Model fitting 1: Only SSC¶

import warnings
warnings.filterwarnings('ignore')

import matplotlib.pylab as plt
import jetset
from jetset.test_data_helper import  test_SEDs
from jetset.data_loader import ObsData,Data
from jetset.plot_sedfit import PlotSED
from jetset.test_data_helper import  test_SEDs

print(jetset.__version__)

1.2.2

test_SEDs

['/Users/orion/anaconda3/envs/jetset/lib/python3.8/site-packages/jetset/test_data/SEDs_data/SED_3C345.ecsv',
 '/Users/orion/anaconda3/envs/jetset/lib/python3.8/site-packages/jetset/test_data/SEDs_data/SED_MW_Mrk421_EBL_DEABS.ecsv',
 '/Users/orion/anaconda3/envs/jetset/lib/python3.8/site-packages/jetset/test_data/SEDs_data/SED_MW_Mrk501_EBL_ABS.ecsv',
 '/Users/orion/anaconda3/envs/jetset/lib/python3.8/site-packages/jetset/test_data/SEDs_data/SED_MW_Mrk501_EBL_DEABS.ecsv']

Loading data¶

see the data_format user guide for further information about loading data

print(test_SEDs[1])
data=Data.from_file(test_SEDs[1])

/Users/orion/anaconda3/envs/jetset/lib/python3.8/site-packages/jetset/test_data/SEDs_data/SED_MW_Mrk421_EBL_DEABS.ecsv

%matplotlib inline
sed_data=ObsData(data_table=data)
sed_data.group_data(bin_width=0.2)

sed_data.add_systematics(0.1,[10.**6,10.**29])
p=sed_data.plot_sed()
#p.setlim(y_min=1E-15,x_min=1E7,x_max=1E29)

================================================================================

*  binning data  *
---> N bins= 89
---> bin_widht= 0.2
================================================================================

../../../_images/Jet_example_model_fit_8_1.png

sed_data.save('Mrk_401.pkl')

phenomenological model constraining¶

see the Phenomenological model constraining: application user guide for further information about phenomenological constraining

spectral indices¶

from jetset.sed_shaper import  SEDShape
my_shape=SEDShape(sed_data)
my_shape.eval_indices(minimizer='lsb',silent=True)
p=my_shape.plot_indices()
p.setlim(y_min=1E-15,y_max=5E-8)

================================================================================

* evaluating spectral indices for data *
================================================================================

../../../_images/Jet_example_model_fit_13_1.png

sed shaper¶

mm,best_fit=my_shape.sync_fit(check_host_gal_template=False,
                  Ep_start=None,
                  minimizer='lsb',
                  silent=True,
                  fit_range=[10.,21.])

================================================================================

* Log-Polynomial fitting of the synchrotron component *
---> first blind fit run,  fit range: [10.0, 21.0]
---> class:  HSP

Table length=4

model name	name	val	bestfit val	err +	err -	start val	fit range min	fit range max	frozen
LogCubic	b	-1.545300e-01	-1.545300e-01	9.534795e-03	--	-1.000000e+00	-1.000000e+01	0.000000e+00	False
LogCubic	c	-1.023245e-02	-1.023245e-02	1.433073e-03	--	-1.000000e+00	-1.000000e+01	1.000000e+01	False
LogCubic	Ep	1.672267e+01	1.672267e+01	4.139942e-02	--	1.667039e+01	0.000000e+00	3.000000e+01	False
LogCubic	Sp	-9.491659e+00	-9.491659e+00	2.515285e-02	--	-1.000000e+01	-3.000000e+01	0.000000e+00	False

---> sync       nu_p=+1.672267e+01 (err=+4.139942e-02)  nuFnu_p=-9.491659e+00 (err=+2.515285e-02) curv.=-1.545300e-01 (err=+9.534795e-03)
================================================================================

my_shape.IC_fit(fit_range=[23.,29.],minimizer='minuit',silent=True)
p=my_shape.plot_shape_fit()
p.setlim(y_min=1E-15,y_max=5E-8)

================================================================================

* Log-Polynomial fitting of the IC component *
---> fit range: [23.0, 29.0]
---> LogCubic fit

Table length=4

model name	name	val	bestfit val	err +	err -	start val	fit range min	fit range max	frozen
LogCubic	b	-2.098186e-01	-2.098186e-01	3.133032e-02	--	-1.000000e+00	-1.000000e+01	0.000000e+00	False
LogCubic	c	-4.661868e-02	-4.661868e-02	2.178352e-02	--	-1.000000e+00	-1.000000e+01	1.000000e+01	False
LogCubic	Ep	2.524926e+01	2.524926e+01	1.147759e-01	--	2.529412e+01	0.000000e+00	3.000000e+01	False
LogCubic	Sp	-1.011085e+01	-1.011085e+01	3.498963e-02	--	-1.000000e+01	-3.000000e+01	0.000000e+00	False

---> IC         nu_p=+2.524926e+01 (err=+1.147759e-01)  nuFnu_p=-1.011085e+01 (err=+3.498963e-02) curv.=-2.098186e-01 (err=+3.133032e-02)
================================================================================

../../../_images/Jet_example_model_fit_16_3.png

Model constraining¶

In this step we are not fitting the model, we are just obtaining the phenomenological pre_fit model, that will be fitted in using minuit ore least-square bound, as shown below

from jetset.obs_constrain import ObsConstrain
from jetset.model_manager import  FitModel
sed_obspar=ObsConstrain(beaming=25,
                        B_range=[0.001,0.1],
                        distr_e='lppl',
                        t_var_sec=3*86400,
                        nu_cut_IR=1E12,
                        SEDShape=my_shape)


prefit_jet=sed_obspar.constrain_SSC_model(electron_distribution_log_values=False,silent=True)
prefit_jet.save_model('prefit_jet.pkl')

================================================================================

*  constrains parameters from observable *

Table length=12

model name	name	par type	units	val	phys. bound. min	phys. bound. max	log	frozen
jet_leptonic	R	region_size	cm	3.112712e+16	1.000000e+03	1.000000e+30	False	False
jet_leptonic	R_H	region_position	cm	1.000000e+17	0.000000e+00	--	False	True
jet_leptonic	B	magnetic_field	gauss	5.050000e-02	0.000000e+00	--	False	False
jet_leptonic	NH_cold_to_rel_e	cold_p_to_rel_e_ratio		1.000000e-01	0.000000e+00	--	False	True
jet_leptonic	beam_obj	beaming	lorentz-factor*	2.500000e+01	1.000000e-04	--	False	False
jet_leptonic	z_cosm	redshift		3.080000e-02	0.000000e+00	--	False	False
jet_leptonic	gmin	low-energy-cut-off	lorentz-factor*	4.697542e+02	1.000000e+00	1.000000e+09	False	False
jet_leptonic	gmax	high-energy-cut-off	lorentz-factor*	1.373160e+06	1.000000e+00	1.000000e+15	False	False
jet_leptonic	N	emitters_density	1 / cm3	9.060842e-01	0.000000e+00	--	False	False
jet_leptonic	gamma0_log_parab	turn-over-energy	lorentz-factor*	3.188500e+04	1.000000e+00	1.000000e+09	False	False
jet_leptonic	s	LE_spectral_slope		2.181578e+00	-1.000000e+01	1.000000e+01	False	False
jet_leptonic	r	spectral_curvature		7.726502e-01	-1.500000e+01	1.500000e+01	False	False

================================================================================

prefit_jet.eval()
pl=prefit_jet.plot_model(sed_data=sed_data)
pl.add_residual_plot(prefit_jet,sed_data)
pl.setlim(y_min=1E-15,x_min=1E7,x_max=1E29)

../../../_images/Jet_example_model_fit_20_0.png

Model fitting procedure¶

We remind that we can use different minimizers for the model fitting. In the following we will use the minuit minimizer and the lsb (least square bound scipy minimizer). Using minuit we notice that sometimes the fit will converge, but the quality will not be enough (valid==false) to run minos. Anyhow, as shown in the MCMC sampling, it still possible to estimate asymmetric errors by means of MCMC sampling

We freeze some parameters, and we also set some fit_range values. Setting fit_range can speed-up the fit convergence but should be judged by the user each time according to the physics of the particular source.

When using minuit the best strategy is to set the fit_range for most of the free parameters

A good strategy is to run first a lsb fit and then, using the same fit_model, run a fit with minuit

Model fitting with LSB¶

see the Composite Models and depending pars user guide for further information about the new implementation of FitModel, in particular for parameter setting

from jetset.minimizer import fit_SED,ModelMinimizer

from jetset.model_manager import  FitModel
from jetset.jet_model import Jet

if you want to fit the prefit_model you can load the saved one (this allows you to save time) ad pass it to the FitModel class

prefit_jet=Jet.load_model('prefit_jet.pkl')
fit_model_lsb=FitModel( jet=prefit_jet, name='SSC-best-fit-lsb',template=None)

Table length=12

model name	name	par type	units	val	phys. bound. min	phys. bound. max	log	frozen
jet_leptonic	gmin	low-energy-cut-off	lorentz-factor*	4.697542e+02	1.000000e+00	1.000000e+09	False	False
jet_leptonic	gmax	high-energy-cut-off	lorentz-factor*	1.373160e+06	1.000000e+00	1.000000e+15	False	False
jet_leptonic	N	emitters_density	1 / cm3	9.060842e-01	0.000000e+00	--	False	False
jet_leptonic	gamma0_log_parab	turn-over-energy	lorentz-factor*	3.188500e+04	1.000000e+00	1.000000e+09	False	False
jet_leptonic	s	LE_spectral_slope		2.181578e+00	-1.000000e+01	1.000000e+01	False	False
jet_leptonic	r	spectral_curvature		7.726502e-01	-1.500000e+01	1.500000e+01	False	False
jet_leptonic	R	region_size	cm	3.112712e+16	1.000000e+03	1.000000e+30	False	False
jet_leptonic	R_H	region_position	cm	1.000000e+17	0.000000e+00	--	False	True
jet_leptonic	B	magnetic_field	gauss	5.050000e-02	0.000000e+00	--	False	False
jet_leptonic	NH_cold_to_rel_e	cold_p_to_rel_e_ratio		1.000000e-01	0.000000e+00	--	False	True
jet_leptonic	beam_obj	beaming	lorentz-factor*	2.500000e+01	1.000000e-04	--	False	False
jet_leptonic	z_cosm	redshift		3.080000e-02	0.000000e+00	--	False	False

OR use the one generated above

fit_model_lsb=FitModel( jet=prefit_jet, name='SSC-best-fit-lsb',template=None)

fit_model_lsb.show_model_components()

--------------------------------------------------------------------------------
Composite model description
--------------------------------------------------------------------------------
name: SSC-best-fit-lsb
type: composite_model
components models:
 -model name: jet_leptonic model type: jet

--------------------------------------------------------------------------------

There is only one component, whit name jet_leptonic, that refers to the prefit_jet model component

We now set the gamma grid size to 200, ad we set composite_expr, anyhow, since we have only one component this step could be skipped

fit_model_lsb.jet_leptonic.set_gamma_grid_size(200)
fit_model_lsb.composite_expr='jet_leptonic'

Freezeing parameters and setting fit_range intervals¶

Note

With the new implementation of composite model (FitModel class) to set parameters you have to specify the model component, this is different from versions<1.1.2, and this holds also for the freeze method and for setting fit_range intervals, and for the methods relate to parameters setting in general. See the Composite Models and depending pars user guide for further information about the new implementation of FitModel, in particular for parameter setting

These methods are alternative and equivalent ways to access a model component for setting parameters state and values

passing as first argument, of the method, the model component name
passing as first argument, of the method, the model component object
accessing the model component member of the composite model class

#a
fit_model_lsb.freeze('jet_leptonic','z_cosm')
fit_model_lsb.freeze('jet_leptonic','R_H')
#b
fit_model_lsb.freeze(prefit_jet,'R')
#c
fit_model_lsb.jet_leptonic.parameters.R.fit_range=[10**15.5,10**17.5]
fit_model_lsb.jet_leptonic.parameters.beam_obj.fit_range=[5., 50.]

Building the ModelMinimizer object¶

Now we build a lsb model minimizer and run the fit method

Note

starting from version 1.1.2 the fit method allows to repeat the fit process, setting the parameter repeat. This will provide a better fit convergence. Setting repeat=3 the fit process will be repeated 3 times

model_minimizer_lsb=ModelMinimizer('lsb')
best_fit_lsb=model_minimizer_lsb.fit(fit_model_lsb,
                                     sed_data,
                                     1E11,
                                     1E29,
                                     fitname='SSC-best-fit-minuit',
                                     repeat=3)

filtering data in fit range = [1.000000e+11,1.000000e+29]
data length 35
================================================================================

* start fit process *
-----
fit run: 0

0it [00:00, ?it/s]

- best chisq=5.32182e+01

fit run: 1
- old chisq=5.32182e+01

0it [00:00, ?it/s]

- best chisq=5.22603e+01

fit run: 2
- old chisq=5.22603e+01

0it [00:00, ?it/s]

- best chisq=5.09003e+01

-------------------------------------------------------------------------
Fit report

Model: SSC-best-fit-minuit

Table length=12

model name	name	par type	units	val	phys. bound. min	phys. bound. max	log	frozen
jet_leptonic	gmin	low-energy-cut-off	lorentz-factor*	4.778915e+02	1.000000e+00	1.000000e+09	False	False
jet_leptonic	gmax	high-energy-cut-off	lorentz-factor*	9.191687e+05	1.000000e+00	1.000000e+15	False	False
jet_leptonic	N	emitters_density	1 / cm3	9.085652e-01	0.000000e+00	--	False	False
jet_leptonic	gamma0_log_parab	turn-over-energy	lorentz-factor*	4.337114e+04	1.000000e+00	1.000000e+09	False	False
jet_leptonic	s	LE_spectral_slope		2.184444e+00	-1.000000e+01	1.000000e+01	False	False
jet_leptonic	r	spectral_curvature		7.644091e-01	-1.500000e+01	1.500000e+01	False	False
jet_leptonic	R	region_size	cm	3.112712e+16	1.000000e+03	1.000000e+30	False	True
jet_leptonic	R_H	region_position	cm	1.000000e+17	0.000000e+00	--	False	True
jet_leptonic	B	magnetic_field	gauss	5.027056e-02	0.000000e+00	--	False	False
jet_leptonic	NH_cold_to_rel_e	cold_p_to_rel_e_ratio		1.000000e-01	0.000000e+00	--	False	True
jet_leptonic	beam_obj	beaming	lorentz-factor*	2.334793e+01	1.000000e-04	--	False	False
jet_leptonic	z_cosm	redshift		3.080000e-02	0.000000e+00	--	False	True

converged=True
calls=63
mesg=

'The relative error between two consecutive iterates is at most 0.000000'

dof=27
chisq=50.900346, chisq/red=1.885198 null hypothesis sig=0.003576

best fit pars

Table length=12

model name	name	val	bestfit val	err +	err -	start val	fit range min	fit range max	frozen
jet_leptonic	gmin	4.778915e+02	4.778915e+02	2.699848e+02	--	4.697542e+02	1.000000e+00	1.000000e+09	False
jet_leptonic	gmax	9.191687e+05	9.191687e+05	1.495007e+05	--	1.373160e+06	1.000000e+00	1.000000e+15	False
jet_leptonic	N	9.085652e-01	9.085652e-01	3.950026e-01	--	9.060842e-01	0.000000e+00	--	False
jet_leptonic	gamma0_log_parab	4.337114e+04	4.337114e+04	2.937759e+04	--	3.188500e+04	1.000000e+00	1.000000e+09	False
jet_leptonic	s	2.184444e+00	2.184444e+00	1.390250e-01	--	2.181578e+00	-1.000000e+01	1.000000e+01	False
jet_leptonic	r	7.644091e-01	7.644091e-01	2.758032e-01	--	7.726502e-01	-1.500000e+01	1.500000e+01	False
jet_leptonic	R	3.112712e+16	--	--	--	3.112712e+16	3.162278e+15	3.162278e+17	True
jet_leptonic	R_H	1.000000e+17	--	--	--	1.000000e+17	0.000000e+00	--	True
jet_leptonic	B	5.027056e-02	5.027056e-02	1.208011e-02	--	5.050000e-02	0.000000e+00	--	False
jet_leptonic	NH_cold_to_rel_e	1.000000e-01	--	--	--	1.000000e-01	0.000000e+00	--	True
jet_leptonic	beam_obj	2.334793e+01	2.334793e+01	3.239464e+00	--	2.500000e+01	5.000000e+00	5.000000e+01	False
jet_leptonic	z_cosm	3.080000e-02	--	--	--	3.080000e-02	0.000000e+00	--	True

-------------------------------------------------------------------------

================================================================================

we can obtain the best fit astropy table

best_fit_lsb.bestfit_table

Table length=12

model name	name	val	bestfit val	err +	err -	start val	fit range min	fit range max	frozen
str12	str16	float64	float64	float64	float64	float64	float64	float64	bool
jet_leptonic	gmin	4.778915e+02	4.778915e+02	2.699848e+02	--	4.697542e+02	1.000000e+00	1.000000e+09	False
jet_leptonic	gmax	9.191687e+05	9.191687e+05	1.495007e+05	--	1.373160e+06	1.000000e+00	1.000000e+15	False
jet_leptonic	N	9.085652e-01	9.085652e-01	3.950026e-01	--	9.060842e-01	0.000000e+00	--	False
jet_leptonic	gamma0_log_parab	4.337114e+04	4.337114e+04	2.937759e+04	--	3.188500e+04	1.000000e+00	1.000000e+09	False
jet_leptonic	s	2.184444e+00	2.184444e+00	1.390250e-01	--	2.181578e+00	-1.000000e+01	1.000000e+01	False
jet_leptonic	r	7.644091e-01	7.644091e-01	2.758032e-01	--	7.726502e-01	-1.500000e+01	1.500000e+01	False
jet_leptonic	R	3.112712e+16	--	--	--	3.112712e+16	3.162278e+15	3.162278e+17	True
jet_leptonic	R_H	1.000000e+17	--	--	--	1.000000e+17	0.000000e+00	--	True
jet_leptonic	B	5.027056e-02	5.027056e-02	1.208011e-02	--	5.050000e-02	0.000000e+00	--	False
jet_leptonic	NH_cold_to_rel_e	1.000000e-01	--	--	--	1.000000e-01	0.000000e+00	--	True
jet_leptonic	beam_obj	2.334793e+01	2.334793e+01	3.239464e+00	--	2.500000e+01	5.000000e+00	5.000000e+01	False
jet_leptonic	z_cosm	3.080000e-02	--	--	--	3.080000e-02	0.000000e+00	--	True

saving fit model, model minimizer¶

We can save all the fit products to be used later.

best_fit_lsb.mesg=None
best_fit_lsb.save_report('SSC-best-fit-lsb.pkl')
model_minimizer_lsb.save_model('model_minimizer_lsb.pkl')
fit_model_lsb.save_model('fit_model_lsb.pkl')

%matplotlib inline
fit_model_lsb.set_nu_grid(1E6,1E30,200)
fit_model_lsb.eval()
p2=fit_model_lsb.plot_model(sed_data=sed_data)
p2.setlim(y_min=1E-14,x_min=1E6,x_max=2E28)

../../../_images/Jet_example_model_fit_48_0.png

Model fitting with Minuit¶

To run the minuit minimizer we will use the best-fit results from lsb to set the boundaries for our parameters.

from jetset.minimizer import fit_SED,ModelMinimizer
from jetset.model_manager import  FitModel
from jetset.jet_model import Jet

jet_minuit=Jet.load_model('prefit_jet.pkl')
jet_minuit.set_gamma_grid_size(200)
#fit_model_minuit=fit_model_lsb
fit_model_minuit=FitModel( jet=jet_minuit, name='SSC-best-fit-minuit',template=None)

Table length=12

model name	name	par type	units	val	phys. bound. min	phys. bound. max	log	frozen
jet_leptonic	gmin	low-energy-cut-off	lorentz-factor*	4.697542e+02	1.000000e+00	1.000000e+09	False	False
jet_leptonic	gmax	high-energy-cut-off	lorentz-factor*	1.373160e+06	1.000000e+00	1.000000e+15	False	False
jet_leptonic	N	emitters_density	1 / cm3	9.060842e-01	0.000000e+00	--	False	False
jet_leptonic	gamma0_log_parab	turn-over-energy	lorentz-factor*	3.188500e+04	1.000000e+00	1.000000e+09	False	False
jet_leptonic	s	LE_spectral_slope		2.181578e+00	-1.000000e+01	1.000000e+01	False	False
jet_leptonic	r	spectral_curvature		7.726502e-01	-1.500000e+01	1.500000e+01	False	False
jet_leptonic	R	region_size	cm	3.112712e+16	1.000000e+03	1.000000e+30	False	False
jet_leptonic	R_H	region_position	cm	1.000000e+17	0.000000e+00	--	False	True
jet_leptonic	B	magnetic_field	gauss	5.050000e-02	0.000000e+00	--	False	False
jet_leptonic	NH_cold_to_rel_e	cold_p_to_rel_e_ratio		1.000000e-01	0.000000e+00	--	False	True
jet_leptonic	beam_obj	beaming	lorentz-factor*	2.500000e+01	1.000000e-04	--	False	False
jet_leptonic	z_cosm	redshift		3.080000e-02	0.000000e+00	--	False	False

fit_model_minuit.show_model_components()

--------------------------------------------------------------------------------
Composite model description
--------------------------------------------------------------------------------
name: SSC-best-fit-minuit
type: composite_model
components models:
 -model name: jet_leptonic model type: jet

--------------------------------------------------------------------------------

fit_model_minuit.freeze('jet_leptonic','z_cosm')
fit_model_minuit.freeze('jet_leptonic','R_H')
fit_model_minuit.jet_leptonic.parameters.R.fit_range=[5E15,1E17]
fit_model_minuit.jet_leptonic.parameters.gmin.fit_range=[10,1000]
fit_model_minuit.jet_leptonic.parameters.gmax.fit_range=[5E5,1E7]
fit_model_minuit.jet_leptonic.parameters.gamma0_log_parab.fit_range=[1E3,1E5]

fit_model_minuit.jet_leptonic.parameters.beam_obj.fit_range=[5,50]

model_minimizer_minuit=ModelMinimizer('minuit')
best_fit_minuit=model_minimizer_minuit.fit(fit_model_minuit,sed_data,10**11.,10**29.0,fitname='SSC-best-fit-minuit',repeat=2)

filtering data in fit range = [1.000000e+11,1.000000e+29]
data length 35
================================================================================

* start fit process *
-----
fit run: 0

0it [00:00, ?it/s]

- best chisq=4.33705e+01

fit run: 1
- old chisq=4.33705e+01

0it [00:00, ?it/s]

- best chisq=3.11482e+01

-------------------------------------------------------------------------
Fit report

Model: SSC-best-fit-minuit

Table length=12

model name	name	par type	units	val	phys. bound. min	phys. bound. max	log	frozen
jet_leptonic	gmin	low-energy-cut-off	lorentz-factor*	4.688283e+02	1.000000e+00	1.000000e+09	False	False
jet_leptonic	gmax	high-energy-cut-off	lorentz-factor*	9.028968e+05	1.000000e+00	1.000000e+15	False	False
jet_leptonic	N	emitters_density	1 / cm3	7.917986e-01	0.000000e+00	--	False	False
jet_leptonic	gamma0_log_parab	turn-over-energy	lorentz-factor*	3.238465e+04	1.000000e+00	1.000000e+09	False	False
jet_leptonic	s	LE_spectral_slope		2.090996e+00	-1.000000e+01	1.000000e+01	False	False
jet_leptonic	r	spectral_curvature		6.854517e-01	-1.500000e+01	1.500000e+01	False	False
jet_leptonic	R	region_size	cm	3.046194e+16	1.000000e+03	1.000000e+30	False	False
jet_leptonic	R_H	region_position	cm	1.000000e+17	0.000000e+00	--	False	True
jet_leptonic	B	magnetic_field	gauss	5.214133e-02	0.000000e+00	--	False	False
jet_leptonic	NH_cold_to_rel_e	cold_p_to_rel_e_ratio		1.000000e-01	0.000000e+00	--	False	True
jet_leptonic	beam_obj	beaming	lorentz-factor*	2.249661e+01	1.000000e-04	--	False	False
jet_leptonic	z_cosm	redshift		3.080000e-02	0.000000e+00	--	False	True

converged=True
calls=2284
mesg=

FCN = 31.15		Nfcn = 2284
EDM = 3.17e+06 (Goal: 0.0002)
INVALID Minimum	Valid Parameters	SOME Parameters at limit
ABOVE EDM threshold (goal x 10)		Below call limit
Covariance	Hesse ok	APPROXIMATE	Pos. def.	Not forced

	Name	Value	Hesse Error	Minos Error-	Minos Error+	Limit-	Limit+	Fixed
0	par_0	0.47e3	0.29e3			10	1E+03
1	par_1	0.9e6	2.0e6			5E+05	1E+07
2	par_2	0.8	0.8			0
3	par_3	0.03e6	0.04e6			1E+03	1E+05
4	par_4	2	8			-10	10
5	par_5	1	12			-15	15
6	par_6	0.030e18	0.035e18			5E+15	1E+17
7	par_7	0.1	0.4			0
8	par_8	22	18			5	50

	par_0	par_1	par_2	par_3	par_4	par_5	par_6	par_7	par_8
par_0	9.3e+04	9.24e+07 (0.160)	-21 (-0.083)	-3.93e+06 (-0.287)	1.75e+03 (0.661)	906 (0.201)	1.8e+18 (0.141)	30.6 (0.333)	-220 (-0.033)
par_1	9.24e+07 (0.160)	3.61e+12	1.29e+04 (0.008)	2.41e+09 (0.028)	-1.07e+06 (-0.065)	-5.55e+05 (-0.020)	-1.1e+21 (-0.014)	-1.87e+04 (-0.033)	1.35e+05 (0.003)
par_2	-21 (-0.083)	1.29e+04 (0.008)	0.686	-548 (-0.015)	0.244 (0.034)	0.126 (0.010)	2.5e+14 (0.007)	0.00426 (0.017)	-0.0307 (-0.002)
par_3	-3.93e+06 (-0.287)	2.41e+09 (0.028)	-548 (-0.015)	2.02e+09	4.56e+04 (0.117)	2.36e+04 (0.036)	4.69e+19 (0.025)	797 (0.059)	-5.75e+03 (-0.006)
par_4	1.75e+03 (0.661)	-1.07e+06 (-0.065)	0.244 (0.034)	4.56e+04 (0.117)	75.3	-10.5 (-0.082)	-2.09e+16 (-0.058)	-0.355 (-0.136)	2.56 (0.013)
par_5	906 (0.201)	-5.55e+05 (-0.020)	0.126 (0.010)	2.36e+04 (0.036)	-10.5 (-0.082)	219	-1.08e+16 (-0.017)	-0.184 (-0.041)	1.32 (0.004)
par_6	1.8e+18 (0.141)	-1.1e+21 (-0.014)	2.5e+14 (0.007)	4.69e+19 (0.025)	-2.09e+16 (-0.058)	-1.08e+16 (-0.017)	1.75e+33	-3.65e+14 (-0.029)	2.63e+15 (0.003)
par_7	30.6 (0.333)	-1.87e+04 (-0.033)	0.00426 (0.017)	797 (0.059)	-0.355 (-0.136)	-0.184 (-0.041)	-3.65e+14 (-0.029)	0.0905	0.0447 (0.007)
par_8	-220 (-0.033)	1.35e+05 (0.003)	-0.0307 (-0.002)	-5.75e+03 (-0.006)	2.56 (0.013)	1.32 (0.004)	2.63e+15 (0.003)	0.0447 (0.007)	481

dof=26
chisq=31.148172, chisq/red=1.198007 null hypothesis sig=0.222797

best fit pars

Table length=12

model name	name	val	bestfit val	err +	err -	start val	fit range min	fit range max	frozen
jet_leptonic	gmin	4.688283e+02	4.688283e+02	2.858882e+02	--	4.697542e+02	1.000000e+01	1.000000e+03	False
jet_leptonic	gmax	9.028968e+05	9.028968e+05	1.970066e+06	--	1.373160e+06	5.000000e+05	1.000000e+07	False
jet_leptonic	N	7.917986e-01	7.917986e-01	7.826153e-01	--	9.060842e-01	0.000000e+00	--	False
jet_leptonic	gamma0_log_parab	3.238465e+04	3.238465e+04	3.814239e+04	--	3.188500e+04	1.000000e+03	1.000000e+05	False
jet_leptonic	s	2.090996e+00	2.090996e+00	7.583699e+00	--	2.181578e+00	-1.000000e+01	1.000000e+01	False
jet_leptonic	r	6.854517e-01	6.854517e-01	1.250924e+01	--	7.726502e-01	-1.500000e+01	1.500000e+01	False
jet_leptonic	R	3.046194e+16	3.046194e+16	3.526875e+16	--	3.112712e+16	5.000000e+15	1.000000e+17	False
jet_leptonic	R_H	1.000000e+17	--	--	--	1.000000e+17	0.000000e+00	--	True
jet_leptonic	B	5.214133e-02	5.214133e-02	3.594802e-01	--	5.050000e-02	0.000000e+00	--	False
jet_leptonic	NH_cold_to_rel_e	1.000000e-01	--	--	--	1.000000e-01	0.000000e+00	--	True
jet_leptonic	beam_obj	2.249661e+01	2.249661e+01	1.845509e+01	--	2.500000e+01	5.000000e+00	5.000000e+01	False
jet_leptonic	z_cosm	3.080000e-02	--	--	--	3.080000e-02	0.000000e+00	--	True

-------------------------------------------------------------------------

================================================================================

%matplotlib inline
fit_model_minuit.eval()
p2=fit_model_minuit.plot_model(sed_data=sed_data)
p2.setlim(y_min=1E-13,x_min=1E6,x_max=2E28)

../../../_images/Jet_example_model_fit_55_0.png

best_fit_minuit.save_report('SSC-best-fit-minuit.pkl')
model_minimizer_minuit.save_model('model_minimizer_minuit.pkl')
fit_model_minuit.save_model('fit_model_minuit.pkl')

You can obtain profile and contours, but this is typically time consuming. In any case, better results can be achieved using the MCMC approach (discussed in next section) For further information regardin minuit please refer to https://iminuit.readthedocs.io

#migrad profile

#access the data
profile_migrad=model_minimizer_minuit.minimizer.mnprofile('s')

#make the plot(no need to run the previous command)
profile_plot_migrad=model_minimizer_minuit.minimizer.draw_mnprofile('s')

#migrad contour
#access the data
contour_migrad=model_minimizer_minuit.minimizer.contour('beam_obj','B')

#make the plot(no need to run the previous command)
contour_plot_migrad=model_minimizer_minuit.minimizer.draw_contour('beam_obj','B')

you can use also minos contour and profile, in this case the computational time is even longer:

profile_migrad=model_minimizer_minuit.minimizer.mnprofile('s')
profile_plot_migrad=model_minimizer_minuit.minimizer.draw_mnprofile('s')

contour_migrad=model_minimizer_minuit.minimizer.mncontour('r','s')
contour_plot_migrad=model_minimizer_minuit.minimizer.draw_mncontour('r','s')

MCMC sampling¶

from jetset.mcmc import McmcSampler
from jetset.minimizer import ModelMinimizer

We used a flat prior centered on the best fit value. Setting bound=5.0 and bound_rel=True means that:

the prior interval will be defined as [best_fit_val - delta_m , best_fit_val + delta_p]
with delta_p=delta_m=best_fit_val*bound

If we set bound_rel=False then delta_p = delta_m = best_fit_err*bound

It is possible to define asymmetric boundaries e.g. bound=[2.0,5.0] meaning that

for bound_rel=True

delta_p = best_fit_val*bound[1]

delta_m =b est_fit_val*bound[0]
for bound_rel=False

delta_p = best_fit_err*bound[1]

delta_m = best_fit_err*bound[0]

In the next release a more flexible prior interface will be added, including different type of priors

Given the large parameter space, we select a sub sample of parameters using the use_labels_dict. If we do not pass the ‘use_labels_dict’ the full set of free parameters will be used

from tqdm.auto import tqdm

model_minimizer_minuit = ModelMinimizer.load_model('model_minimizer_minuit.pkl')

mcmc=McmcSampler(model_minimizer_minuit)

labels=['N','B','beam_obj','s','gamma0_log_parab']
model_name='jet_leptonic'
use_labels_dict={model_name:labels}

mcmc.run_sampler(nwalkers=128,burnin=10,steps=50,bound=5.0,bound_rel=True,threads=None,walker_start_bound=0.005,use_labels_dict=use_labels_dict,progress='notebook')

mcmc run starting

0%|          | 0/50 [00:00<?, ?it/s]

mcmc run done, with 1 threads took 341.59 seconds

print(mcmc.acceptance_fraction)

0.5459375

p=mcmc.plot_model(sed_data=sed_data,fit_range=[1E11,2E28],size=100)
p.setlim(y_min=1E-14,x_min=1E6,x_max=2E28)

../../../_images/Jet_example_model_fit_67_0.png

p=mcmc.plot_model(sed_data=sed_data,fit_range=[1E11, 2E27],size=100,quantiles=[0.05,0.95])
p.setlim(y_min=1E-14,x_min=1E6,x_max=2E28)

../../../_images/Jet_example_model_fit_68_0.png

f=mcmc.plot_chain('s',log_plot=False)

../../../_images/Jet_example_model_fit_69_0.png

f=mcmc.corner_plot()

../../../_images/Jet_example_model_fit_70_0.png

mcmc.get_par('N')

(array([0.78402592, 0.79701842, 0.7860921 , ..., 0.68880801, 0.70229451,
        0.65232144]),
 0)

f=mcmc.plot_par('beam_obj')

../../../_images/Jet_example_model_fit_72_0.png

f=mcmc.plot_par('gamma0_log_parab',log_plot=True)

../../../_images/Jet_example_model_fit_73_0.png

Save and reuse MCMC¶

mcmc.save('mcmc_sampler.pkl')

from jetset.mcmc import McmcSampler
from jetset.data_loader import ObsData
from jetset.plot_sedfit import PlotSED
from jetset.test_data_helper import  test_SEDs

sed_data=ObsData.load('Mrk_401.pkl')

ms=McmcSampler.load('mcmc_sampler.pkl')

ms.model.name

'SSC-best-fit-minuit'

p=ms.plot_model(sed_data=sed_data,fit_range=[1E11, 2E27],size=100)
p.setlim(y_min=1E-14,x_min=1E6,x_max=2E28)

../../../_images/Jet_example_model_fit_78_0.png

p=ms.plot_model(sed_data=sed_data,fit_range=[1E11, 2E27],size=100,quantiles=[0.05,0.95])
p.setlim(y_min=1E-14,x_min=1E6,x_max=2E28)

../../../_images/Jet_example_model_fit_79_0.png

f=ms.plot_par('beam_obj',log_plot=False)

../../../_images/Jet_example_model_fit_80_0.png

f=ms.plot_par('B',log_plot=True)

../../../_images/Jet_example_model_fit_81_0.png

f=ms.plot_chain('s',log_plot=False)

../../../_images/Jet_example_model_fit_82_0.png

f=ms.corner_plot()

../../../_images/Jet_example_model_fit_83_0.png