Simulating Chandra ACIS-S LETG Spectra with Sherpa: Establishing the Source Model Expression
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Sherpa Threads (CIAO 4.16 Sherpa)
Overview
Synopsis:
The thread "Simulating Chandra ACIS-S LETG Spectra with Sherpa" illustrates the use of the Sherpa fake_pha command to simulate a spectrum of a point source with the ACIS-S/LETG detector/grating configuration aboard Chandra, using a source model tailored to an existing ACIS-S/HETG data set. This thread details the steps taken to define this source model expression.
Last Update: 13 Dec 2019 - Updated for CIAO 4.12: switched from ChIPS to Matplotlib.
Contents
- Defining the Instrument Response for the Real Data
- Filtering the Data
- Defining the Source and Background Models
- Examining Method & Statistic Settings
- Fitting
- Examining Fit Results
- Saving and Quitting the Session
- History
- Images
Defining the Instrument Response for the Real Data
We begin by loading the ACIS-S/HETG data sets which will serve as the guide for our simulated ACIS-S/LETG data. The Sherpa command load_pha will automatically establish the instrument responses and backgrounds associated with each of the four spectral orders, provided their associated filenames are recorded in the header of the PHA files (if not, they should be loaded manually with the load_arf and load_rmf commands).
sherpa> load_pha(1, "459_heg_m1_bin10.pha") WARNING: systematic errors were not found in file '459_heg_m1_bin10.pha' statistical errors were found in file '459_heg_m1_bin10.pha' but not used; to use them, re-read with use_errors=True read background_up into a dataset from file 459_heg_m1_bin10.pha read background_down into a dataset from file 459_heg_m1_bin10.pha sherpa> load_pha(2, "459_heg_p1_bin10.pha") WARNING: systematic errors were not found in file '459_heg_p1_bin10.pha' statistical errors were found in file '459_heg_p1_bin10.pha' but not used; to use them, re-read with use_errors=True read background_up into a dataset from file 459_heg_p1_bin10.pha read background_down into a dataset from file 459_heg_p1_bin10.pha sherpa> load_pha(3, "459_meg_m1_bin10.pha") WARNING: systematic errors were not found in file '459_meg_m1_bin10.pha' statistical errors were found in file '459_meg_m1_bin10.pha' but not used; to use them, re-read with use_errors=True read background_up into a dataset from file 459_meg_m1_bin10.pha read background_down into a dataset from file 459_meg_m1_bin10.pha sherpa> load_pha(4, "459_meg_p1_bin10.pha") WARNING: systematic errors were not found in file '459_meg_p1_bin10.pha' statistical errors were found in file '459_meg_p1_bin10.pha' but not used; to use them, re-read with use_errors=True read background_up into a dataset from file 459_meg_p1_bin10.pha read background_down into a dataset from file 459_meg_p1_bin10.pha sherpa> load_arf(1, "459_heg_m1.arf") sherpa> load_arf(2, "459_heg_p1.arf") sherpa> load_arf(3, "459_meg_m1.arf") sherpa> load_arf(4, "459_meg_p1.arf") sherpa> load_arf(1, "459_heg_m1.arf", bkg_id=1) sherpa> load_arf(1, "459_heg_m1.arf", bkg_id=2) sherpa> load_arf(2, "459_heg_p1.arf", bkg_id=1) sherpa> load_arf(2, "459_heg_p1.arf", bkg_id=2) sherpa> load_arf(3, "459_meg_m1.arf", bkg_id=1) sherpa> load_arf(3, "459_meg_m1.arf", bkg_id=2) sherpa> load_arf(4, "459_meg_p1.arf", bkg_id=1) sherpa> load_arf(4, "459_meg_p1.arf", bkg_id=2)
Sherpa now refers to the spectra as follows:
- HEG, -1 order = dataset 1
- HEG, +1 order = dataset 2
- MEG, -1 order = dataset 3
- MEG, +1 order = dataset 4
The current definition of the instrument response, along with information about the loaded data sets, may be examined using the commands show_data and show_bkg (only show_data is shown below since it includes the output from show_bkg):
sherpa> show_data() Data Set: 1 Filter: 0.5775-12.3676 Energy (keV) Bkg Scale: 0.248829 Noticed Channels: 1-8192 name = 459_heg_m1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 1.0 areascal = 1.0 grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [1, 2] ARF Data Set: 1:1 name = 459_heg_m1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38565.284798602 ethresh = 1e-10 Background Data Set: 1:1 Filter: 0.5775-12.3676 Energy (keV) Noticed Channels: 1-8192 name = 459_heg_m1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 1:1 name = 459_heg_m1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38565.284798602 ethresh = 1e-10 Background Data Set: 1:2 Filter: 0.5775-12.3676 Energy (keV) Noticed Channels: 1-8192 name = 459_heg_m1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 1:2 name = 459_heg_m1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38565.284798602 ethresh = 1e-10 Data Set: 2 Filter: 0.5775-12.3676 Energy (keV) Bkg Scale: 0.248829 Noticed Channels: 1-8192 name = 459_heg_p1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 1.0 areascal = 1.0 grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [1, 2] ARF Data Set: 2:1 name = 459_heg_p1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38563.013342202 ethresh = 1e-10 Background Data Set: 2:1 Filter: 0.5775-12.3676 Energy (keV) Noticed Channels: 1-8192 name = 459_heg_p1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 2:1 name = 459_heg_p1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38563.013342202 ethresh = 1e-10 Background Data Set: 2:2 Filter: 0.5775-12.3676 Energy (keV) Noticed Channels: 1-8192 name = 459_heg_p1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 2:2 name = 459_heg_p1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38563.013342202 ethresh = 1e-10 Data Set: 3 Filter: 0.2957-12.3370 Energy (keV) Bkg Scale: 0.248829 Noticed Channels: 1-8192 name = 459_meg_m1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 1.0 areascal = 1.0 grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [1, 2] ARF Data Set: 3:1 name = 459_meg_m1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38565.285573929 ethresh = 1e-10 Background Data Set: 3:1 Filter: 0.2957-12.3370 Energy (keV) Noticed Channels: 1-8192 name = 459_meg_m1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 3:1 name = 459_meg_m1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38565.285573929 ethresh = 1e-10 Background Data Set: 3:2 Filter: 0.2957-12.3370 Energy (keV) Noticed Channels: 1-8192 name = 459_meg_m1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 3:2 name = 459_meg_m1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38565.285573929 ethresh = 1e-10 Data Set: 4 Filter: 0.2957-12.3370 Energy (keV) Bkg Scale: 0.248829 Noticed Channels: 1-8192 name = 459_meg_p1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 1.0 areascal = 1.0 grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [1, 2] ARF Data Set: 4:1 name = 459_meg_p1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38563.014116893 ethresh = 1e-10 Background Data Set: 4:1 Filter: 0.2957-12.3370 Energy (keV) Noticed Channels: 1-8192 name = 459_meg_p1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 4:1 name = 459_meg_p1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38563.014116893 ethresh = 1e-10 Background Data Set: 4:2 Filter: 0.2957-12.3370 Energy (keV) Noticed Channels: 1-8192 name = 459_meg_p1_bin10.pha channel = Float64[8192] counts = Float64[8192] staterror = None syserror = None bin_lo = Float64[8192] bin_hi = Float64[8192] grouping = Int16[8192] quality = Int16[8192] exposure = 38564.608926889 backscal = 4.0188284 areascal = None grouped = True subtracted = False units = energy rate = True plot_fac = 0 response_ids = [1] background_ids = [] Background ARF Data Set: 4:2 name = 459_meg_p1.arf energ_lo = Float64[8192] energ_hi = Float64[8192] specresp = Float64[8192] bin_lo = Float64[8192] bin_hi = Float64[8192] exposure = 38563.014116893 ethresh = 1e-10
To visualize the four orders of data in wavelength space before establishing a source model expression to fit, we ensure that the units field of each data set is set to "wavelength" with the set_analysis command, and then plot the orders with the plot_data command:
sherpa> set_analysis("wave")
sherpa> plot_data(1, ylog=True)
sherpa> plot_data(2, overplot=True)
sherpa> plot_data(3, overplot=True)
sherpa> plot_data(4, overplot=True)
The resulting plot is shown in Figure 1
Figure 1: Plotting the four orders
![[The four datasets are drawn in blue and orange (these look similar) for the HEG -1 and +1 orders, and green and red for the MEG -1 and +1 orders. The MEG has more counts longward of about 5 Angstrom, and the MEG -1 order shows more counts than the +1 order at longer wavelengths than about 15 Angstroms.]](all-data.png)
Figure 1: Plotting the four orders
The blue and orange data points are for the HEG -1 and +1 orders, the green and red are the MEG -1 and +1 orders.
Now that Sherpa data sets and associated instrument responses for the ACIS-S/HETG data have been established, we are ready to define the source and background model expressions which best describe this data. The resulting source model will be applied to the simulated ACIS-S/LETG data in the section "Running the Simulation with fake_pha" of the thread "Simulating Chandra ACIS-S LETG Spectra with Sherpa."
Filtering the Data
To get the best fit to the data, we first choose to filter the data in wavelength space to exclude data above 15.0 Angstroms and below 1.0 Angstrom.
sherpa> show_filter() Data Set Filter: 1 1.0025-21.4675 Wavelength (Angstrom) Data Set Filter: 2 1.0025-21.4675 Wavelength (Angstrom) Data Set Filter: 3 1.0050-41.9350 Wavelength (Angstrom) Data Set Filter: 4 1.0050-41.9350 Wavelength (Angstrom) sherpa> ignore() sherpa> notice(1.0, 15.) sherpa> show_filter() Data Set Filter: 1 1.0025-14.9925 Wavelength (Angstrom) Data Set Filter: 2 1.0025-14.9925 Wavelength (Angstrom) Data Set Filter: 3 1.0050-14.9850 Wavelength (Angstrom) Data Set Filter: 4 1.0050-14.9850 Wavelength (Angstrom)
A plot of the four orders filtered in wavelength space is shown in Figure 2.
sherpa> plt.clf()
sherpa> plt.subplot(2, 1, 1)
sherpa> plot_data(1, ylog=True, clearwindow=False)
sherpa> plot_data(2, overplot=True)
sherpa> plt.title('HEG -1/+1')
sherpa> plt.subplot(2, 1, 2)
sherpa> plot_data(3, ylog=True, clearwindow=False)
sherpa> plot_data(4, overplot=True)
sherpa> plt.subplots_adjust(hspace=0.5)
sherpa> plt.title('MEG -1/+1')
Figure 2: Filtering the data sets
![[Plot of filtered ACIS HEG and MEG +/- 1 orders for 3C 273 Cygni in energy space]](data-filtered.png)
Figure 2: Filtering the data sets
The top plot shows the HEG and the bottom plot the MEG, where blue is for the -1 orders (datasets 1 and 3) and orange for the +1 orders (datasets 2 and 4).
This plot was created by combining Sherpa plot_data calls with Matplotlib calls for creating and adjusting plot areas.
Defining the Source and Background Models
We plan on simultaneously fitting the background data (rather than subtracting it), so we need to create a model expression for the source and the background. We model this source with a broken power law (bpl1d) absorbed by the interstellar medium (atten). The background will be modeled by a one-dimensional power law (powlaw1d), also absorbed by the ISM (the same atten model).
First, we set up the model components with create_model_component, and set some initial parameter values. The absorption model is referred to as "abs1", the broken power law as "bpow1", and the 1-D power law as "pow1d":
sherpa> create_model_component("atten", "abs1")
sherpa> abs1.integrate = False
sherpa> abs1.hcol = 1.8e+20
sherpa> abs1.heiRatio = 0.1
sherpa> abs1.heiiRatio = 0.01
sherpa> freeze(abs1)
sherpa> create_model_component("bpl1d", "bpow1")
sherpa> bpow1.gamma1 = 0
sherpa> bpow1.gamma2 = 0
sherpa> bpow1.eb = 7.99625
sherpa> freeze(bpow1.ref)
sherpa> bpow1.ampl = 0.001
sherpa> create_model_component("powlaw1d","pow1d")
sherpa> pow1d.gamma = 1
sherpa> pow1d.ampl = 1e-5
We freeze the normalization parameters for bpow1 and pow1d (bpow1.ref and pow1d.ref) without changing the default values. For the bpow1 and pow1d parameters for which we did set initial values, we could have used the Sherpa guess() function to estimate reasonable starting values, based on the data input to the Sherpa session. To have Sherpa automatically query for initial parameter values when a model is established, set 'paramprompt(True)' (it is 'False' by default).
The model parameter values can be listed with print() command (note that show_model/show_source is appropriate once the full model expression has been assigned to the data with set_source:
sherpa> print(abs1) atten.abs1 Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 sherpa> print(bpow1) bpl1d.bpow1 Param Type Value Min Max Units ----- ---- ----- --- --- ----- bpow1.gamma1 thawed 0 -10 10 bpow1.gamma2 thawed 0 -10 10 bpow1.eb thawed 7.99625 0 1e+03 bpow1.ref frozen 1 -3.4e+38 3.4e+38 bpow1.ampl thawed 0.001 1e-20 3.4e+38 sherpa> print(pow1d) powlaw1d.pow1d Param Type Value Min Max Units ----- ---- ----- --- --- ----- pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.4e+38 3.4e+38 pow1d.ampl thawed 1e-05 0 3.4e+38
Now that the model components have been established, the product of abs1 and bpow1 is assigned as the source model for all data sets :
sherpa> src_mdl = ans1 * bpow1 sherpa> set_source(1, src_mdl) sherpa> set_source(2, src_mdl) sherpa> set_source(3, src_mdl) sherpa> set_source(4, src_mdl)
while the background model is set as the product of abs1 and pow1d (a Python loop is used instead of editing the previous command, to make sure we don't miss anything):
sherpa> bkg_mdl = abs1 * pow1d
sherpa> for sid in [1, 2, 3, 4]:
...: for bid in [1, 2]:
...: set_bkg_model(sid, bkg_mdl, bkg_id=bid)
The source and background model definitions can be listed with show_model and show_bkg_model:
sherpa> show_model() Model: 1 apply_arf((38564.608926889 * ((atten.abs1 * bpl1d.bpow1) + 0.248829 * ((atten.abs1 * powlaw1d.pow1d) + (atten.abs1 * powlaw1d.pow1d))))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 bpow1.gamma1 thawed 0 -10 10 bpow1.gamma2 thawed 0 -10 10 bpow1.eb thawed 7.99625 0 1000 bpow1.ref frozen 1 -3.40282e+38 3.40282e+38 bpow1.ampl thawed 0.001 1e-20 3.40282e+38 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 Model: 2 apply_arf((38564.608926889 * ((atten.abs1 * bpl1d.bpow1) + 0.248829 * ((atten.abs1 * powlaw1d.pow1d) + (atten.abs1 * powlaw1d.pow1d))))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 bpow1.gamma1 thawed 0 -10 10 bpow1.gamma2 thawed 0 -10 10 bpow1.eb thawed 7.99625 0 1000 bpow1.ref frozen 1 -3.40282e+38 3.40282e+38 bpow1.ampl thawed 0.001 1e-20 3.40282e+38 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 Model: 3 apply_arf((38564.608926889 * ((atten.abs1 * bpl1d.bpow1) + 0.248829 * ((atten.abs1 * powlaw1d.pow1d) + (atten.abs1 * powlaw1d.pow1d))))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 bpow1.gamma1 thawed 0 -10 10 bpow1.gamma2 thawed 0 -10 10 bpow1.eb thawed 7.99625 0 1000 bpow1.ref frozen 1 -3.40282e+38 3.40282e+38 bpow1.ampl thawed 0.001 1e-20 3.40282e+38 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 Model: 4 apply_arf((38564.608926889 * ((atten.abs1 * bpl1d.bpow1) + 0.248829 * ((atten.abs1 * powlaw1d.pow1d) + (atten.abs1 * powlaw1d.pow1d))))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 bpow1.gamma1 thawed 0 -10 10 bpow1.gamma2 thawed 0 -10 10 bpow1.eb thawed 7.99625 0 1000 bpow1.ref frozen 1 -3.40282e+38 3.40282e+38 bpow1.ampl thawed 0.001 1e-20 3.40282e+38 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 sherpa> show_bkg_model() Background Model: 1:1 apply_arf((38564.608926889 * (atten.abs1 * powlaw1d.pow1d))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 Background Model: 1:2 apply_arf((38564.608926889 * (atten.abs1 * powlaw1d.pow1d))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 Background Model: 2:2 apply_arf((38564.608926889 * (atten.abs1 * powlaw1d.pow1d))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 Background Model: 3:2 apply_arf((38564.608926889 * (atten.abs1 * powlaw1d.pow1d))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38 Background Model: 4:2 apply_arf((38564.608926889 * (atten.abs1 * powlaw1d.pow1d))) Param Type Value Min Max Units ----- ---- ----- --- --- ----- abs1.hcol frozen 1.81e+20 1e+17 1e+24 abs1.heiRatio frozen 0.1 0 1 abs1.heiiRatio frozen 0.01 0 1 pow1d.gamma thawed 1 -10 10 pow1d.ref frozen 1 -3.40282e+38 3.40282e+38 pow1d.ampl thawed 1e-05 0 3.40282e+38
Examining Method & Statistic Settings
Next we check the current method and statistics settings:
sherpa> show_method()
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
sherpa> get_stat_name()
'chi2gehrels'
The Sherpa default fitting statistic and optimization method are Chi2Gehrels and LevMar, respectively. For this fit, we will use the Neldermead method and the CStat statistic - Chi2Gehrels could bias the fit results and yield an artificially low reduced statistic for this data, and the CStat (and Cash) statistic is appropriate for simultaneously fitting source and background data. For a list of all the available methods and statistic settings, see the Sherpa Statistics and Optimization Methods pages.
To change the current method and statistic, we use set_method and set_stat
sherpa> set_method("neldermead")
sherpa> set_stat("cstat")
Fitting
The data sets are now fit:
sherpa> fit() Datasets = 1, 2, 3, 4 Method = neldermead Statistic = cstat Initial fit statistic = 100816 Final fit statistic = 5976.06 at function evaluation 1548 Data points = 5052 Degrees of freedom = 5046 Probability [Q-value] = 1.06549e-18 Reduced statistic = 1.18432 Change in statistic = 94840 bpow1.gamma1 0.424705 bpow1.gamma2 0.0496687 bpow1.eb 10.6201 bpow1.ampl 0.00548922 pow1d.gamma 0.33154 pow1d.ampl 8.87688e-05
To plot the fits (the result is Figure 3):
sherpa> pprefs = get_data_plot_prefs() sherpa> pprefs['yerrorbars'] = False sherpa> plot("fit", 1, "fit", 2, "fit", 3, "fit", 4) sherpa> plt.subplots_adjust(hspace=0.5, wspace=0.4) sherpa> axes = plt.gcf().axes sherpa> axes[0].set_title('HEG -1') sherpa> axes[1].set_title('HEG +1') sherpa> axes[2].set_title('MEG -1') sherpa> axes[3].set_title('MEG +1')
Figure 3: Results of Simultaneous Fit
![[Labeled plots of the simultaneous fit on ACIS HEG and MEG +/- 1 orders of 3C 273]](all-fits.png)
Figure 3: Results of Simultaneous Fit
Labeled plots of the simultaneous fit on ACIS HEG and MEG +/- 1 orders of 3C 273.
Note that the cstat statistic does not calculate errors for the data points (which is why the yerrorbars preference setting was set to False). Since it is useful to do so, we change the fit statistic to something suitable for calculating errors, and view the residuals of the fit with the plot_fit_delchi command (see Figure 4):
sherpa> set_stat('chi2datavar')
sherpa> pprefs['yerrorbars'] = True
sherpa> plot_fit_delchi(4)
Figure 4: Fit and residuals for the MEG +1 order spectrum
![[Plot of the fit and residuals for the MEG +1 order spectrum]](fit-delchi-4.png)
Figure 4: Fit and residuals for the MEG +1 order spectrum
Examining Fit Results
The output of the fit command returns information on the best-fit statistic, and any associated "goodness of fit" (for chi-squared-like statistics). This information can also be retrieced with get_fit_results and show_fit.
sherpa> show_fit()
Optimization Method: NelderMead
name = simplex
ftol = 1.1920928955078125e-07
maxfev = None
initsimplex = 0
finalsimplex = 9
step = None
iquad = 1
verbose = 0
Statistic: Chi2DataVar
Chi Squared with data variance.
The variance in each bin is estimated from the data value in that
bin.
If the number of counts in each bin is large, then the shape of
the Poisson distribution from which the counts are sampled tends
asymptotically towards that of a Gaussian distribution, with
variance
sigma(i)^2 = N(i,S) + [A(S)/A(B)]^2 N(i,B)
where N is the number of on-source (and off-source) bins included
in the fit. The background term appears only if an estimate of the
background has been subtracted from the data.
See Also
--------
Chi2Gehrels, Chi2ModVar, Chi2XspecVar
Fit:Datasets = 1, 2, 3, 4
Method = neldermead
Statistic = cstat
Initial fit statistic = 100816
Final fit statistic = 5976.06 at function evaluation 1548
Data points = 5052
Degrees of freedom = 5046
Probability [Q-value] = 1.06549e-18
Reduced statistic = 1.18432
Change in statistic = 94840
bpow1.gamma1 0.424705
bpow1.gamma2 0.0496687
bpow1.eb 10.6201
bpow1.ampl 0.00548922
pow1d.gamma 0.33154
pow1d.ampl 8.87688e-05
The number of bins in the fit (Data points), the number of degrees of freedom (i.e. the number of bins minus the number of free parameters), and the final fit statistic value are reported. If the chosen statistic is one of the chi-square statistics, as in this example, the reduced statistic, i.e. the statistic value divided by the number of degrees of freedom, and the probability (Q-value) are included as well.
The confidence (conf), covariance (covar) and projection (proj) commands can be used to estimate confidence intervals for the thawed parameters (in this particular example the runtime of the conf call is long):
sherpa> set_stat("cstat")
sherpa> conf()
pow1d.ampl lower bound: -5.50626e-06
bpow1.gamma1 -: WARNING: The confidence level lies within (4.157243e-01, 4.202145e-01)
bpow1.gamma1 lower bound: -0.00673539
bpow1.eb lower bound: -0.318481
pow1d.gamma lower bound: -0.0325211
bpow1.eb upper bound: 0.281067
pow1d.gamma upper bound: 0.0322908
bpow1.ampl -: WARNING: The confidence level lies within (5.395501e-03, 5.442362e-03)
bpow1.ampl lower bound: -7.02911e-05
pow1d.ampl upper bound: 5.84816e-06
bpow1.gamma1 upper bound: 0.00708619
bpow1.ampl upper bound: 7.32199e-05
bpow1.gamma2 lower bound: -0.0636349
bpow1.gamma2 upper bound: 0.0683684
Datasets = 1, 2, 3, 4
Confidence Method = confidence
Iterative Fit Method = None
Fitting Method = neldermead
Statistic = cstat
confidence 1-sigma (68.2689%) bounds:
Param Best-Fit Lower Bound Upper Bound
----- -------- ----------- -----------
bpow1.gamma1 0.424705 -0.00673539 0.00708619
bpow1.gamma2 0.0496687 -0.0636349 0.0683684
bpow1.eb 10.6201 -0.318481 0.281067
bpow1.ampl 0.00548922 -7.02911e-05 7.32199e-05
pow1d.gamma 0.33154 -0.0325211 0.0322908
pow1d.ampl 8.87688e-05 -5.50626e-06 5.84816e-06
Saving and Quitting the Session
Before exiting Sherpa, you may wish to save the session in order to return to the analysis at a later point:
sherpa> save("459_fitting_session.save") sherpa> save("459_fitting_session.ascii")
The save function records all the information about the current session to the binary file 459_fitting_session.save, and the save_all function records the session settings to an editable ASCII file.
To restore the session that was saved to the binary file 459_fitting_session.save or ASCII file 459_fitting_session.ascii:
sherpa> restore("session1.save")
sherpa> %run -i session1.ascii
Finally, quit the session:
sherpa> quit
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.
History
| 02 Apr 2009 | Created for CIAO/Sherpa 4.1 |
| 15 Jan 2010 | updated for CIAO 4.2 |
| 13 Jul 2010 | updated for CIAO 4.2 Sherpa v2: removal of S-Lang version of thread. |
| 13 Dec 2019 | Updated for CIAO 4.12: switched from ChIPS to Matplotlib. |