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")
22:20:36 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)
22:21:51 INFO      Auto-determined polynomial order: 0                                binned_spectrum_series.py:389
22:22:06 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
22:22:07 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
22:22:11 INFO      Now using 120 bins                                                          SpectrumLike.py:1739
22:22:13 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389
22:22:31 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
22:22:32 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
22:22:33 INFO      Range 9-900 translates to channels 5-123                                    SpectrumLike.py:1247
         INFO      Now using 119 bins                                                          SpectrumLike.py:1739
22:22:35 INFO      Auto-determined polynomial order: 1                                binned_spectrum_series.py:389
22:22:48 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
22:22:50 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)
22:22:51 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
 *****************************************************
 ln(ev)=  -3101.1683501056382      +/-  0.22364591399313155
 Total Likelihood Evaluations:        26021
 Sampling finished. Exiting MultiNest
  analysing data from chains/fit-.txt
22:23:20 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
GRB080916009...K (1.460 -0.009 +0.028) x 10^-2 1 / (cm2 keV s)
GRB080916009...alpha -1.100 +0.006 +0.05
GRB080916009...break_energy (1.93 +0.08 +0.7) x 10^2 keV
GRB080916009...break_scale (0.0 +1.7 +3.4) x 10^-1
GRB080916009...beta -1.985 -0.20 -0.021
Values of -log(posterior) at the minimum:

-log(posterior)
b0 -1049.754437
n3 -1019.729631
n4 -1010.819791
total -3080.303860
Values of statistical measures:

statistical measures
AIC 6170.778174
BIC 6190.010384
DIC 6179.924198
PDIC 4.415255
log(Z) -1346.820302

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)
22:26:27 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)
22:26:28 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
22:26:29 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
22:26:31 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)=  -792.08146805331899      +/-  0.18730854093483151
 Total Likelihood Evaluations:        17575
 Sampling finished. Exiting MultiNest

22:26:54 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (3.6 -0.4 +0.8) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-5.9 -0.7 +1.6) x 10^-1
grb.spectrum.main.Band.xp (3.0 -0.7 +0.4) x 10^2 keV
grb.spectrum.main.Band.beta -1.93 +0.06 +0.17
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval0 -285.823958
n3_interval0 -250.217985
n4_interval0 -267.821247
total -803.863189
Values of statistical measures:

statistical measures
AIC 1615.839693
BIC 1631.248510
DIC 1571.526776
PDIC 1.615174
log(Z) -343.996611
         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)=  -1943.1120786496945      +/-  0.21175776784445846
 Total Likelihood Evaluations:        22856
 Sampling finished. Exiting MultiNest

22:27:13 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 (4.13 -0.15 +0.11) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.59 +/- 0.27) x 10^-1
grb.spectrum.main.Band.xp (6.2 -0.4 +0.6) x 10^2 keV
grb.spectrum.main.Band.beta -2.12 -0.13 +0.05
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval1 -673.868258
n3_interval1 -641.898348
n4_interval1 -645.307329
total -1961.073935
Values of statistical measures:

statistical measures
AIC 3930.261183
BIC 3945.670001
DIC 3873.542520
PDIC 3.765382
log(Z) -843.882853
         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)=  -908.11747218055962      +/-  0.20249622540226878
 Total Likelihood Evaluations:        19752
 Sampling finished. Exiting MultiNest

22:27:29 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 (2.47 -0.06 +0.26) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.068 -0.017 +0.06
grb.spectrum.main.Band.xp (6.4 -1.4 +1.1) x 10^2 keV
grb.spectrum.main.Band.beta -2.009 -0.15 -0.028
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval2 -324.896288
n3_interval2 -289.551469
n4_interval2 -312.562137
total -927.009893
Values of statistical measures:

statistical measures
AIC 1862.133101
BIC 1877.541919
DIC 1803.994234
PDIC 1.771256
log(Z) -394.390407
         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)=  -789.34743473236176      +/-  0.18254457039820565
 Total Likelihood Evaluations:        17178
 Sampling finished. Exiting MultiNest

22:27:42 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 -0.14 +0.4) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.4 -0.4 +0.9) x 10^-1
grb.spectrum.main.Band.xp (3.5 -0.7 +0.4) x 10^2 keV
grb.spectrum.main.Band.beta -2.4 +/- 0.4
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval3 -298.582737
n3_interval3 -242.934924
n4_interval3 -262.738485
total -804.256146
Values of statistical measures:

statistical measures
AIC 1616.625607
BIC 1632.034424
DIC 1571.047683
PDIC 3.066061
log(Z) -342.809235
         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)=  -2272.4582432846742      +/-  0.20395918490392356
 Total Likelihood Evaluations:        19990
 Sampling finished. Exiting MultiNest

22:27:58 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 (2.06 -0.10 +0.20) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.87 -0.35 +0.7) x 10^-1
grb.spectrum.main.Band.xp (4.0 -0.7 +0.5) x 10^2 keV
grb.spectrum.main.Band.beta -1.97 -0.12 +0.08
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval4 -778.455816
n3_interval4 -757.333003
n4_interval4 -746.653370
total -2282.442189
Values of statistical measures:

statistical measures
AIC 4572.997693
BIC 4588.406510
DIC 4528.293503
PDIC 3.247572
log(Z) -986.916075
         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)=  -1573.5488480269184      +/-  0.19462444682275437
 Total Likelihood Evaluations:        19047
 Sampling finished. Exiting MultiNest

22:28:11 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 (2.82 -0.20 +0.18) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-9.0 -0.6 +0.4) x 10^-1
grb.spectrum.main.Band.xp (4.1 -0.4 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.09 -0.25 +0.04
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval5 -536.763116
n3_interval5 -523.668412
n4_interval5 -527.412708
total -1587.844236
Values of statistical measures:

statistical measures
AIC 3183.801786
BIC 3199.210604
DIC 3135.712641
PDIC 2.932422
log(Z) -683.383582
         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)=  -1755.2666430220588      +/-  0.19305821162533141
 Total Likelihood Evaluations:        18879
 Sampling finished. Exiting MultiNest

22:28:24 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.95 -0.11 +0.14) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.01 -0.05 +0.04
grb.spectrum.main.Band.xp (4.5 -0.6 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -2.34 -0.4 +0.13
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval6 -609.541735
n3_interval6 -584.279703
n4_interval6 -576.807411
total -1770.628849
Values of statistical measures:

statistical measures
AIC 3549.371011
BIC 3564.779829
DIC 3500.709670
PDIC 3.060047
log(Z) -762.302617
         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)=  -1940.2692099613960      +/-  0.19928571331533931
 Total Likelihood Evaluations:        20914
 Sampling finished. Exiting MultiNest

22:28:38 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 (1.708 -0.12 +0.011) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha -1.039 -0.04 +0.013
grb.spectrum.main.Band.xp (4.10 -0.08 +0.9) x 10^2 keV
grb.spectrum.main.Band.beta -2.4161 -0.27 -0.0033
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval7 -662.018270
n3_interval7 -642.127038
n4_interval7 -649.898686
total -1954.043993
Values of statistical measures:

statistical measures
AIC 3916.201301
BIC 3931.610118
DIC 3865.840324
PDIC 1.459761
log(Z) -842.648211
         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)=  -2053.6659707356666      +/-  0.18598700757708717
 Total Likelihood Evaluations:        20376
 Sampling finished. Exiting MultiNest

22:28:50 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
         INFO      fit restored to maximum of posterior                                         sampler_base.py:178
Maximum a posteriori probability (MAP) point:

result unit
parameter
grb.spectrum.main.Band.K (1.56 -0.16 +0.09) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.4 -0.8 +0.5) x 10^-1
grb.spectrum.main.Band.xp (3.67 -0.30 +0.8) x 10^2 keV
grb.spectrum.main.Band.beta -2.29 -0.4 +0.12
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval8 -702.177520
n3_interval8 -698.537758
n4_interval8 -665.930852
total -2066.646131
Values of statistical measures:

statistical measures
AIC 4141.405576
BIC 4156.814393
DIC 4098.527342
PDIC 3.462529
log(Z) -891.895799
         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)=  -1879.4586173419830      +/-  0.14727869751747899
 Total Likelihood Evaluations:        13488
 Sampling finished. Exiting MultiNest

22:28:59 INFO      fit restored to maximum of posterior                                         sampler_base.py:178
22:29:00 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 (1.04 -0.14 +2.5) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-8.8 -0.9 +5) x 10^-1
grb.spectrum.main.Band.xp (1.19 -0.6 +0.07) x 10^2 keV
grb.spectrum.main.Band.beta -1.87 -0.05 +0.20
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval9 -648.498454
n3_interval9 -617.060548
n4_interval9 -616.304404
total -1881.863405
Values of statistical measures:

statistical measures
AIC 3771.840125
BIC 3787.248942
DIC 3683.368392
PDIC -63.779047
log(Z) -816.238506
         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)=  -1322.3765215170556      +/-  0.16793115577943218
 Total Likelihood Evaluations:        15163
 Sampling finished. Exiting MultiNest

22:29:13 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 (2.1 -0.4 +0.6) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-7.3 -1.1 +1.9) x 10^-1
grb.spectrum.main.Band.xp (2.1 -0.4 +0.7) x 10^2 keV
grb.spectrum.main.Band.beta -1.93 -0.6 +0.04
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval10 -460.786545
n3_interval10 -437.848066
n4_interval10 -432.874692
total -1331.509303
Values of statistical measures:

statistical measures
AIC 2671.131920
BIC 2686.540738
DIC 2634.065523
PDIC -0.135360
log(Z) -574.300826
         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)=  -812.55499448671151      +/-  0.14939285761245039
 Total Likelihood Evaluations:        12492
 Sampling finished. Exiting MultiNest

22:29:23 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
grb.spectrum.main.Band.K (2.73 -0.31 +4) x 10^-2 1 / (cm2 keV s)
grb.spectrum.main.Band.alpha (-4.9 -1.3 +4) x 10^-1
grb.spectrum.main.Band.xp (1.312 -0.4 +0.009) x 10^2 keV
grb.spectrum.main.Band.beta -2.09 -0.06 +0.28
Values of -log(posterior) at the minimum:

-log(posterior)
b0_interval11 -292.387537
n3_interval11 -272.337305
n4_interval11 -255.949019
total -820.673860
Values of statistical measures:

statistical measures
AIC 1649.461035
BIC 1664.869852
DIC 1616.249716
PDIC -0.682543
log(Z) -352.888150

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
22:29:24 INFO      fit restored to median of posterior                                          sampler_base.py:164
22:29:28 INFO      fit restored to median of posterior                                          sampler_base.py:164
         INFO      fit restored to median of posterior                                          sampler_base.py:164
22:29:29 INFO      fit restored to median of posterior                                          sampler_base.py:164
22:29:30 INFO      fit restored to median of posterior                                          sampler_base.py:164
         INFO      fit restored to median of posterior                                          sampler_base.py:164
22:29:31 INFO      fit restored to median of posterior                                          sampler_base.py:164
22:29:32 INFO      fit restored to median of posterior                                          sampler_base.py:164
22:29:33 INFO      fit restored to median of posterior                                          sampler_base.py:164
         INFO      fit restored to median of posterior                                          sampler_base.py:164
22:29:34 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.