# 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.

[5]:

%matplotlib inline
import matplotlib.pyplot as plt
from jupyterthemes import jtplot

jtplot.style(context="talk", fscale=1, ticks=True, grid=False)
plt.style.use("mike")

from threeML import *
from threeML.io.package_data import get_path_of_data_file

import os

import warnings

warnings.simplefilter("ignore")


## 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¶

[7]:

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()

Auto-probed noise models:
- observation: gaussian
- background: None
Range 15-150 translates to channels 3-62
Now using 60 channels out of 80


### Fermi GBM¶

[8]:

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()

Auto-probed noise models:
- observation: poisson
- background: gaussian
Range 8-900 translates to channels 2-124
Now using 123 channels out of 128
Auto-probed noise models:
- observation: poisson
- background: gaussian
Range 250-30000 translates to channels 1-119
Now using 119 channels out of 128


## 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.

[9]:

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¶

[10]:

data_list = DataList(bat, nai6, bgo0)

jl_no_eac = JointLikelihood(model_no_eac, data_list)

jl_no_eac.fit()

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 0.9 -0.85 0.14 0.9 1 -0.73 0.08 -0.85 -0.73 1 -0.32 0.14 0.08 -0.32 1

Values of -log(likelihood) at the minimum:


-log(likelihood)
BAT 53.044658
b0 553.071650
n6 761.174809
total 1367.291118

Values of statistical measures:


statistical measures
AIC 2742.716915
BIC 2757.423943

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.

[11]:

threeML_config["ogip"]["model plot cmap"] = "Set1"

[12]:

display_spectrum_model_counts(
jl_no_eac, step=False, min_rate=[0.01, 10.0, 10.0], data_colors=["grey", "k", "k"]
)


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.

[14]:

gof_object = GoodnessOfFit(jl_no_eac)

with parallel_computation():

gof, res_frame, lh_frame = gof_object.by_mc(n_iterations=8000)

Starting ipyparallel cluster with this command line:
/Users/jburgess/.environs/3ml/bin/ipcluster start
Waiting for connection file: ~/.ipython/profile_default/security/ipcontroller-client.json
12 engines are active


Shutting down ipcluster...

[15]:

import pandas as pd
pd.Series(gof)

[15]:

total    0.000000
BAT      0.000125
n6       0.000000
b0       0.560125
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

[16]:

# 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()

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 0.95 -0.92 0.31 0.29 0.62 0.95 1 -0.82 0.26 0.26 0.52 -0.92 -0.82 1 -0.36 -0.45 -0.77 0.31 0.26 -0.36 1 0.04 -0.03 0.29 0.26 -0.45 0.04 1 0.47 0.62 0.52 -0.77 -0.03 0.47 1

Values of -log(likelihood) at the minimum:


-log(likelihood)
BAT 40.078874
b0 544.747650
n6 644.705650
total 1229.532174

Values of statistical measures:


statistical measures
AIC 2471.349094
BIC 2493.326910

Now we have a much better fit to all data sets

[17]:

display_spectrum_model_counts(
jl_eac, step=False, min_rate=[0.01, 10.0, 10.0], data_colors=["grey", "k", "k"]
)

[18]:

gof_object = GoodnessOfFit(jl_eac)

with parallel_computation():

gof, res_frame, lh_frame = gof_object.by_mc(
n_iterations=8000, continue_on_failure=True
)

Starting ipyparallel cluster with this command line:
/Users/jburgess/.environs/3ml/bin/ipcluster start
Waiting for connection file: ~/.ipython/profile_default/security/ipcontroller-client.json
12 engines are active


Shutting down ipcluster...

[19]:

import pandas as pd
pd.Series(gof)

[19]:

total    0.048750
BAT      0.028375
n6       0.017500
b0       0.841625
dtype: float64


## Examining the differences¶

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

[20]:

plot_spectra(
jl_eac.results,
jl_no_eac.results,
fit_cmap="Set1",
contour_cmap="Set1",
flux_unit="erg2/(keV s cm2)",
equal_tailed=True,
)


We can easily see that the models are different