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")
10:00:06 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)
10:01:07 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:391
10:01:23 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
         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
10:01:26 INFO      Now using 120 bins                                                          SpectrumLike.py:1735
10:01:29 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:391
10:01:45 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
10:01:46 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: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
10:01:48 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:391
10:02:05 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
10:02:06 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)
10:02:07 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 (1.469 +/- 0.019) x 10^-2 1 / (cm2 keV s)
GRB080916009...alpha -1.072 -0.019 +0.020
GRB080916009...break_energy (2.36 +/- 0.35) x 10^2 keV
GRB080916009...break_scale (2.6 -0.9 +1.0) x 10^-1
GRB080916009...beta -2.12 +/- 0.11

Values of -log(posterior) at the minimum:

-log(posterior)
b0 -1049.502620
n3 -1020.026001
n4 -1011.194640
total -3080.723261

Values of statistical measures:

statistical measures
AIC 6171.616977
BIC 6190.849187
DIC 6180.791749
PDIC 4.656708
log(Z) -1346.747617
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  400
 dimensionality =    5
 *****************************************************
 ln(ev)=  -3101.0009868699144      +/-  0.22246454214953432
 Total Likelihood Evaluations:        24985
 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)
10:04:39 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)
10:04:40 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:291
         INFO      Created 12 bins via custom                                            time_series_builder.py:708
10:04:41 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
10:04:42 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 -1.0 +0.7) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-4.8 -2.3 +1.3) x 10^-1
grb.spectrum.main.Band.xp (2.9 -0.5 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.00 -0.13 +0.12

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval0 -285.690748
n3_interval0 -250.073782
n4_interval0 -267.853329
total -803.617860

Values of statistical measures:

statistical measures
AIC 1615.349034
BIC 1630.757852
DIC 1569.650561
PDIC 1.492236
log(Z) -343.259111
10:05:02 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 (4.31 +/- 0.09) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.25 +/- 0.18) x 10^-1
grb.spectrum.main.Band.xp (5.56 -0.28 +0.27) x 10^2 keV
grb.spectrum.main.Band.beta -2.06 +/- 0.05

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval1 -674.532847
n3_interval1 -641.444570
n4_interval1 -644.933931
total -1960.911348

Values of statistical measures:

statistical measures
AIC 3929.936011
BIC 3945.344828
DIC 3872.882278
PDIC 2.594158
log(Z) -844.858520
10:05:26 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 (2.60 +/- 0.21) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.04 -0.06 +0.07
grb.spectrum.main.Band.xp (5.9 -1.5 +1.6) x 10^2 keV
grb.spectrum.main.Band.beta -1.90 +/- 0.13

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval2 -324.632414
n3_interval2 -289.211122
n4_interval2 -312.740749
total -926.584285

Values of statistical measures:

statistical measures
AIC 1861.281885
BIC 1876.690703
DIC 1804.135306
PDIC 2.372451
log(Z) -393.569431
10:05:48 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 (3.07 -0.33 +0.31) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.1 +/- 0.8) x 10^-1
grb.spectrum.main.Band.xp (3.2 +/- 0.5) x 10^2 keV
grb.spectrum.main.Band.beta -2.26 +/- 0.29

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval3 -298.422234
n3_interval3 -242.622298
n4_interval3 -262.621840
total -803.666371

Values of statistical measures:

statistical measures
AIC 1615.446057
BIC 1630.854875
DIC 1570.011202
PDIC 2.590830
log(Z) -342.596521
10:06:07 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 (2.08 +/- 0.10) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.85 -0.31 +0.29) x 10^-1
grb.spectrum.main.Band.xp (3.8 +/- 0.4) x 10^2 keV
grb.spectrum.main.Band.beta -1.887 -0.04 +0.035

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval4 -778.683054
n3_interval4 -756.943471
n4_interval4 -746.717139
total -2282.343665

Values of statistical measures:

statistical measures
AIC 4572.800644
BIC 4588.209462
DIC 4527.407063
PDIC 2.734281
log(Z) -987.209836
10:06:33 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 (2.99 -0.30 +0.21) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.7 -0.8 +0.6) x 10^-1
grb.spectrum.main.Band.xp (3.8 -0.5 +0.6) x 10^2 keV
grb.spectrum.main.Band.beta -2.10 -0.14 +0.12

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval5 -537.042222
n3_interval5 -523.083400
n4_interval5 -528.185590
total -1588.311212

Values of statistical measures:

statistical measures
AIC 3184.735738
BIC 3200.144555
DIC 3137.436745
PDIC 3.463493
log(Z) -684.143061
10:06:54 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 (2.04 +/- 0.11) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.8 +/- 0.4) x 10^-1
grb.spectrum.main.Band.xp (4.2 +/- 0.5) x 10^2 keV
grb.spectrum.main.Band.beta -2.39 -0.26 +0.28

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval6 -609.501113
n3_interval6 -584.272914
n4_interval6 -576.759868
total -1770.533895

Values of statistical measures:

statistical measures
AIC 3549.181104
BIC 3564.589922
DIC 3500.823439
PDIC 3.070759
log(Z) -762.534847
10:07:15 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 (1.67 +/- 0.11) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.04 +/- 0.04
grb.spectrum.main.Band.xp (4.4 +/- 0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.33 +/- 0.23

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval7 -662.454905
n3_interval7 -640.923108
n4_interval7 -650.519265
total -1953.897278

Values of statistical measures:

statistical measures
AIC 3915.907870
BIC 3931.316687
DIC 3868.334215
PDIC 3.176513
log(Z) -842.379198
10:07:35 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 (1.56 -0.17 +0.21) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.7 -0.7 +1.1) x 10^-1
grb.spectrum.main.Band.xp (3.7 +/- 0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.23 -0.15 +0.13

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval8 -702.722401
n3_interval8 -698.312513
n4_interval8 -667.525366
total -2068.560280

Values of statistical measures:

statistical measures
AIC 4145.233875
BIC 4160.642692
DIC 4099.237362
PDIC 3.449312
log(Z) -893.158487
10:07:55 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 (1.47 -0.7 +0.28) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.2 -2.0 +1.8) x 10^-1
grb.spectrum.main.Band.xp (1.18 +/- 0.35) x 10^2 keV
grb.spectrum.main.Band.beta -1.97 +/- 0.22

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval9 -648.437912
n3_interval9 -616.999789
n4_interval9 -616.271583
total -1881.709284

Values of statistical measures:

statistical measures
AIC 3771.531881
BIC 3786.940699
DIC 3719.141680
PDIC -27.645710
log(Z) -816.026810
10:08:08 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 -0.5 +0.4) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-7.4 +/- 1.4) x 10^-1
grb.spectrum.main.Band.xp (2.3 +/- 0.5) x 10^2 keV
grb.spectrum.main.Band.beta -2.12 -0.28 +0.25

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval10 -460.872428
n3_interval10 -437.623744
n4_interval10 -433.287253
total -1331.783426

Values of statistical measures:

statistical measures
AIC 2671.680166
BIC 2687.088984
DIC 2634.444174
PDIC 0.639310
log(Z) -574.120202
10:08:25 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 (3.4 +/- 1.5) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-4.4 +/- 2.6) x 10^-1
grb.spectrum.main.Band.xp (1.30 +/- 0.30) x 10^2 keV
grb.spectrum.main.Band.beta -2.21 -0.35 +0.32

Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval11 -292.361522
n3_interval11 -272.173182
n4_interval11 -256.058584
total -820.593289

Values of statistical measures:

statistical measures
AIC 1649.299892
BIC 1664.708709
DIC 1616.951163
PDIC -0.296568
log(Z) -352.495106
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -790.38331211600575      +/-  0.18420561188202758
 Total Likelihood Evaluations:        16416
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1945.3586331724277      +/-  0.21795535376199693
 Total Likelihood Evaluations:        23355
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -906.22710401662300      +/-  0.19225340806447377
 Total Likelihood Evaluations:        21180
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -788.85764164210411      +/-  0.17766284505127389
 Total Likelihood Evaluations:        18129
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -2273.1346509207533      +/-  0.20791361591376825
 Total Likelihood Evaluations:        20527
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1575.2976134911346      +/-  0.19781488394338823
 Total Likelihood Evaluations:        19003
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1755.8013706945428      +/-  0.19299180494349946
 Total Likelihood Evaluations:        21213
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1939.6497837491625      +/-  0.19347304613748229
 Total Likelihood Evaluations:        20971
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -2056.5734182271458      +/-  0.19567219720618326
 Total Likelihood Evaluations:        19865
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1878.9711683610135      +/-  0.14766857410998710
 Total Likelihood Evaluations:        12517
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -1321.9606197593064      +/-  0.16630571859754445
 Total Likelihood Evaluations:        15246
 Sampling finished. Exiting MultiNest
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
 ln(ev)=  -811.64997565716908      +/-  0.14470614285406944
 Total Likelihood Evaluations:        12688
 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.