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")

22:53:35 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

name	ra	dec	trigger_time	t90
object	float64	float64	float64	float64
GRB080916009	119.800	-56.600	54725.0088613	62.977

To aid in quickly replicating the catalog analysis, and thanks to the tireless efforts of the Fermi-GBM team, we have added the ability to extract the analysis parameters from the catalog as well as build an astromodels model with the best fit parameters baked in. Using this information, one can quickly run through the catalog an replicate the entire analysis with a script. Let’s give it a try.

[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)

22:54:33 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:389

22:54:42 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

22:54:45 INFO      Now using 120 bins                                                          SpectrumLike.py:1739

22:54:46 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389

22:54:55 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

22:54:56 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

22:54:57 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389

22:55:06 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

22:55:07 INFO      Successfully restored fit from b0_bkg.h5                              time_series_builder.py:171

         INFO      Interval set to 1.28-64.257 for b0                                    time_series_builder.py:290

         INFO      Auto-probed noise models:                                                    SpectrumLike.py: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)

22:55:08 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)=  -3101.4147604990776      +/-  0.22583221277586474
 Total Likelihood Evaluations:        21832
 Sampling finished. Exiting MultiNest

22:55:23 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	(1.457 -0.007 +0.031) x 10^-2	1 / (keV s cm2)
GRB080916009...alpha	-1.100 +0.006 +0.04
GRB080916009...break_energy	(1.96 +0.06 +0.7) x 10^2	keV
GRB080916009...break_scale	(0.0 +1.7 +3.5) x 10^-1
GRB080916009...beta	-2.001 -0.20 -0.009

Values of -log(posterior) at the minimum:

	-log(posterior)
n3	-1019.445895
n4	-1010.659579
b0	-1049.337488
total	-3079.442963

Values of statistical measures:

	statistical measures
AIC	6169.056380
BIC	6188.288590
DIC	6179.765394
PDIC	4.262033
log(Z)	-1346.927317

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)

22:57:18 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")

22:57:19 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

         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

22:57:20 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

22:57:21 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)=  -788.79338610399054      +/-  0.17689370891985945
 Total Likelihood Evaluations:        17226
 Sampling finished. Exiting MultiNest

22:57:31 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(3.49 -0.04 +0.9) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-5.64 -0.27 +1.9) x 10^-1
grb.spectrum.main.Band.xp	(3.24 -0.7 +0.06) x 10^2	keV
grb.spectrum.main.Band.beta	-2.08 -0.34 +0.19

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval0	-250.272823
n4_interval0	-268.094477
b0_interval0	-285.698594
total	-804.065894

Values of statistical measures:

	statistical measures
AIC	1616.245103
BIC	1631.653920
DIC	1571.154073
PDIC	2.991606
log(Z)	-342.568615

         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)=  -1943.3619960405456      +/-  0.21381446103098059
 Total Likelihood Evaluations:        22372
 Sampling finished. Exiting MultiNest

22:57:45 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(4.14 -0.15 +0.11) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-8.54 -0.28 +0.20) x 10^-1
grb.spectrum.main.Band.xp	(6.1 -0.4 +0.6) x 10^2	keV
grb.spectrum.main.Band.beta	-2.12 -0.13 +0.04

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval1	-641.688221
n4_interval1	-645.266394
b0_interval1	-674.051042
total	-1961.005657

Values of statistical measures:

	statistical measures
AIC	3930.124628
BIC	3945.533446
DIC	3872.691790
PDIC	3.335875
log(Z)	-843.991391

         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)=  -910.77774603071293      +/-  0.19994929621980959
 Total Likelihood Evaluations:        18968
 Sampling finished. Exiting MultiNest

22:57:57 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	(2.547 +0.019 +0.6) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.06 -0.05 +0.11
grb.spectrum.main.Band.xp	(5.20 -2.1 -0.18) x 10^2	keV
grb.spectrum.main.Band.beta	-1.77 -0.05 +0.10

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval2	-289.412447
n4_interval2	-311.185694
b0_interval2	-324.649771
total	-925.247913

Values of statistical measures:

	statistical measures
AIC	1858.609140
BIC	1874.017957
DIC	1806.181643
PDIC	1.822992
log(Z)	-395.545749

         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)=  -789.42386373107661      +/-  0.18029091349770524
 Total Likelihood Evaluations:        17201
 Sampling finished. Exiting MultiNest

22:58:08 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.87 +0.07 +0.7) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.31 +0.05 +1.3) x 10^-1
grb.spectrum.main.Band.xp	(3.38 -0.8 +0.16) x 10^2	keV
grb.spectrum.main.Band.beta	-2.20 -0.4 +0.19

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval3	-242.472076
n4_interval3	-262.423537
b0_interval3	-298.430546
total	-803.326160

Values of statistical measures:

	statistical measures
AIC	1614.765634
BIC	1630.174452
DIC	1569.562460
PDIC	2.229847
log(Z)	-342.842428

         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)=  -2275.4704032178838      +/-  0.20935514975514879
 Total Likelihood Evaluations:        20404
 Sampling finished. Exiting MultiNest

22:58:22 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	(2.17 +0.04 +0.15) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.469 -0.006 +0.5) x 10^-1
grb.spectrum.main.Band.xp	(3.58 -0.4 -0.06) x 10^2	keV
grb.spectrum.main.Band.beta	-1.982 -0.11 +0.020

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval4	-757.966297
n4_interval4	-745.857398
b0_interval4	-778.296240
total	-2282.119935

Values of statistical measures:

	statistical measures
AIC	4572.353185
BIC	4587.762002
DIC	4529.849238
PDIC	2.089757
log(Z)	-988.224240

         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
 *****************************************************
 ln(ev)=  -1573.4935501546256      +/-  0.19339000303499421
 Total Likelihood Evaluations:        21719
 Sampling finished. Exiting MultiNest
  analysing data from chains/fit-.txt

22:58:34 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(2.80 -0.12 +0.21) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-9.1 -0.4 +0.5) x 10^-1
grb.spectrum.main.Band.xp	(4.2 -0.4 +0.5) x 10^2	keV
grb.spectrum.main.Band.beta	-2.10 -0.29 +0.06

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval5	-523.641205
n4_interval5	-527.509941
b0_interval5	-536.779211
total	-1587.930357

Values of statistical measures:

	statistical measures
AIC	3183.974028
BIC	3199.382846
DIC	3136.252219
PDIC	3.128852
log(Z)	-683.359566

         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)=  -1757.1252056269975      +/-  0.19832115961372146
 Total Likelihood Evaluations:        19110
 Sampling finished. Exiting MultiNest

22:58:48 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	(1.988 +0.015 +0.20) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-10.00 +0.08 +0.8) x 10^-1
grb.spectrum.main.Band.xp	(4.277 -0.7 +0.006) x 10^2	keV
grb.spectrum.main.Band.beta	-2.34 -0.10 +0.15

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval6	-584.351170
n4_interval6	-576.613364
b0_interval6	-609.313130
total	-1770.277664

Values of statistical measures:

	statistical measures
AIC	3548.668642
BIC	3564.077460
DIC	3499.506361
PDIC	2.174323
log(Z)	-763.109781

         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)=  -1939.9286572811206      +/-  0.19313111166414290
 Total Likelihood Evaluations:        20479
 Sampling finished. Exiting MultiNest

22:59:00 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	(1.69 -0.07 +0.12) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	-1.041 -0.027 +0.06
grb.spectrum.main.Band.xp	(4.2 -0.6 +0.4) x 10^2	keV
grb.spectrum.main.Band.beta	-2.24 -0.26 +0.16

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval7	-641.183275
n4_interval7	-650.021787
b0_interval7	-662.159851
total	-1953.364912

Values of statistical measures:

	statistical measures
AIC	3914.843139
BIC	3930.251957
DIC	3868.246841
PDIC	3.113085
log(Z)	-842.500311

         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)=  -2053.5691851408351      +/-  0.18521479853648190
 Total Likelihood Evaluations:        21680
 Sampling finished. Exiting MultiNest

22:59:12 INFO      fit restored to maximum of posterior                                         sampler_base.py:178

         INFO      fit restored to maximum of posterior                                         sampler_base.py:178

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
grb.spectrum.main.Band.K	(1.54 -0.14 +0.11) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-8.5 -0.7 +0.5) x 10^-1
grb.spectrum.main.Band.xp	(3.69 -0.34 +0.8) x 10^2	keV
grb.spectrum.main.Band.beta	-2.25 -0.5 +0.08

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval8	-698.295064
n4_interval8	-666.204615
b0_interval8	-702.077771
total	-2066.577450

Values of statistical measures:

	statistical measures
AIC	4141.268214
BIC	4156.677032
DIC	4098.702490
PDIC	3.564441
log(Z)	-891.853765

22:59:13 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)=  -1879.5451611770859      +/-  0.14879154675457013
 Total Likelihood Evaluations:        12750
 Sampling finished. Exiting MultiNest

22:59:21 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	(1.11 -0.23 +1.8) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-8.6 -1.5 +4) x 10^-1
grb.spectrum.main.Band.xp	(1.12 -0.5 +0.16) x 10^2	keV
grb.spectrum.main.Band.beta	-1.86 -0.10 +0.15

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval9	-616.905387
n4_interval9	-616.282143
b0_interval9	-648.356373
total	-1881.543902

Values of statistical measures:

	statistical measures
AIC	3771.201119
BIC	3786.609936
DIC	3704.503269
PDIC	-42.190356
log(Z)	-816.276092

         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)=  -1322.8381247031502      +/-  0.17372093758967122
 Total Likelihood Evaluations:        14944
 Sampling finished. Exiting MultiNest

22:59:30 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	(2.05 -0.4 +0.14) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-7.4 -1.5 +0.5) x 10^-1
grb.spectrum.main.Band.xp	(2.20 -0.08 +0.8) x 10^2	keV
grb.spectrum.main.Band.beta	-1.95 -0.5 -0.13

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval10	-437.705516
n4_interval10	-433.199172
b0_interval10	-460.869759
total	-1331.774447

Values of statistical measures:

	statistical measures
AIC	2671.662209
BIC	2687.071027
DIC	2635.635894
PDIC	2.018347
log(Z)	-574.501298

         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)=  -812.76265073548689      +/-  0.14966112866848980
 Total Likelihood Evaluations:        12504
 Sampling finished. Exiting MultiNest

22:59:38 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	(2.9 -1.1 +4) x 10^-2	1 / (keV s cm2)
grb.spectrum.main.Band.alpha	(-4.8 -3.1 +4) x 10^-1
grb.spectrum.main.Band.xp	(1.26 -0.4 +0.30) x 10^2	keV
grb.spectrum.main.Band.beta	-2.07 -0.35 +0.25

Values of -log(posterior) at the minimum:

	-log(posterior)
n3_interval11	-272.544506
n4_interval11	-255.657502
b0_interval11	-292.301143
total	-820.503151

Values of statistical measures:

	statistical measures
AIC	1649.119616
BIC	1664.528433
DIC	1608.672364
PDIC	-9.176611
log(Z)	-352.978334

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

22:59:40 INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:41 INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:44 INFO      fit restored to median of posterior                                          sampler_base.py:164

         INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:45 INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:46 INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:47 INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:48 INFO      fit restored to median of posterior                                          sampler_base.py:164

         INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:49 INFO      fit restored to median of posterior                                          sampler_base.py:164

22:59:50 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.