Fitting the model components to the data
Now you are ready to fit the model to the data and to determine its maximum likelihood parameters.
You do this with the :ref:`ctlike` tool that adjusts all parameters in the :ref:`model definition file <glossary_moddef>` that have the attribute free set to "1". In the current example, the free model parameters are the positions and spectral parameters of the two point sources and the spectral normalisation of the background component. You run the :ref:`ctlike` tool as follows:
$ ctlike
Input event list, counts cube or observation definition XML file [events.fits] obs_selected.xml
Input model definition XML file [$CTOOLS/share/models/crab.xml] models.xml
Output model definition XML file [crab_results.xml] results.xml
The tool will take a few minutes (on Mac OS X) to perform the model fitting, and will write the results into an updated :ref:`model definition file <glossary_moddef>` containing the fitted model parameters and their statistical uncertainties. You may inspect the log file ctlike.log to verify that the model fit converged properly, as illustrated in the example below:
2017-03-03T18:06:30: +=================================+
2017-03-03T18:06:30: | Maximum likelihood optimisation |
2017-03-03T18:06:30: +=================================+
2017-03-03T18:06:56: >Iteration 0: -logL=18221386.597, Lambda=1.0e-03
2017-03-03T18:07:19: >Iteration 1: -logL=18206762.996, Lambda=1.0e-03, delta=14623.601, max(|grad|)=45198.417696 [Index:13]
2017-03-03T18:07:42: >Iteration 2: -logL=18203712.440, Lambda=1.0e-04, delta=3050.557, max(|grad|)=10013.593417 [Index:3]
2017-03-03T18:08:04: >Iteration 3: -logL=18202302.390, Lambda=1.0e-05, delta=1410.050, max(|grad|)=7461.719387 [Index:3]
2017-03-03T18:08:27: >Iteration 4: -logL=18201825.720, Lambda=1.0e-06, delta=476.669, max(|grad|)=4076.653047 [RA:0]
2017-03-03T18:08:50: >Iteration 5: -logL=18201757.606, Lambda=1.0e-07, delta=68.114, max(|grad|)=1547.953907 [RA:0]
2017-03-03T18:09:13: >Iteration 6: -logL=18201754.932, Lambda=1.0e-08, delta=2.674, max(|grad|)=388.888160 [RA:0]
2017-03-03T18:09:36: >Iteration 7: -logL=18201754.869, Lambda=1.0e-09, delta=0.063, max(|grad|)=81.488909 [RA:0]
2017-03-03T18:09:59: >Iteration 8: -logL=18201754.866, Lambda=1.0e-10, delta=0.003, max(|grad|)=16.367278 [RA:0]
2017-03-03T18:10:22:
2017-03-03T18:10:22: +=========================================+
2017-03-03T18:10:22: | Maximum likelihood optimisation results |
2017-03-03T18:10:22: +=========================================+
2017-03-03T18:10:22: === GOptimizerLM ===
2017-03-03T18:10:22: Optimized function value ..: 18201754.866
2017-03-03T18:10:22: Absolute precision ........: 0.005
2017-03-03T18:10:22: Acceptable value decrease .: 2
2017-03-03T18:10:22: Optimization status .......: converged
2017-03-03T18:10:22: Number of parameters ......: 16
2017-03-03T18:10:22: Number of free parameters .: 10
2017-03-03T18:10:22: Number of iterations ......: 8
2017-03-03T18:10:22: Lambda ....................: 1e-11
2017-03-03T18:10:22: Maximum log likelihood ....: -18201754.866
2017-03-03T18:10:22: Observed events (Nobs) ...: 2841472.000
2017-03-03T18:10:22: Predicted events (Npred) ..: 2841471.996 (Nobs - Npred = 0.00435844948515296)
You may also convert the fitted model positions into a ds9 region file using the :ref:`csmodelinfo` script so that you can overlay the fit results over a sky map:
$ csmodelinfo pnt_type=circle free_color=black show_labels=no
Input model definition XML file [model.xml] results.xml
Output DS9 region file [ds9.reg] positions.reg
The command line arguments pnt_type, free_color and show_labels enable to fine tune the parameters in the ds9 region file. In this case, the positions are marked by black circles without showing the source names.
The following image shows a zoom of the sky map that comprises both point sources, with the initial source positions determined by :ref:`cssrcdetect` as green crosses and the positions fitted by :ref:`ctlike` as black circles. Obviously, the initial positions were already near the fitted positions, which is required to assure the proper convergence of the fit.
Background subtracted sky map of the events recorded around the Galactic Centre during the Galactic Centre Survey with the fitted positions of the sources shown as black circles
You can also convert the spectral parameters of the point sources into a butterfly diagram for each source using the :ref:`ctbutterfly` tool. The butterfly diagram shows the envelope of all power laws that are statistically compatible with the data. You create the butterfly diagram for the first source using
$ ctbutterfly
Input event list, counts cube or observation definition XML file [events.fits] obs_selected.xml
Source of interest [Crab] Src001
Input model definition XML file [$CTOOLS/share/models/crab.xml] results.xml
Start value for first energy bin in TeV [0.1]
Stop value for last energy bin in TeV [100.0]
Output ASCII file [butterfly.txt] butterfly_src001.txt
and for the second source using
$ ctbutterfly
Input event list, counts cube or observation definition XML file [events.fits] obs_selected.xml
Source of interest [Crab] Src002
Input model definition XML file [$CTOOLS/share/models/crab.xml] results.xml
Start value for first energy bin in TeV [0.1]
Stop value for last energy bin in TeV [100.0]
Output ASCII file [butterfly.txt] butterfly_src002.txt
The butterfly diagrams for both sources are displayed in the figure below:
Butterfly diagrams determined with ctbutterfly and spectral points determined with csspec for Src001 (top) and Src002 (bottom)
The figure also shows spectral points for each source that were determined using the :ref:`csspec` script. You create the spectrum for the first source using
$ csspec
Input event list, counts cube, or observation definition XML file [events.fits] obs_selected.xml
Input model definition XML file [$CTOOLS/share/models/crab.xml] results.xml
Source name [Crab] Src001
Binning algorithm (LIN|LOG|FILE) [LOG]
Lower energy limit (TeV) [0.1]
Upper energy limit (TeV) [100.0]
Number of energy bins (0=unbinned) [20] 10
Output spectrum file [spectrum.fits] spectrum_src001.fits
and for the second source using
$ csspec
Input event list, counts cube, or observation definition XML file [events.fits] obs_selected.xml
Input model definition XML file [$CTOOLS/share/models/crab.xml] results.xml
Source name [Crab] Src002
Binning algorithm (LIN|LOG|FILE) [LOG]
Lower energy limit (TeV) [0.1]
Upper energy limit (TeV) [100.0]
Number of energy bins (0=unbinned) [20] 10
Output spectrum file [spectrum.fits] spectrum_src002.fits
The :ref:`csspec` script divided here the data into ten logarithmically spaced energy bins and determined the source flux in each of the bins using a maximum likelihood model fit.