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")
15:11:38 INFO      The cache for fermigbrst does not yet exist. We will try to    get_heasarc_table_as_pandas.py:66
                  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)
15:12:42 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:391
15:12:57 INFO      None 0-order polynomial fit with the mle method                               time_series.py:459
         INFO      Saved fitted background to n3_bkg.h5                                         time_series.py:1064
         INFO      Saved background to n3_bkg.h5                                         time_series_builder.py:471
15:12:58 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:291
         INFO      Auto-probed noise models:                                                    SpectrumLike.py:491
         INFO      - observation: poisson                                                       SpectrumLike.py:492
         INFO      - background: gaussian                                                       SpectrumLike.py:493
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
15:13:01 INFO      Now using 120 bins                                                          SpectrumLike.py:1735
15:13:03 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:391
15:13:20 INFO      None 1-order polynomial fit with the mle method                               time_series.py:459
         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
15:13:21 INFO      Interval set to 1.28-64.257 for n4                                    time_series_builder.py:291
         INFO      Auto-probed noise models:                                                    SpectrumLike.py:491
         INFO      - observation: poisson                                                       SpectrumLike.py:492
         INFO      - background: gaussian                                                       SpectrumLike.py:493
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
15:13:23 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:391
15:13:41 INFO      None 1-order polynomial fit with the mle method                               time_series.py:459
         INFO      Saved fitted background to b0_bkg.h5                                         time_series.py:1064
         INFO      Saved background to b0_bkg.h5                                         time_series_builder.py:471
         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:291
15:13:42 INFO      Auto-probed noise models:                                                    SpectrumLike.py:491
         INFO      - observation: poisson                                                       SpectrumLike.py:492
         INFO      - background: gaussian                                                       SpectrumLike.py:493
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
../_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)
15:13:43 INFO      sampler set to multinest                                                bayesian_analysis.py:233

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

result unit
parameter
GRB080916009...K (1.476 +/- 0.018) x 10^-2 1 / (cm2 keV s)
GRB080916009...alpha -1.074 -0.019 +0.018
GRB080916009...break_energy (2.12 +/- 0.14) x 10^2 keV
GRB080916009...break_scale (2.3 +/- 0.5) x 10^-1
GRB080916009...beta -2.04 +/- 0.04

Values of -log(posterior) at the minimum:

-log(posterior)
b0 -1055.268956
n3 -1025.892420
n4 -1017.209415
total -3098.370791

Values of statistical measures:

statistical measures
AIC 6206.912036
BIC 6226.144247
DIC 6178.566314
PDIC 3.820675
log(Z) -1347.558361
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  400
 dimensionality =    5
 *****************************************************
 ln(ev)=  -3102.8677936379740      +/-  0.23422151611540976
 Total Likelihood Evaluations:        21941
 Sampling finished. Exiting MultiNest

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)
../_images/notebooks_grb080916C_22_0.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)
15:16:11 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")
15:16:12 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)
         INFO      Created 12 bins via custom                                            time_series_builder.py:708
15:16:13 INFO      Interval set to 1.28-64.257 for n3                                    time_series_builder.py:291
         INFO      Created 12 bins via custom                                            time_series_builder.py:708
         INFO      Interval set to 1.28-64.257 for n4                                    time_series_builder.py:291
         INFO      Created 12 bins via custom                                            time_series_builder.py:708
15:16:15 INFO      Interval set to 1.28-64.257 for b0                                    time_series_builder.py:291

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:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 107 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (4.1 -0.6 +0.7) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-4.7 -1.0 +1.2) x 10^-1
grb.spectrum.main.Band.xp (2.8 +/- 0.4) x 10^2 keV
grb.spectrum.main.Band.beta -2.00 -0.11 +0.14

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval0 -285.638192
n3_interval0 -250.103678
n4_interval0 -267.984257
total -803.726127

Values of statistical measures:

statistical measures
AIC 1615.565568
BIC 1630.974385
DIC 1570.136946
PDIC 2.242703
log(Z) -342.979197
15:16:34 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (4.57 +/- 0.08) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-7.85 -0.16 +0.17) x 10^-1
grb.spectrum.main.Band.xp (4.88 +/- 0.18) x 10^2 keV
grb.spectrum.main.Band.beta -1.992 -0.032 +0.021

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval1 -676.667114
n3_interval1 -641.672176
n4_interval1 -643.809494
total -1962.148784

Values of statistical measures:

statistical measures
AIC 3932.410883
BIC 3947.819700
DIC 3878.249903
PDIC 2.171098
log(Z) -846.785482
15:16:59 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 115 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.82 -0.26 +0.23) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.01 +/- 0.09
grb.spectrum.main.Band.xp (4.8 +/- 1.0) x 10^2 keV
grb.spectrum.main.Band.beta -1.94 -0.20 +0.15

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval2 -324.117732
n3_interval2 -288.894914
n4_interval2 -311.319906
total -924.332552

Values of statistical measures:

statistical measures
AIC 1856.778418
BIC 1872.187236
DIC 1806.144398
PDIC 2.449228
log(Z) -394.542243
15:17:21 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 109 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.84 -0.34 +0.35) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.6 +/- 0.8) x 10^-1
grb.spectrum.main.Band.xp (3.6 +/- 0.8) x 10^2 keV
grb.spectrum.main.Band.beta -2.29 -0.29 +0.30

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval3 -298.522448
n3_interval3 -242.670087
n4_interval3 -262.706932
total -803.899467

Values of statistical measures:

statistical measures
AIC 1615.912248
BIC 1631.321066
DIC 1570.287792
PDIC 2.889649
log(Z) -342.517463
15:17:37 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
15:17:38 INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.05 -0.11 +0.10) 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) x 10^2 keV
grb.spectrum.main.Band.beta -1.99 +/- 0.09

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval4 -778.612324
n3_interval4 -757.026640
n4_interval4 -746.938591
total -2282.577555

Values of statistical measures:

statistical measures
AIC 4573.268424
BIC 4588.677241
DIC 4527.814424
PDIC 3.468919
log(Z) -986.088368
15:17:58 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.93 -0.13 +0.14) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.77 -0.4 +0.35) x 10^-1
grb.spectrum.main.Band.xp (3.92 +/- 0.35) x 10^2 keV
grb.spectrum.main.Band.beta -2.16 -0.18 +0.16

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval5 -536.756419
n3_interval5 -523.452621
n4_interval5 -527.668616
total -1587.877657

Values of statistical measures:

statistical measures
AIC 3183.868629
BIC 3199.277446
DIC 3135.988422
PDIC 2.976020
log(Z) -683.599543
15:18:18 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
15:18:19 INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.96 +/- 0.11) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.01 +/- 0.04
grb.spectrum.main.Band.xp (4.5 +/- 0.6) x 10^2 keV
grb.spectrum.main.Band.beta -2.46 +/- 0.28

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval6 -609.393344
n3_interval6 -584.241599
n4_interval6 -576.643030
total -1770.277974

Values of statistical measures:

statistical measures
AIC 3548.669262
BIC 3564.078080
DIC 3500.601980
PDIC 3.070200
log(Z) -762.301115
15:18:38 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.92 -0.09 +0.10) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.54 -0.30 +0.32) x 10^-1
grb.spectrum.main.Band.xp (3.23 -0.30 +0.29) x 10^2 keV
grb.spectrum.main.Band.beta -1.95 +/- 0.04

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval7 -662.511266
n3_interval7 -640.775074
n4_interval7 -649.494822
total -1952.781162

Values of statistical measures:

statistical measures
AIC 3913.675638
BIC 3929.084455
DIC 3870.805433
PDIC 1.923403
log(Z) -844.722229
15:18:59 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.53 +/- 0.12) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.5 +/- 0.6) x 10^-1
grb.spectrum.main.Band.xp (3.9 +/- 0.5) x 10^2 keV
grb.spectrum.main.Band.beta -2.42 -0.25 +0.24

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval8 -702.242492
n3_interval8 -698.321892
n4_interval8 -666.119802
total -2066.684186

Values of statistical measures:

statistical measures
AIC 4141.481686
BIC 4156.890503
DIC 4097.588223
PDIC 3.037414
log(Z) -892.093657
15:19:18 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.7 +/- 0.9) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-7.7 -2.5 +2.9) x 10^-1
grb.spectrum.main.Band.xp (1.2 +/- 0.5) x 10^2 keV
grb.spectrum.main.Band.beta -1.98 +/- 0.26

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval9 -648.316123
n3_interval9 -616.807621
n4_interval9 -616.165365
total -1881.289109

Values of statistical measures:

statistical measures
AIC 3770.691532
BIC 3786.100350
DIC 3690.716368
PDIC -56.485708
log(Z) -815.863832
15:19:30 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (2.1 +/- 0.4) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-7.5 -1.3 +1.4) x 10^-1
grb.spectrum.main.Band.xp (2.4 +/- 0.6) x 10^2 keV
grb.spectrum.main.Band.beta -2.15 -0.32 +0.30

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval10 -460.711263
n3_interval10 -437.883795
n4_interval10 -433.009881
total -1331.604939

Values of statistical measures:

statistical measures
AIC 2671.323192
BIC 2686.732010
DIC 2634.721724
PDIC 0.870390
log(Z) -574.078280
15:19:46 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1245
         INFO      Now using 120 bins                                                          SpectrumLike.py:1735
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
         INFO      Now using 119 bins                                                          SpectrumLike.py:1735
         INFO      sampler set to multinest                                                bayesian_analysis.py:233
  analysing data from chains/fit-.txt
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (3.0 +/- 1.1) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-4.7 +/- 2.2) x 10^-1
grb.spectrum.main.Band.xp (1.34 +/- 0.27) x 10^2 keV
grb.spectrum.main.Band.beta -2.27 +/- 0.33

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval11 -292.358205
n3_interval11 -272.258089
n4_interval11 -255.948404
total -820.564697

Values of statistical measures:

statistical measures
AIC 1649.242708
BIC 1664.651525
DIC 1618.967626
PDIC 1.880922
log(Z) -352.603487
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -789.73878681930694      +/-  0.18098405505713330
 Total Likelihood Evaluations:        16759
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1949.7956268792047      +/-  0.22213034758664543
 Total Likelihood Evaluations:        22513
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -908.46708721095990      +/-  0.19413893084600245
 Total Likelihood Evaluations:        20443
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -788.67560544621301      +/-  0.17933485226552201
 Total Likelihood Evaluations:        16675
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -2270.5523762280877      +/-  0.19736235937198920
 Total Likelihood Evaluations:        22380
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1574.0461167805324      +/-  0.19463807849438255
 Total Likelihood Evaluations:        20058
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1755.2631849014340      +/-  0.19319692080192116
 Total Likelihood Evaluations:        19450
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1945.0448129296092      +/-  0.20242702214109917
 Total Likelihood Evaluations:        20422
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -2054.1215563050641      +/-  0.18895117192294758
 Total Likelihood Evaluations:        19259
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1878.5958984166969      +/-  0.14414964171602374
 Total Likelihood Evaluations:        12969
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1321.8640907865163      +/-  0.16616817213469348
 Total Likelihood Evaluations:        15528
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -811.89953242139006      +/-  0.14686043381892885
 Total Likelihood Evaluations:        11871
 Sampling finished. Exiting MultiNest

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)
../_images/notebooks_grb080916C_41_0.png
../_images/notebooks_grb080916C_41_1.png
../_images/notebooks_grb080916C_41_2.png
../_images/notebooks_grb080916C_41_3.png
../_images/notebooks_grb080916C_41_4.png
../_images/notebooks_grb080916C_41_5.png
../_images/notebooks_grb080916C_41_6.png
../_images/notebooks_grb080916C_41_7.png
../_images/notebooks_grb080916C_41_8.png
../_images/notebooks_grb080916C_41_9.png
../_images/notebooks_grb080916C_41_10.png
../_images/notebooks_grb080916C_41_11.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.