1.0 +4) x 10^-1
grb.spectrum.main.Band.xp (1.25 -0.6 +0.29) x 10^2 keV
grb.spectrum.main.Band.beta -1.90 -0.30 +0.17
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval9 -648.563589
n3_interval9 -617.237883
n4_interval9 -616.343799
total -1882.145270
Values of statistical measures:

statistical measures
AIC 3772.403855
BIC 3787.812673
DIC 3698.897974
PDIC -48.261186
log(Z) -815.961952
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -1325.5403570596727      +/-  0.18216481686582917
 Total Likelihood Evaluations:        16348
 Sampling finished. Exiting MultiNest

08:53:37 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.36 -0.8 -0.08) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-6.9 -2.6 -0.8) x 10^-1
grb.spectrum.main.Band.xp (1.836 +0.009 +1.0) x 10^2 keV
grb.spectrum.main.Band.beta -1.85 -0.05 +0.09
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval10 -460.687514
n3_interval10 -437.981263
n4_interval10 -432.295029
total -1330.963806
Values of statistical measures:

statistical measures
AIC 2670.040927
BIC 2685.449744
DIC 2636.795556
PDIC 1.917882
log(Z) -575.674863
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1210
         INFO      Now using 120 bins                                                          SpectrumLike.py:1673
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1210
         INFO      Now using 119 bins                                                          SpectrumLike.py:1673
         INFO      sampler set to multinest                                                bayesian_analysis.py:197
 *****************************************************
 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)=  -813.19796939849527      +/-  0.15840240500119140
 Total Likelihood Evaluations:        12563
 Sampling finished. Exiting MultiNest

08:53:50 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.6 -0.5 +1.5) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-5.6 -0.7 +2.3) x 10^-1
grb.spectrum.main.Band.xp (1.32 -0.18 +0.30) x 10^2 keV
grb.spectrum.main.Band.beta -2.44 -0.20 +0.08
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval11 -292.776978
n3_interval11 -272.670073
n4_interval11 -256.114739
total -821.561791
Values of statistical measures:

statistical measures
AIC 1651.236896
BIC 1666.645713
DIC 1617.681992
PDIC 0.998917
log(Z) -353.167391

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:164
08:53:51 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:52 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:53 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:54 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:55 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:56 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:57 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:58 INFO      fit restored to median of posterior                                          sampler_base.py:164
         INFO      fit restored to median of posterior                                          sampler_base.py:164
08:53:59 INFO      fit restored to median of posterior                                          sampler_base.py:164
08:54:00 INFO      fit restored to median of posterior                                          sampler_base.py:164
../_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.