Doing an unbinned analysisΒΆ
What you will learn
You will learn how to adjust a parametrised model to the events without binning the data.
Gamma-ray events are rare, hence the counts cubes generated by ctbin may be sparsly populated, having many empty pixels, in particular at high energies. In that case it may be worth to analyse the events directly with an unbinned maximum likelihood analysis.
An unbinned analysis is generally preferred over a binned analysis for short observation times (a few tens of hours) or if you want to assure that the analysis results are not biased by the selected binning.
An alternative analysis technique consists of working directly on the event list without binning the events in a counts cube. You do this with the ctlike tool as follows:
$ ctlike
Input event list, counts cube or observation definition XML file [cntcube.fits] selected_events.fits
Calibration database [prod2]
Instrument response function [South_0.5h]
Input model definition XML file [models.xml] $CTOOLS/share/models/crab.xml
Output model definition XML file [crab_results.xml]
You will recognise that ctlike runs much faster in unbinned mode compared to binned mode. This is understandable as the selected event list contains only 20877 events, while the binned counts cube you used before had 200 x 200 x 20 = 800000 bins. As unbinned maximum likelihood fitting loops over the events (while binned maximum likelihood loops over the bins), there are much less operations to perform in unbinned than in binned mode (there is some additional overhead in unbinned mode that comes from integrating the models over the region of interest, yet this is negligible compared to the operations needed when looping over all pixels). So as long as you work with small event lists, unbinned mode is faster (this typically holds up to few tens of hours of observing time). Unbinned ctlike should also be more precise as no binning is performed, hence there is no loss of information due to histogramming.
Below you see the corresponding output from the ctlike.log file. The fitted
parameters are essentially identical to the ones found in binned mode.
The slight difference with respect to the binned analysis may be explained
by the different event sample that were used for the analysis: while
binned likelihood works on rectangular counts cubes, unbinned likelihood works
on circular event selection regions. It is thus not possible to select exactly
the same events for both analyses.
2019-04-02T14:15:10: +=================================+
2019-04-02T14:15:10: | Maximum likelihood optimisation |
2019-04-02T14:15:10: +=================================+
2019-04-02T14:15:10: >Iteration 0: -logL=143782.837, Lambda=1.0e-03
2019-04-02T14:15:10: >Iteration 1: -logL=143779.346, Lambda=1.0e-03, delta=3.491, step=1.0e+00, max(|grad|)=5.346881 [Index:7]
2019-04-02T14:15:10: >Iteration 2: -logL=143779.343, Lambda=1.0e-04, delta=0.003, step=1.0e+00, max(|grad|)=-0.055672 [Index:3]
2019-04-02T14:15:10:
2019-04-02T14:15:10: +=========================================+
2019-04-02T14:15:10: | Maximum likelihood optimisation results |
2019-04-02T14:15:10: +=========================================+
2019-04-02T14:15:10: === GOptimizerLM ===
2019-04-02T14:15:10: Optimized function value ..: 143779.343
2019-04-02T14:15:10: Absolute precision ........: 0.005
2019-04-02T14:15:10: Acceptable value decrease .: 2
2019-04-02T14:15:10: Optimization status .......: converged
2019-04-02T14:15:10: Number of parameters ......: 10
2019-04-02T14:15:10: Number of free parameters .: 4
2019-04-02T14:15:10: Number of iterations ......: 2
2019-04-02T14:15:10: Lambda ....................: 1e-05
2019-04-02T14:15:10: Maximum log likelihood ....: -143779.343
2019-04-02T14:15:10: Observed events (Nobs) ...: 22708.000
2019-04-02T14:15:10: Predicted events (Npred) ..: 22707.995 (Nobs - Npred = 0.00519125738355797)
2019-04-02T14:15:10: === GModels ===
2019-04-02T14:15:10: Number of models ..........: 2
2019-04-02T14:15:10: Number of parameters ......: 10
2019-04-02T14:15:10: === GModelSky ===
2019-04-02T14:15:10: Name ......................: Crab
2019-04-02T14:15:10: Instruments ...............: all
2019-04-02T14:15:10: Observation identifiers ...: all
2019-04-02T14:15:10: Model type ................: PointSource
2019-04-02T14:15:10: Model components ..........: "PointSource" * "PowerLaw" * "Constant"
2019-04-02T14:15:10: Number of parameters ......: 6
2019-04-02T14:15:10: Number of spatial par's ...: 2
2019-04-02T14:15:10: RA .......................: 83.6331 [-360,360] deg (fixed,scale=1)
2019-04-02T14:15:10: DEC ......................: 22.0145 [-90,90] deg (fixed,scale=1)
2019-04-02T14:15:10: Number of spectral par's ..: 3
2019-04-02T14:15:10: Prefactor ................: 5.88338676901458e-16 +/- 1.02452856089807e-17 [1e-23,1e-13] ph/cm2/s/MeV (free,scale=1e-16,gradient)
2019-04-02T14:15:10: Index ....................: -2.49375950219757 +/- 0.0149889370322137 [-0,-5] (free,scale=-1,gradient)
2019-04-02T14:15:10: PivotEnergy ..............: 300000 [10000,1000000000] MeV (fixed,scale=1000000,gradient)
2019-04-02T14:15:10: Number of temporal par's ..: 1
2019-04-02T14:15:10: Normalization ............: 1 (relative value) (fixed,scale=1,gradient)
2019-04-02T14:15:10: Number of scale par's .....: 0
2019-04-02T14:15:10: === GCTAModelIrfBackground ===
2019-04-02T14:15:10: Name ......................: CTABackgroundModel
2019-04-02T14:15:10: Instruments ...............: CTA
2019-04-02T14:15:10: Observation identifiers ...: all
2019-04-02T14:15:10: Model type ................: "PowerLaw" * "Constant"
2019-04-02T14:15:10: Number of parameters ......: 4
2019-04-02T14:15:10: Number of spectral par's ..: 3
2019-04-02T14:15:10: Prefactor ................: 1.0018169793538 +/- 0.0133053833141539 [0.001,1000] ph/cm2/s/MeV (free,scale=1,gradient)
2019-04-02T14:15:10: Index ....................: -0.00708154249642314 +/- 0.00805278228449961 [-5,5] (free,scale=1,gradient)
2019-04-02T14:15:10: PivotEnergy ..............: 1000000 [10000,1000000000] MeV (fixed,scale=1000000,gradient)
2019-04-02T14:15:10: Number of temporal par's ..: 1
2019-04-02T14:15:10: Normalization ............: 1 (relative value) (fixed,scale=1,gradient)
Note
Many tools or scripts can also be used in unbinned mode, including csresmap, ctbutterfly and csspec that were used earlier. It is sufficient to replace the input counts cube by an event list to activate unbinned mode for these tools.