# Simultaneously Fitting Source and Background Spectra

Sherpa Threads (CIAO 4.12 Sherpa v1)

## Overview

#### Synopsis:

If you would like to fit a background-subtracted source spectrum using a common RMF and ARF for source and background, simply read the source spectrum fits file into Sherpa, subtract the background, and fit it. To fit source and background spectra simultaneously with proper and distinct RMFs and ARFs, load the source and background as different data sets. This thread illustrates both cases.

The sample data files used in this thread are available in sherpa.tar.gz; they can be generated by following the CIAO thread Extract ACIS Spectra and Response Files for Pointlike Sources.

Last Update: 11 Dec 2019 - Updated to use Matplotlib in CIAO 4.12 and to take advantage of changes to the plot support (e.g. plot_fit(ylog=True)); the text and commands have been revised slightly.

## Statistical Issues: Background Subtraction

A typical dataset may contain multiple spectra, one of which contains contributions from the source and the background, and one or more which contain background counts alone. (The background itself may contain contributions from the cosmic X-ray background, the particle background, and so on, but we ignore this complication.)

The proper way to treat background data is to model them. However, many X-ray astronomers subtract background data from the raw data.

Why should one not subtract background?

• It reduces the amount of statistical information in the analysis—the final fit parameter values will be a less accurate estimate of the true values.
• The background-subtracted data are not Poisson-distributed; one cannot fit them with the Poisson likelihood or the Cash statistic, even in the low-counts limit. For example, subtracting a background can give negative counts; this is definitely not Poissonian!
• Fluctuations, particularly in the vicinity of localized features, can adversely affect analysis.

## Fit a Background-Subtracted Source

### Reading FITS data for source and background

By default, specextract updates the header keywords BACKFILE, RESPFILE, and ANCRFILE in the source spectrum file:

unix% dmkeypar 3c273.pi BACKFILE echo+
3c273_bg.pi

unix% dmkeypar 3c273.pi RESPFILE echo+
3c273.rmf

unix% dmkeypar 3c273.pi ANCRFILE echo+
3c273.arf


On account of these keywords, Sherpa automatically reads in the (ungrouped) background spectrum "3c273_bg.pi" and the source response matrices "3c273.rmf" and "3c273.arf" when the source spectrum is read:

sherpa> load_pha("3c273.pi")
statistical errors were found in file '3c273.pi'
but not used; to use them, re-read with use_errors=True
statistical errors were found in file '3c273_bg.pi'
but not used; to use them, re-read with use_errors=True


If Sherpa does not automatically read in the background data, then it can be input as follows:

sherpa> load_bkg("3c273_bg.pi")
statistical errors were found in file '3c273_bg.pi'
but not used; to use them, re-read with use_errors=True


### Subtracting the background

The user can then subtract the background from the source spectrum and fit the background-subtracted source spectrum using response matrices for the source only. The following commands may be used as an example script for subtracting the background and then visualizing the 0.3-3 keV background-subtracted source data in Sherpa:

sherpa> subtract()
sherpa> set_xlog()
sherpa> set_ylog()
sherpa> plot_data()

sherpa> notice(0.3, 3.)
sherpa> plot_data(xlog=False, ylog=False)


Which produces Figure 2.

The get_bkg_scale function—not to be confused with get_backscal—returns the value of the coefficient which is used to scale background counts during background subtraction of a source spectrum, or a simultaneous fit of source and background spectra. The complete scaling factor used to scale the background counts in these cases consists of the product of the source-to-background exposure and backscale (extraction region area).

The background-subtracted spectrum is then:

and the background scaling factor is:

Note

The get_backscal function returns the value associated with the OGIP PHA header keyword BACKSCAL, which is only one component of the complete scaling factor.

The background scaling factor is also returned by the show_data and show_all functions. The value returned by the get_bkg_scale function can be checked by doing the following in Sherpa:

sherpa> get_bkg_scale()
0.13492064388809599

sherpa> manual_bkg_scale_check = get_exposure()*get_backscal()/get_exposure(bkg_id=1)/get_backscal(bkg_id=1)

sherpa> print(manual_bkg_scale_check)
0.134920643888


### Defining the source instrument response

Since the header of the input source spectrum file referenced the instrument response files, an instrument response was automatically defined:

sherpa> print(get_arf())
name     = 3c273.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 38570.6924812
ethresh  = 1e-10

sherpa> print(get_rmf())
name     = 3c273.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[2004]
n_chan   = UInt32[2004]
matrix   = Float64[58564]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10

sherpa> print(get_bkg_arf())
name     = 3c273.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 38570.6924812
ethresh  = 1e-10

sherpa> print(get_bkg_rmf())
name     = 3c273.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[2004]
n_chan   = UInt32[2004]
matrix   = Float64[58564]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10


Note that the instrument response was automatically set up for both the source and background. However, unless a background model is defined for fitting using the set_bkg_model command, then the background data will be ignored.

If you manually read in a background file after reading the source data, the background instrument response will be reset. To set it to be identical to the source instrument response, use load_bkg_arf and load_bkg_rmf with the source response files:

sherpa> load_bkg_rmf("3c273.rmf")

sherpa> print(get_bkg_arf())
name     = 3c273.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 38570.6924812
ethresh  = 1e-10

sherpa> print(get_bkg_rmf())
name     = 3c273.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[2004]
n_chan   = UInt32[2004]
matrix   = Float64[58564]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10


### Defining and fitting a source model

The following commands may be used as an example script for fitting the background-subtracted data with a source model consisting of an absorbed power-law:

sherpa> set_source(xswabs.abs1 * powlaw1d.p1)
sherpa> show_source()
Model: 1
(xswabs.abs1 * powlaw1d.p1)
Param        Type          Value          Min          Max      Units
-----        ----          -----          ---          ---      -----
abs1.nh      thawed            1            0       100000 10^22 atoms / cm^2
p1.gamma     thawed            1          -10           10
p1.ref       frozen            1 -3.40282e+38  3.40282e+38
p1.ampl      thawed            1            0  3.40282e+38

sherpa> abs1.nh = 0.1
sherpa> guess(p1)
sherpa> print(p1)
powlaw1d.p1
Param        Type          Value          Min          Max      Units
-----        ----          -----          ---          ---      -----
p1.gamma     thawed            1          -10           10
p1.ref       frozen            1 -3.40282e+38  3.40282e+38
p1.ampl      thawed  0.000144301  1.44301e-07     0.144301

sherpa> fit()
Dataset               = 1
Method                = levmar
Statistic             = chi2gehrels
Initial fit statistic = 89.8323
Final fit statistic   = 15.4612 at function evaluation 184
Data points           = 33
Degrees of freedom    = 30
Probability [Q-value] = 0.98686
Reduced statistic     = 0.515375
Change in statistic   = 74.3711
abs1.nH        0.000216916  +/- 0.0149809
p1.gamma       1.70584      +/- 0.153887
p1.ampl        0.000161164  +/- 1.53492e-05

sherpa> plot_fit_resid()
sherpa> plt.yscale('linear')


The results are shown Figure 3.

The confidence function may be used to estimate the errors on the individual parameters of the fit; in this example, we choose to calculate the 1σ errors:

sherpa> get_conf_opt()
{'sigma': 1,
'eps': 0.01,
'maxiters': 200,
'soft_limits': False,
'remin': 0.01,
'fast': False,
'parallel': True,
'numcores': 4,
'maxfits': 5,
'max_rstat': 3,
'tol': 0.2,
'verbose': False,
'openinterval': False}

sherpa> set_conf_opt("sigma", 1)
sherpa> conf()
abs1.nH lower bound:	-----
abs1.nH upper bound:	0.0113339
p1.gamma lower bound:	-0.106925
p1.gamma upper bound:	0.123277
p1.ampl lower bound:	-8.44533e-06
p1.ampl upper bound:	1.43411e-05
Dataset               = 1
Confidence Method     = confidence
Iterative Fit Method  = None
Fitting Method        = levmar
Statistic             = chi2gehrels
confidence 1-sigma (68.2689%) bounds:
Param            Best-Fit  Lower Bound  Upper Bound
-----            --------  -----------  -----------
abs1.nH       0.000216916        -----    0.0113339
p1.gamma          1.70584    -0.106925     0.123277
p1.ampl       0.000161164 -8.44533e-06  1.43411e-05


## Simultaneously Fit Source and Background with the Same Responses

Instead of subtracting the background, the user may choose to simultaneously fit the source and background spectra, using the same (source) RMF and ARF.

### Reading FITS data for source and background

Again, Sherpa automatically reads in the (ungrouped) background spectrum and the source response matrices when the source spectrum is read:

sherpa> clean()
statistical errors were found in file '3c273.pi'
but not used; to use them, re-read with use_errors=True
statistical errors were found in file '3c273_bg.pi'
but not used; to use them, re-read with use_errors=True


### Defining the source and background instrument response

Since the header of the input source spectrum file referenced the instrument response files, an instrument response was automatically defined:

sherpa> print(get_arf())
name     = 3c273.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 38570.6924812

sherpa> print(get_rmf())
name     = 3c273.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[2004]
n_chan   = UInt32[2004]
matrix   = Float64[58564]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10

sherpa> print(get_bkg_arf())
name     = 3c273.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 38570.6924812
ethresh  = 1e-10

sherpa> print(get_bkg_rmf())
name     = 3c273.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[2004]
n_chan   = UInt32[2004]
matrix   = Float64[58564]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10


The instrument response was automatically set up for both the source and background. However, unless a background model is defined for fitting using the set_bkg_model command, then the background data will be ignored. In order to fit the background simultaneously with the source, the set_bkg_model() command will be issued in the next step.

As with the Defining the source instrument response section, the load_bkg_arf and load_bkg_rmf routines can be used to set the responses.

### Defining and fitting source and background models

The fit_bkg function is available (and the equivalent 'bkg_only' option of the fit command) allows you to fit models to just the background components. It fits the defined background model(s) to the background data set(s) by ID. It may be called with no arguments, in which case a fit is done simultaneously on all background data sets for which the user has defined a model to be fit; source data sets ready for fitting will be ignored in this case.

Here, we choose to simultaneously fit the source and background with the fit function, using an absorbed power-law model for the source and an unabsorbed power-law model for the background. The source and background models are defined using the set_source and set_bkg_model commands, respectively.

sherpa> set_source(xswabs.abs1 * powlaw1d.srcp1)
sherpa> set_bkg_model(abs1 * powlaw1d.bkgp1)
sherpa> abs1.nh = 0.1
sherpa> guess(srcp1)
sherpa> guess(bkgp1)

sherpa> show_model()
Model: 1
apply_rmf(apply_arf((38564.608926889 * ((xswabs.abs1 * powlaw1d.srcp1) + 0.134921 * ((xswabs.abs1 * powlaw1d.bkgp1))))))
Param        Type          Value          Min          Max      Units
-----        ----          -----          ---          ---      -----
abs1.nH      thawed          0.1            0       100000 10^22 atoms / cm^2
srcp1.gamma  thawed            1          -10           10
srcp1.ref    frozen            1 -3.40282e+38  3.40282e+38
srcp1.ampl   thawed  0.000144301  1.44301e-07     0.144301
bkgp1.gamma  thawed            1          -10           10
bkgp1.ref    frozen            1 -3.40282e+38  3.40282e+38
bkgp1.ampl   thawed  0.000144301  1.44301e-07     0.144301

sherpa> show_bkg_model()
Background Model: 1:1
apply_rmf(apply_arf((38564.608926889 * (xswabs.abs1 * powlaw1d.bkgp1))))
Param        Type          Value          Min          Max      Units
-----        ----          -----          ---          ---      -----
abs1.nH      thawed          0.1            0       100000 10^22 atoms / cm^2
bkgp1.gamma  thawed            1          -10           10
bkgp1.ref    frozen            1 -3.40282e+38  3.40282e+38
bkgp1.ampl   thawed  0.000144301  1.44301e-07     0.144301

sherpa> fit()
Dataset               = 1
Method                = levmar
Statistic             = chi2gehrels
Initial fit statistic = 4604.01
Final fit statistic   = 137.257 at function evaluation 86
Data points           = 92
Degrees of freedom    = 87
Probability [Q-value] = 0.000478885
Reduced statistic     = 1.57766
Change in statistic   = 4466.75
abs1.nH        0.0178333    +/- 0.00594283
srcp1.gamma    1.92493      +/- 0.0776745
srcp1.ampl     0.000174607  +/- 1.27987e-05
bkgp1.gamma    -0.167143    +/- 0.517068
bkgp1.ampl     2.46653e-06  +/- 2.08853e-06

sherpa> plot_fit_resid(xlog=True, ylog=True)
sherpa> plt.yscale('linear')
sherpa> plt.xlim(0.2, 3.5)
sherpa> plt.sca(plt.gcf().axes[0])
sherpa> plt.ylim(7e-4, 2e-2)


The results are shown Figure 4.

The calc_stat_info command (see also get_stat_info) may be used to access the goodness-of-fit statistics for the individual source and background data set included in the simultaneous fit, plus the simultaneous fit, without having to re-run the fit.

sherpa> calc_stat_info()
Dataset               = 1
Statistic             = chi2gehrels
Fit statistic value   = 36.8354
Data points           = 46
Degrees of freedom    = 41
Probability [Q-value] = 0.656205
Reduced statistic     = 0.898423

Background 1 in Dataset = 1
Statistic             = chi2gehrels
Fit statistic value   = 100.421
Data points           = 46
Degrees of freedom    = 43
Probability [Q-value] = 1.71508e-06
Reduced statistic     = 2.33538

Dataset               = 1
Statistic             = chi2gehrels
Fit statistic value   = 137.257
Data points           = 92
Degrees of freedom    = 87
Probability [Q-value] = 0.000478885
Reduced statistic     = 1.57766


The confidence 1σ error estimates on the individual parameters of the fit are:

sherpa> conf()
bkgp1.ampl lower bound:	-2.0436e-06
abs1.nH lower bound:	-0.0052732
bkgp1.gamma lower bound:	-1.03539
srcp1.ampl lower bound:	-1.24576e-05
bkgp1.gamma upper bound:	0.659018
bkgp1.ampl upper bound:	4.28518e-06
abs1.nH upper bound:	0.00672868
srcp1.gamma lower bound:	-0.0750823
srcp1.ampl upper bound:	1.28531e-05
srcp1.gamma upper bound:	0.0763325
Dataset               = 1
Confidence Method     = confidence
Iterative Fit Method  = None
Fitting Method        = levmar
Statistic             = chi2gehrels
confidence 1-sigma (68.2689%) bounds:
Param            Best-Fit  Lower Bound  Upper Bound
-----            --------  -----------  -----------
abs1.nH         0.0178333   -0.0052732   0.00672868
srcp1.gamma       1.92493   -0.0750823    0.0763325
srcp1.ampl    0.000174607 -1.24576e-05  1.28531e-05
bkgp1.gamma     -0.167143     -1.03539     0.659018
bkgp1.ampl    2.46653e-06  -2.0436e-06  4.28518e-06


## Simultaneously Fit Source and Background with Independent Responses

Instead of subtracting the background, the user may choose to simultaneously fit the source and background spectra, each with its own RMF and ARF.

### Reading FITS data for source and background

In this thread, we wish to fit spectral data from the FITS (ungrouped) data files source.pi and back.pi. These data sets are input into Sherpa with load_* commands:

sherpa> clean()

statistical errors were found in file 'source.pi'
but not used; to use them, re-read with use_errors=True

statistical errors were found in file 'source.pi'
but not used; to use them, re-read with use_errors=True


The source and background data sets can be plotted in separate windows with plot_data() and plot_bkg(), or together in one window with the following syntax:

sherpa> plot("data", "bkg")


The plots are displayed in Figure 5. The data are plotted in bin (i.e. channel) space, since we have not yet defined the instrument response (RMFs and ARFs).

### Defining source and background instrument responses

Here we establish 1-D instrument responses by loading the source and background response files with load_* commands:

sherpa> load_arf("source.arf")

sherpa> load_bkg_arf("back.arf")


Now when we plot the data the X axis is in energy, rather than channel (Figure 6).

The current definition of the instrument response may be examined using show_all or get commands:

sherpa> print(get_arf())
name     = source.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 49470.9477594
ethresh  = 1e-10

sherpa> print(get_rmf())
name     = source.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[1090]
n_chan   = UInt32[1090]
matrix   = Float64[572598]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10

sherpa> print(get_bkg_arf())
name     = back.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 49470.9477594
ethresh  = 1e-10

sherpa> print(get_bkg_rmf())
name     = back.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[1090]
n_chan   = UInt32[1090]
matrix   = Float64[552166]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10


The output shows that source.arf and source.rmf currently define the source instrument response.

### Defining and fitting source and background models

We now need to define a source model for our source data set and a background model for our background data set. For these data, we find that absorbed blackbody spectra are appropriate. We define a model which includes an absorbing column and a blackbody for both the source and background:

sherpa> set_source(xswabs.a1 * xsbbody.b1)
sherpa> set_bkg_model(a1 * xsbbody.b2)


Note that we also ignore the high and low energy regions of the spectrum, as they generally have lower quality data. The plots of the data and model (with the default parameter values) produced by plot commands are shown in Figure 7.

sherpa> notice(0.3, 10.)
sherpa> plot_fit(xlog=True, ylog=True, color='green')
sherpa> plot_bkg_fit(overplot=True, color='orange')


Clearly, the default values are not a very good fit to the data. We may choose to begin the fit now, using the fit() command, or we can refine the values of the fit-by-eye, prior to fitting. By varying parameter values, while plotting and replotting the data, we can help the fitting algorithm find the best minimum. By setting the parameters to values near what we expect, we also help avoid local minima in the parameter space. (Fitting spectra is something of an art; one generally gets better fits when they have a priori knowledge of the source.) We set the values of the parameters as follows:

sherpa> a1.nH = 0.0336676
sherpa> b1.kT = 20
sherpa> b1.norm = 1e-02
sherpa> b2.kT = 0.5
sherpa> b2.norm.min = 1e-6
sherpa> b2.norm = 1e-05


Note that the normalization of b2, b2.norm, appears to have a better fit at values less than the default minimum. We set the b2.norm.min to a new, smaller value to give us the optimal fit. Now we have a reasonable start to the fit:

sherpa> plt.subplot(2, 1, 1)
sherpa> plot_fit(xlog=True, ylog=True, clearwindow=False)
sherpa> plt.subplot(2, 1, 2)
sherpa> plot_bkg_fit(xlog=True, ylog=True, clearwindow=False)


The plot is visible Figure 8.

In general, we may also want to change our fit statistic or optimization method with set_stat and set_method; here, we choose to run the fit with the current fit statisic and method:

sherpa> Dataset               = 1
Method                = levmar
Statistic             = chi2gehrels
Initial fit statistic = 828.712
Final fit statistic   = 731.835 at function evaluation 196
Data points           = 1056
Degrees of freedom    = 1051
Probability [Q-value] = 1
Reduced statistic     = 0.696322
Change in statistic   = 96.877
a1.nH          0.0352536    +/- 0.0122578
b1.kT          4.23927      +/- 0.636896
b1.norm        0.000146661  +/- 5.15993e-05
b2.kT          0.556463     +/- 0.0103887
b2.norm        1.12408e-05  +/- 2.44213e-07



Once the fit is complete, Sherpa will display the fit values to the screen. You may also display the overall status of the Sherpa session with the show_all and show_bkg command:

sherpa> show_all()
Data Set: 1
Filter: 0.2993-7.9935 Energy (keV)
Bkg Scale: 0.694444
Noticed Channels: 21-548
name           = source.pi
channel        = Float64[1024]
counts         = Float64[1024]
staterror      = None
syserror       = None
bin_lo         = None
bin_hi         = None
grouping       = None
quality        = None
exposure       = 49429.233467924
backscal       = 1.872535141462e-05
areascal       = 1.0
grouped        = False
subtracted     = False
units          = energy
rate           = True
plot_fac       = 0
response_ids   = [1]
background_ids = [1]

RMF Data Set: 1:1
name     = source.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[1090]
n_chan   = UInt32[1090]
matrix   = Float64[572598]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10

ARF Data Set: 1:1
name     = source.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 49470.94775939
ethresh  = 1e-10

Background Data Set: 1:1
Filter: 0.2993-7.9935 Energy (keV)
Noticed Channels: 21-548
name           = back.pi
channel        = Float64[1024]
counts         = Float64[1024]
staterror      = None
syserror       = None
bin_lo         = None
bin_hi         = None
grouping       = None
quality        = None
exposure       = 49429.233467924
backscal       = 2.6964506037052e-05
areascal       = 1.0
grouped        = False
subtracted     = False
units          = energy
rate           = True
plot_fac       = 0
response_ids   = [1]
background_ids = []

Background RMF Data Set: 1:1
name     = back.rmf
detchans = 1024
energ_lo = Float64[1090]
energ_hi = Float64[1090]
n_grp    = UInt64[1090]
f_chan   = UInt32[1090]
n_chan   = UInt32[1090]
matrix   = Float64[552166]
offset   = 1
e_min    = Float64[1024]
e_max    = Float64[1024]
ethresh  = 1e-10

Background ARF Data Set: 1:1
name     = back.arf
energ_lo = Float64[1090]
energ_hi = Float64[1090]
specresp = Float64[1090]
bin_lo   = None
bin_hi   = None
exposure = 49470.94775939
ethresh  = 1e-10

Model: 1
apply_rmf(apply_arf((49429.233467924 * ((xswabs.a1 * xsbbody.b1) + 0.694444 * ((xswabs.a1 * xsbbody.b2))))))
Param        Type          Value          Min          Max      Units
-----        ----          -----          ---          ---      -----
a1.nH        thawed    0.0352536            0       100000 10^22 atoms / cm^2
b1.kT        thawed      4.23927         0.01          100        keV
b1.norm      thawed  0.000146661            0        1e+24 L39 / (D10)**2
b2.kT        thawed     0.556463         0.01          100        keV
b2.norm      thawed  1.12408e-05        1e-06        1e+24 L39 / (D10)**2

Optimization Method: LevMar
name     = levmar
ftol     = 1.1920928955078125e-07
xtol     = 1.1920928955078125e-07
gtol     = 1.1920928955078125e-07
maxfev   = None
epsfcn   = 1.1920928955078125e-07
factor   = 100.0
numcores = 1
verbose  = 0

Statistic: Chi2Gehrels
Chi Squared with Gehrels variance.

The variance is estimated from the number of counts in each bin,
but unlike Chi2DataVar, the Gaussian approximation is not
used. This makes it more-suitable for use with low-count data.

The standard deviation for each bin is calculated using the
approximation from [1]_:

sigma(i,S) = 1 + sqrt(N(i,s) + 0.75)

where the higher-order terms have been dropped. This is accurate
to approximately one percent. For data where the background has
not been subtracted then the error term is:

sigma(i) = sigma(i,S)

whereas with background subtraction,

sigma(i)^2 = sigma(i,S)^2 + [A(S)/A(B)]^2 sigma(i,B)^2

--------
Chi2DataVar, Chi2ModVar, Chi2XspecVar

Notes
-----
The accuracy of the error term when the background has been
subtracted has not been determined. A preferable approach to
background subtraction is to model the background as well as the
source signal.

References
----------

.. [1] "Confidence limits for small numbers of events in
astrophysical data", Gehrels, N. 1986, ApJ, vol 303,
p. 336-346.

Fit:Dataset               = 1
Method                = levmar
Statistic             = chi2gehrels
Initial fit statistic = 828.712
Final fit statistic   = 731.835 at function evaluation 196
Data points           = 1056
Degrees of freedom    = 1051
Probability [Q-value] = 1
Reduced statistic     = 0.696322
Change in statistic   = 96.877
a1.nH          0.0352536    +/- 0.0122578
b1.kT          4.23927      +/- 0.636896
b1.norm        0.000146661  +/- 5.15993e-05
b2.kT          0.556463     +/- 0.0103887
b2.norm        1.12408e-05  +/- 2.44213e-07



One may visually examine the fit with the Sherpa plot functions:

sherpa> set_xlog()
sherpa> set_ylog()
sherpa> plot_fit_delchi()
sherpa> plt.yscale('linear')

sherpa> plot_bkg_fit_resid()
sherpa> plt.yscale('linear')


We may also calculate the confidence 1σ error estimates on the individual parameters of the fit:

sherpa> conf()
b2.norm lower bound:	-2.55525e-07
b1.norm lower bound:	-4.66583e-05
b2.kT lower bound:	-0.0107563
b2.kT upper bound:	0.0106946
a1.nH lower bound:	-0.0108897
b1.norm upper bound:	9.71841e-05
b2.norm upper bound:	2.57521e-07
a1.nH upper bound:	0.0114366
b1.kT lower bound:	-0.656143
b1.kT upper bound:	1.00158
Dataset               = 1
Confidence Method     = confidence
Iterative Fit Method  = None
Fitting Method        = levmar
Statistic             = chi2gehrels
confidence 1-sigma (68.2689%) bounds:
Param            Best-Fit  Lower Bound  Upper Bound
-----            --------  -----------  -----------
a1.nH           0.0352536   -0.0108897    0.0114366
b1.kT             4.23927    -0.656143      1.00158
b1.norm       0.000146661 -4.66583e-05  9.71841e-05
b2.kT            0.556463   -0.0107563    0.0106946
b2.norm       1.12408e-05 -2.55525e-07  2.57521e-07


### Manually defining source and background models

We may choose to explicitly set the complete convolved model expression to be used for fitting a source data set and its associated background, using the new functions set_full_model and set_bkg_full_model, together with get_response or get_arf/get_rmf. These functions offer an alternative to using the functions set_source and set_bkg_model, which automatically convolve the appropriate instrument response with a defined source and background model expression.

The following set of commands represents the manual definition of the chosen source and background models, which were automatically (and equivalently) defined above with set_source and set_bkg_model.

sherpa> rsp = get_response()
sherpa> bkg_rsp = get_response(bkg_id=1)
sherpa> bkg_scale = get_bkg_scale()
sherpa> src_model = rsp(xswabs.a1 * xsbbody.b1)
sherpa> bkg_model = bkg_rsp(a1 * xsbbody.b2)
sherpa> set_full_model(src_model + bkg_scale * bkg_model)
sherpa> set_bkg_full_model(bkg_model)


You could also use get_arf and get_rmf in place of get_response, as follows:

sherpa> arf = get_arf()
sherpa> rmf = get_rmf()
sherpa> bgarf = get_arf(bkg_id=1)
sherpa> bgrmf = get_rmf(bkg_id=1)

sherpa> bkg_scale = get_bkg_scale()
sherpa> src_model = rmf(arf(xswabs.a1 * xsbbody.b1))
sherpa> bkg_model = bgrmf(bgarf(a1 * xsbbody.b2))
sherpa> set_full_model(src_model + bkg_scale * bkg_model)
sherpa> set_bkg_full_model(bkg_model)


Note that the Sherpa functions which are related to a source or background model defined with set_source or set_bkg_source, such as plot_source/plot_bkg_source or calc_energy_flux, are not compatible with the complete model expression defined by the set_full_model or set_bkg_full_model functions. In order to use these Sherpa functions, users should define source and background models in the usual way with the automatic functions set_source and set_bkg_source.

## Scripting It

The file fit.py is a Python script which performs the primary commands used above; it can be executed by typing %run -i fit.py on the Sherpa command line.

The Sherpa script command may be used to save everything typed on the command line in a Sherpa session:

sherpa> script(filename="sherpa.log", clobber=False)


(Note that restoring a Sherpa session from such a file could be problematic since it may include syntax errors, unwanted fitting trials, et cetera.)

## Summary

This thread is complete, so we can exit the Sherpa session:

sherpa> quit()


## History

 14 Jan 2005 updated for CIAO 3.2: minor change in fit results 21 Dec 2005 reviewed for CIAO 3.3: no changes 01 Dec 2006 reviewed for CIAO 3.4: no changes 08 Dec 2008 updated for Sherpa 4.1 29 Apr 2009 new script command is available with CIAO 4.1.2 17 Dec 2009 updated for CIAO 4.2: the fit_bkg function is available 14 Jun 2010 updated to include confidence 1-sigma error estimates on individual parameters of fits 14 Jun 2010 updated with an example using the CIAO 4.2 Sherpa v2 functions set_full_model, set_bkg_full_model, get_response, and a new usage of the functions get_arf and get_rmf. S-Lang version of thread removed. 15 Dec 2010 updated for CIAO 4.3: new functions get_bkg_scale, calc_stat_info, and set_xlog/set_ylog are available 22 Jun 2011 renamed thread from "Independent Background Responses" to "Simultaneously Fitting Source and Background Spectra" 15 Dec 2011 reviewed for CIAO 4.4: added a warning about filtering/grouping source and background 13 Dec 2012 reviewed for CIAO 4.5: removed an outdated warning about filtering/grouping both source and background data, as the associated bug was fixed 04 Dec 2013 reviewed for CIAO 4.6: removed references to deprecated tools, updated fit results 03 Dec 2015 updated for CIAO 4.8: no content change 09 Nov 2016 reviewed for CIAO 4.9: updated fits, no content change. 25 May 2018 reviewed for CIAO 4.10: no content change; typeset equations with LaTeX. 10 Dec 2018 reviewed for CIAO 4.11, screen output revised 11 Dec 2019 Updated to use Matplotlib in CIAO 4.12 and to take advantage of changes to the plot support (e.g. plot_fit(ylog=True)); the text and commands have been revised slightly.