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")
22:18:29 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)
22:19:29 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:389
22:19:40 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:490
         INFO      - observation: poisson                                                       SpectrumLike.py:491
         INFO      - background: gaussian                                                       SpectrumLike.py:492
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
22:19:43 INFO      Now using 120 bins                                                          SpectrumLike.py:1739
22:19:45 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389
22:19:56 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
22:19:57 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:490
         INFO      - observation: poisson                                                       SpectrumLike.py:491
         INFO      - background: gaussian                                                       SpectrumLike.py:492
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
22:19:59 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389
22:20:10 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
         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
22:20:11 INFO      Auto-probed noise models:                                                    SpectrumLike.py:490
         INFO      - observation: poisson                                                       SpectrumLike.py:491
         INFO      - background: gaussian                                                       SpectrumLike.py:492
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
../_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)
22:20:12 INFO      sampler set to multinest                                                bayesian_analysis.py:202

Examine at the catalog fitted model

We can quickly examine how well the catalog fit matches the data. There appears to be a discrepancy between the data and the model! Let’s refit to see if we can fix it.

[11]:
fig = display_spectrum_model_counts(bayes, min_rate=20, step=False)
../_images/notebooks_grb080916C_18_0.png

Run the sampler

We let MultiNest condition the model on the data

[12]:
bayes.sampler.setup(n_live_points=400)
bayes.sample()
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  400
 dimensionality =    5
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -3100.8170891549435      +/-  0.22221871250931144
 Total Likelihood Evaluations:        27465
 Sampling finished. Exiting MultiNest

22:20:32 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
GRB080916009...K (1.463 -0.012 +0.025) x 10^-2 1 / (cm2 keV s)
GRB080916009...alpha -1.087 -0.007 +0.032
GRB080916009...break_energy (1.92 +0.11 +0.7) x 10^2 keV
GRB080916009...break_scale (0.0 +1.8 +3.4) x 10^-1
GRB080916009...beta -1.988 -0.22 -0.024
Values of -log(posterior) at the minimum:

-log(posterior)
b0 -1049.714150
n3 -1019.455246
n4 -1010.744288
total -3079.913684
Values of statistical measures:

statistical measures
AIC 6169.997822
BIC 6189.230032
DIC 6180.114864
PDIC 4.417556
log(Z) -1346.667751

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)
22:22:28 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)
         INFO      Created 12 bins via custom                                            time_series_builder.py:708
22:22:29 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
         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
22:22:31 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 = (None, None)
band.K.prior = Log_uniform_prior(lower_bound=1e-10, upper_bound=1e3)
ps = PointSource("grb", 0, 0, spectral_shape=band)
band_model = Model(ps)

Perform the fits

One way to perform Bayesian spectral fits to all the intervals is to loop through each one. There can are many ways to do this, so find an analysis pattern that works for you.

[23]:
models = []
results = []
analysis = []
for interval in range(12):
    # clone the model above so that we have a separate model
    # for each fit

    this_model = clone_model(band_model)

    # for each detector set up the plugin
    # for this time interval

    this_data_list = []
    for k, v in time_resolved_plugins.items():
        pi = v[interval]

        if k.startswith("b"):
            pi.set_active_measurements("250-30000")
        else:
            pi.set_active_measurements("9-900")

        pi.rebin_on_background(1.0)

        this_data_list.append(pi)

    # create a data list

    dlist = DataList(*this_data_list)

    # set up the sampler and fit

    bayes = BayesianAnalysis(this_model, dlist)

    # get some speed with share spectrum
    bayes.set_sampler("multinest", share_spectrum=True)
    bayes.sampler.setup(n_live_points=500)
    bayes.sample()

    # at this stage we coudl also
    # save the analysis result to
    # disk but we will simply hold
    # onto them in memory

    analysis.append(bayes)
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 107 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -790.02833697304425      +/-  0.18003123741649052
 Total Likelihood Evaluations:        16175
 Sampling finished. Exiting MultiNest

22:22:43 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 (3.87 -0.33 +0.9) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-5.0 -0.9 +1.6) x 10^-1
grb.spectrum.main.Band.xp (2.9 -0.5 +0.4) x 10^2 keV
grb.spectrum.main.Band.beta -2.05 -0.5 +0.09
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval0 -285.577350
n3_interval0 -249.932248
n4_interval0 -267.990356
total -803.499954
Values of statistical measures:

statistical measures
AIC 1615.113223
BIC 1630.522040
DIC 1571.915657
PDIC 2.374273
log(Z) -343.104947
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -2015.6020396904735      +/-  0.22639007204651804
 Total Likelihood Evaluations:        23147
 Sampling finished. Exiting MultiNest

22:23:00 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 (7.67 -0.04 +0.11) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-4.68 +0.06 +0.4) x 10^-1
grb.spectrum.main.Band.xp (2.302 -0.06 -0.019) x 10^2 keV
grb.spectrum.main.Band.beta -1.709 -0.008 +0.014
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval1 -703.226509
n3_interval1 -660.028408
n4_interval1 -654.800380
total -2018.055297
Values of statistical measures:

statistical measures
AIC 4044.223908
BIC 4059.632726
DIC 4002.778293
PDIC 1.903423
log(Z) -875.364844
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 115 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -906.31719628797725      +/-  0.19595980518206735
 Total Likelihood Evaluations:        21067
 Sampling finished. Exiting MultiNest

22:23:15 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 (2.56 -0.28 +0.14) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.04 -0.08 +0.04
grb.spectrum.main.Band.xp (5.6 -0.4 +4) x 10^2 keV
grb.spectrum.main.Band.beta -1.86 -0.5 -0.11
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval2 -324.364071
n3_interval2 -289.003719
n4_interval2 -312.054373
total -925.422162
Values of statistical measures:

statistical measures
AIC 1858.957639
BIC 1874.366457
DIC 1805.015158
PDIC 2.041820
log(Z) -393.608557
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 109 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -788.90752511757785      +/-  0.17720859571104447
 Total Likelihood Evaluations:        17331
 Sampling finished. Exiting MultiNest

22:23:26 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 (2.86 -0.26 +0.5) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.3 -0.9 +1.2) x 10^-1
grb.spectrum.main.Band.xp (3.5 -0.8 +0.5) x 10^2 keV
grb.spectrum.main.Band.beta -2.22 -0.08 +0.28
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval3 -298.463220
n3_interval3 -242.557091
n4_interval3 -262.595481
total -803.615793
Values of statistical measures:

statistical measures
AIC 1615.344900
BIC 1630.753717
DIC 1570.396298
PDIC 2.891963
log(Z) -342.618185
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -2270.6204387420898      +/-  0.19773126391792753
 Total Likelihood Evaluations:        20444
 Sampling finished. Exiting MultiNest

22:23:39 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 (2.09 -0.16 +0.05) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.70 -0.5 +0.19) x 10^-1
grb.spectrum.main.Band.xp (3.84 -0.18 +0.8) x 10^2 keV
grb.spectrum.main.Band.beta -1.942 -0.18 +0.032
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval4 -778.396449
n3_interval4 -757.236883
n4_interval4 -746.414263
total -2282.047595
Values of statistical measures:

statistical measures
AIC 4572.208505
BIC 4587.617322
DIC 4527.894882
PDIC 3.415310
log(Z) -986.117927
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -1575.3369769139256      +/-  0.19852821962465281
 Total Likelihood Evaluations:        18603
 Sampling finished. Exiting MultiNest

22:23:53 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 (2.85 +/- 0.29) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.0 +/- 0.8) x 10^-1
grb.spectrum.main.Band.xp (4.1 -0.6 +1.0) x 10^2 keV
grb.spectrum.main.Band.beta -2.16 -0.4 +0.15
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval5 -536.435099
n3_interval5 -523.717418
n4_interval5 -527.723255
total -1587.875773
Values of statistical measures:

statistical measures
AIC 3183.864860
BIC 3199.273677
DIC 3138.074068
PDIC 3.311957
log(Z) -684.160156
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -1759.9220764020010      +/-  0.19858403442282160
 Total Likelihood Evaluations:        19820
 Sampling finished. Exiting MultiNest

22:24:06 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 (2.163 +0.016 +0.22) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.51 -0.22 +0.6) x 10^-1
grb.spectrum.main.Band.xp (3.562 -0.5 +0.004) x 10^2 keV
grb.spectrum.main.Band.beta -2.08 -0.15 +0.10
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval6 -609.042504
n3_interval6 -584.293769
n4_interval6 -575.867130
total -1769.203404
Values of statistical measures:

statistical measures
AIC 3546.520122
BIC 3561.928939
DIC 3503.412642
PDIC 2.506699
log(Z) -764.324446
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -1938.9883124928601      +/-  0.18950256933185303
 Total Likelihood Evaluations:        21100
 Sampling finished. Exiting MultiNest

22:24:18 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 (1.67 -0.11 +0.10) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.04 -0.06 +0.04
grb.spectrum.main.Band.xp (4.3 -0.4 +0.9) x 10^2 keV
grb.spectrum.main.Band.beta -2.27 -0.4 +0.14
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval7 -662.242364
n3_interval7 -640.779345
n4_interval7 -650.562833
total -1953.584541
Values of statistical measures:

statistical measures
AIC 3915.282397
BIC 3930.691215
DIC 3869.354759
PDIC 3.553184
log(Z) -842.091925
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 MultiNest v3.10
 Copyright Farhan Feroz & Mike Hobson
 Release Jul 2015

 no. of live points =  500
 dimensionality =    4
 *****************************************************
  analysing data from chains/fit-.txt ln(ev)=  -2054.1984688834614      +/-  0.18828794846549968
 Total Likelihood Evaluations:        18927
 Sampling finished. Exiting MultiNest

22:24:30 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 (1.54 -0.11 +0.14) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.4 -0.7 +0.6) x 10^-1
grb.spectrum.main.Band.xp (3.7 -0.4 +0.5) x 10^2 keV
grb.spectrum.main.Band.beta -2.30 -0.31 +0.16
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval8 -702.199107
n3_interval8 -698.433899
n4_interval8 -666.140065
total -2066.773070
Values of statistical measures:

statistical measures
AIC 4141.659455
BIC 4157.068272
DIC 4097.941347
PDIC 3.260445
log(Z) -892.127060
22:24:31 INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -1880.2421158807092      +/-  0.15228882294307208
 Total Likelihood Evaluations:        12421
 Sampling finished. Exiting MultiNest

22:24:39 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 (1.1 -0.4 +3.4) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.5 -2.2 +5) x 10^-1
grb.spectrum.main.Band.xp (1.08 -0.6 +0.30) x 10^2 keV
grb.spectrum.main.Band.beta -1.83 -0.12 +0.15
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval9 -648.458047
n3_interval9 -616.805699
n4_interval9 -616.134905
total -1881.398650
Values of statistical measures:

statistical measures
AIC 3770.910615
BIC 3786.319433
DIC 3584.983587
PDIC -162.092411
log(Z) -816.578776
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -1323.2094660475673      +/-  0.17574782447782100
 Total Likelihood Evaluations:        15008
 Sampling finished. Exiting MultiNest

22:24:48 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 (2.05 -0.5 +0.05) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-7.26 -1.8 +0.15) x 10^-1
grb.spectrum.main.Band.xp (2.22396 -0.00008 +0.9) x 10^2 keV
grb.spectrum.main.Band.beta -2.09 -0.6 -0.10
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval10 -460.609259
n3_interval10 -438.037863
n4_interval10 -433.535092
total -1332.182215
Values of statistical measures:

statistical measures
AIC 2672.477744
BIC 2687.886561
DIC 2636.550302
PDIC 2.103052
log(Z) -574.662570
         INFO      Range 9-900 translates to channels 5-124                                    SpectrumLike.py:1247
         INFO      Now using 120 bins                                                          SpectrumLike.py:1739
         INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
         INFO      sampler set to multinest                                                bayesian_analysis.py:202
 *****************************************************
 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)=  -811.75183122626038      +/-  0.14563134237021672
 Total Likelihood Evaluations:        12375
 Sampling finished. Exiting MultiNest

22:24:56 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 (2.8 -0.7 +2.5) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-5.1 -1.7 +4) x 10^-1
grb.spectrum.main.Band.xp (1.25 -0.26 +0.29) x 10^2 keV
grb.spectrum.main.Band.beta -2.06 -0.5 +0.13
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval11 -292.200272
n3_interval11 -272.187398
n4_interval11 -255.876508
total -820.264178
Values of statistical measures:

statistical measures
AIC 1648.641671
BIC 1664.050488
DIC 1615.659897
PDIC -1.786278
log(Z) -352.539341

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
         INFO      fit restored to median of posterior                                          sampler_base.py:164
22:24:57 INFO      fit restored to median of posterior                                          sampler_base.py:164
22:24:58 INFO      fit restored to median of posterior                                          sampler_base.py:164
         INFO      fit restored to median of posterior                                          sampler_base.py:164
22:24:59 INFO      fit restored to median of posterior                                          sampler_base.py:164
         INFO      fit restored to median of posterior                                          sampler_base.py:164
22:25:00 INFO      fit restored to median of posterior                                          sampler_base.py:164
22:25:01 INFO      fit restored to median of posterior                                          sampler_base.py:164
         INFO      fit restored to median of posterior                                          sampler_base.py:164
22:25:02 INFO      fit restored to median of posterior                                          sampler_base.py:164
22:25:03 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.