Example joint fit between GBM and Swift BAT

One of the key features of 3ML is the abil ity to fit multi-messenger data properly. A simple example of this is the joint fitting of two instruments whose data obey different likelihoods. Here, we have GBM data which obey a Poisson-Gaussian profile likelihoog ( PGSTAT in XSPEC lingo) and Swift BAT which data which are the result of a “fit” via a coded mask and hence obey a Gaussian ( \(\chi^2\) ) likelihood.

[1]:
import warnings

warnings.simplefilter("ignore")
import numpy as np

np.seterr(all="ignore")
[1]:
{'divide': 'warn', 'over': 'warn', 'under': 'ignore', 'invalid': 'warn'}
[2]:
%%capture
import matplotlib.pyplot as plt

np.random.seed(12345)
from threeML import *
from threeML.io.package_data import get_path_of_data_file
from threeML.io.logging import silence_console_log
[3]:
from jupyterthemes import jtplot

%matplotlib inline
jtplot.style(context="talk", fscale=1, ticks=True, grid=False)
set_threeML_style()
silence_warnings()

Plugin setup

We have data from the same time interval from Swift BAT and a GBM NAI and BGO detector. We have preprocessed GBM data to so that it is OGIP compliant. (Remember that we can handle the raw data with the TimeSeriesBuilder). Thus, we will use the OGIPLike plugin to read in each dataset, make energy selections and examine the raw count spectra.

Swift BAT

[4]:
bat_pha = get_path_of_data_file("datasets/bat/gbm_bat_joint_BAT.pha")
bat_rsp = get_path_of_data_file("datasets/bat/gbm_bat_joint_BAT.rsp")

bat = OGIPLike("BAT", observation=bat_pha, response=bat_rsp)

bat.set_active_measurements("15-150")
bat.view_count_spectrum()
10:12:07 INFO      Auto-probed noise models:                                                    SpectrumLike.py:491
         INFO      - observation: gaussian                                                      SpectrumLike.py:492
         INFO      - background: None                                                           SpectrumLike.py:493
         INFO      Range 15-150 translates to channels 3-62                                    SpectrumLike.py:1245
[4]:
../_images/notebooks_joint_BAT_gbm_demo_7_4.png
../_images/notebooks_joint_BAT_gbm_demo_7_5.png

Fermi GBM

[5]:
nai6 = OGIPLike(
    "n6",
    get_path_of_data_file("datasets/gbm/gbm_bat_joint_NAI_06.pha"),
    get_path_of_data_file("datasets/gbm/gbm_bat_joint_NAI_06.bak"),
    get_path_of_data_file("datasets/gbm/gbm_bat_joint_NAI_06.rsp"),
    spectrum_number=1,
)


nai6.set_active_measurements("8-900")
nai6.view_count_spectrum()

bgo0 = OGIPLike(
    "b0",
    get_path_of_data_file("datasets/gbm/gbm_bat_joint_BGO_00.pha"),
    get_path_of_data_file("datasets/gbm/gbm_bat_joint_BGO_00.bak"),
    get_path_of_data_file("datasets/gbm/gbm_bat_joint_BGO_00.rsp"),
    spectrum_number=1,
)

bgo0.set_active_measurements("250-30000")
bgo0.view_count_spectrum()
10:12:10 INFO      Auto-probed noise models:                                                    SpectrumLike.py:491
         INFO      - observation: poisson                                                       SpectrumLike.py:492
         INFO      - background: gaussian                                                       SpectrumLike.py:493
         INFO      Range 8-900 translates to channels 2-124                                    SpectrumLike.py:1245
         INFO      Auto-probed noise models:                                                    SpectrumLike.py:491
         INFO      - observation: poisson                                                       SpectrumLike.py:492
         INFO      - background: gaussian                                                       SpectrumLike.py:493
         INFO      Range 250-30000 translates to channels 1-119                                SpectrumLike.py:1245
[5]:
../_images/notebooks_joint_BAT_gbm_demo_9_8.png
../_images/notebooks_joint_BAT_gbm_demo_9_9.png
../_images/notebooks_joint_BAT_gbm_demo_9_10.png

Model setup

We setup up or spectrum and likelihood model and combine the data. 3ML will automatically assign the proper likelihood to each data set. At first, we will assume a perfect calibration between the different detectors and not a apply a so-called effective area correction.

[6]:
band = Band()

model_no_eac = Model(PointSource("joint_fit_no_eac", 0, 0, spectral_shape=band))

Spectral fitting

Now we simply fit the data by building the data list, creating the joint likelihood and running the fit.

No effective area correction

[7]:
data_list = DataList(bat, nai6, bgo0)

jl_no_eac = JointLikelihood(model_no_eac, data_list)

jl_no_eac.fit()
10:12:13 INFO      set the minimizer to minuit                                             joint_likelihood.py:1042
Best fit values:

result unit
parameter
joint_fit_no_eac.spectrum.main.Band.K (2.75 +/- 0.06) x 10^-2 1 / (cm2 keV s)
joint_fit_no_eac.spectrum.main.Band.alpha -1.029 +/- 0.017
joint_fit_no_eac.spectrum.main.Band.xp (5.7 +/- 0.4) x 10^2 keV
joint_fit_no_eac.spectrum.main.Band.beta -2.41 +/- 0.18

Correlation matrix:

1.000.90-0.850.14
0.901.00-0.730.08
-0.85-0.731.00-0.32
0.140.08-0.321.00

Values of -log(likelihood) at the minimum:

-log(likelihood)
BAT 53.039963
b0 553.071103
n6 761.180052
total 1367.291118

Values of statistical measures:

statistical measures
AIC 2742.716917
BIC 2757.423945
[7]:
(                                                value  negative_error  \
 joint_fit_no_eac.spectrum.main.Band.K        0.027472       -0.000556
 joint_fit_no_eac.spectrum.main.Band.alpha   -1.029161       -0.016648
 joint_fit_no_eac.spectrum.main.Band.xp     567.674652      -37.852371
 joint_fit_no_eac.spectrum.main.Band.beta    -2.409618       -0.178302

                                            positive_error      error  \
 joint_fit_no_eac.spectrum.main.Band.K            0.000581   0.000568
 joint_fit_no_eac.spectrum.main.Band.alpha        0.016915   0.016782
 joint_fit_no_eac.spectrum.main.Band.xp          34.829552  36.340961
 joint_fit_no_eac.spectrum.main.Band.beta         0.185679   0.181990

                                                       unit
 joint_fit_no_eac.spectrum.main.Band.K      1 / (cm2 keV s)
 joint_fit_no_eac.spectrum.main.Band.alpha
 joint_fit_no_eac.spectrum.main.Band.xp                 keV
 joint_fit_no_eac.spectrum.main.Band.beta                    ,
        -log(likelihood)
 BAT           53.039963
 b0           553.071103
 n6           761.180052
 total       1367.291118)

The fit has resulted in a very typical Band function fit. Let’s look in count space at how good of a fit we have obtained.

[8]:
threeML_config.plugins.ogip.fit_plot.model_cmap = "Set1"
threeML_config.plugins.ogip.fit_plot.n_colors = 3
display_spectrum_model_counts(
    jl_no_eac,
    min_rate=[0.01, 10.0, 10.0],
    data_colors=["grey", "k", "k"],
    show_background=False,
    source_only=True,
)
[8]:
../_images/notebooks_joint_BAT_gbm_demo_16_0.png
../_images/notebooks_joint_BAT_gbm_demo_16_1.png

It seems that the effective areas between GBM and BAT do not agree! We can look at the goodness of fit for the various data sets.

[9]:
gof_object = GoodnessOfFit(jl_no_eac)

gof, res_frame, lh_frame = gof_object.by_mc(n_iterations=100)
[10]:
import pandas as pd

pd.Series(gof)
[10]:
total    0.00
BAT      0.00
n6       0.00
b0       0.57
dtype: float64

Both the GBM NaI detector and Swift BAT exhibit poor GOF.

With effective are correction

Now let’s add an effective area correction between the detectors to see if this fixes the problem. The effective area is a nuissance parameter that attempts to model systematic problems in a instruments calibration. It simply scales the counts of an instrument by a multiplicative factor. It cannot handle more complicated energy dependent

[11]:
# turn on the effective area correction and set it's bounds
nai6.use_effective_area_correction(0.2, 1.8)
bgo0.use_effective_area_correction(0.2, 1.8)

model_eac = Model(PointSource("joint_fit_eac", 0, 0, spectral_shape=band))

jl_eac = JointLikelihood(model_eac, data_list)

jl_eac.fit()
10:12:55 INFO      n6 is using effective area correction (between 0.2 and 1.8)                 SpectrumLike.py:2232
         INFO      b0 is using effective area correction (between 0.2 and 1.8)                 SpectrumLike.py:2232
         INFO      set the minimizer to minuit                                             joint_likelihood.py:1042
Best fit values:

result unit
parameter
joint_fit_eac.spectrum.main.Band.K (2.98 +/- 0.12) x 10^-2 1 / (cm2 keV s)
joint_fit_eac.spectrum.main.Band.alpha (-9.84 +/- 0.26) x 10^-1
joint_fit_eac.spectrum.main.Band.xp (3.31 +/- 0.32) x 10^2 keV
joint_fit_eac.spectrum.main.Band.beta -2.36 +/- 0.15
cons_n6 1.56 +/- 0.04
cons_b0 1.41 +/- 0.10

Correlation matrix:

1.000.95-0.920.310.290.62
0.951.00-0.820.260.260.52
-0.92-0.821.00-0.36-0.45-0.77
0.310.26-0.361.000.04-0.03
0.290.26-0.450.041.000.47
0.620.52-0.77-0.030.471.00

Values of -log(likelihood) at the minimum:

-log(likelihood)
BAT 40.078495
b0 544.747683
n6 644.705993
total 1229.532171

Values of statistical measures:

statistical measures
AIC 2471.349089
BIC 2493.326905
[11]:
(                                             value  negative_error  \
 joint_fit_eac.spectrum.main.Band.K        0.029814       -0.001162
 joint_fit_eac.spectrum.main.Band.alpha   -0.983921       -0.026346
 joint_fit_eac.spectrum.main.Band.xp     330.869922      -31.878541
 joint_fit_eac.spectrum.main.Band.beta    -2.357474       -0.152273
 cons_n6                                   1.562854       -0.036967
 cons_b0                                   1.408466       -0.093774

                                         positive_error      error  \
 joint_fit_eac.spectrum.main.Band.K            0.001171   0.001167
 joint_fit_eac.spectrum.main.Band.alpha        0.025927   0.026137
 joint_fit_eac.spectrum.main.Band.xp          31.039415  31.458978
 joint_fit_eac.spectrum.main.Band.beta         0.154106   0.153189
 cons_n6                                       0.039063   0.038015
 cons_b0                                       0.098213   0.095993

                                                    unit
 joint_fit_eac.spectrum.main.Band.K      1 / (cm2 keV s)
 joint_fit_eac.spectrum.main.Band.alpha
 joint_fit_eac.spectrum.main.Band.xp                 keV
 joint_fit_eac.spectrum.main.Band.beta
 cons_n6
 cons_b0                                                  ,
        -log(likelihood)
 BAT           40.078495
 b0           544.747683
 n6           644.705993
 total       1229.532171)

Now we have a much better fit to all data sets

[12]:
display_spectrum_model_counts(
    jl_eac, step=False, min_rate=[0.01, 10.0, 10.0], data_colors=["grey", "k", "k"]
)
[12]:
../_images/notebooks_joint_BAT_gbm_demo_24_0.png
../_images/notebooks_joint_BAT_gbm_demo_24_1.png
[13]:
gof_object = GoodnessOfFit(jl_eac)

# for display purposes we are keeping the output clear
# with silence_console_log(and_progress_bars=False):
gof, res_frame, lh_frame = gof_object.by_mc(n_iterations=100, continue_on_failure=True)
[14]:
import pandas as pd

pd.Series(gof)
[14]:
total    0.17
BAT      0.10
n6       0.17
b0       0.17
dtype: float64

Examining the differences

Let’s plot the fits in model space and see how different the resulting models are.

[15]:
plot_spectra(
    jl_eac.results,
    jl_no_eac.results,
    fit_cmap="Set1",
    contour_cmap="Set1",
    flux_unit="erg2/(keV s cm2)",
    equal_tailed=True,
)
[15]:
../_images/notebooks_joint_BAT_gbm_demo_28_3.png
../_images/notebooks_joint_BAT_gbm_demo_28_4.png

We can easily see that the models are different