PIG 9 - Event Sampling¶
- Author: Fabio Pintore, Andrea Giuliani, Axel Donath
- Created: May 03, 2019
- Accepted: August 30, 2019
- Status: accepted
- Discussion: GH_2136
An event sampler for gamma events is an important part of the science tools of the future Cherenkov Telescope Array (CTA) observatory. It will allow users to simulate observations of sources with different spectral, morphological and temporal properties and predict the performance of CTA on the simulated events e.g. to support observation proposals or study the sensitivity of future observations. For this reason, we propose to implement a framework for event simulation in Gammapy.
The proposed framework consists of individual building blocks, represented by classes and methods, that can be chained together to achieve a full simulation of an event list corresponding to a given observation. This includes the simulation of source events, background events, effects of instrument response functions (IRF) and arrival times. As underlying method for the actual event sampling we propose to use inverse cumulative distribution function (CDF) sampling (inverseCDF) with finely binned discrete source and background. Temporal models will be also taken into account and time will be sampled separately in a 1D analysis, assuming that the temporal dependency of the input source models factorizes.
Inverse CDF sampling (inverseCDF) is an established method to sample from discrete
probability mass functions. It is used by
ASTRIsim (astrisim), the event simulator
of the AGILE collaboration. However it is not the method of choice for other existing
event samplers such as the the Fermi-LAT Science Tools (gtobsim) and CTOOLS (ctobssim).
The latter uses a combination of analytical sampling for models, where a solution is
known (e.g. power-laws) and the rejection sampling method (rej_sampl), where the sampling
has to be done numerically (see an example here gammalib).
As rejection sampling can directly sample from continuous probability density functions, it is expected to yield very precise results. However an enveloping distribution is needed, which should be adapted to the target distribution to be efficient (see also rejection sampling in Python for an example implementation), otherwise a lot of computation time is spend in rejecting drawn samples.
For this reason we favour the inverse CDF sampling method, as a simple to implement and general sampling method. The precision of the inverse CDF sampling method can be controlled by the resolution of the input probability mass function (PMF) and is in practice only limited by the available memory. We will study the required bin-size of the PMFs to reach sufficient precision. If we find the inverse CDF sampling method to be not precise enough, it is still possible to achieve better precision adopting the rejection sampling. This will not have a strong impact on the structure of the event-sampler.
We propose to include in
gammapy.cube an high level interface (HLI) class, labelled as
MapDataset.sample method. This class handles the complete
event sampling process, including the corrections related to the IRF and source temporal
variability, for a given GTI / observation.
The dataset will be computed using the standard data reduction procedure of Gammapy, as illustrated in the following example:
obs = Observation(pointing, gti, aeff, psf, edisp, expomap) maker = MapDatasetMaker(geom, geom_irf, ...) dataset = maker.run(obs) model = SkyModels.read("model.yaml") dataset.model = model sampler = MapDatasetEventSampler(dataset) events = sampler.sample() events.write()
After data reduction, the Dataset object should contain all the needed information,
such as the pointing sky coordinates, the GTI, and the setup of all the models (spectra,
spatial morphology, temporal model) for any given source, and it is passed
as input parameter to the
MapDatasetEventSampler. It is important to note that the
MapDataset object can store information for more than one source. Then, a
will draw the sampled events and will provide an output
~astropy.table.Table object. The
latter will containt the reconstructed sky positions, energies, times, and an
EVENT_ID is a unique number or a string to identify the sampled event, while
MC_ID is a unique ID (number or string) to identify the model component the event was sampled
MapDatasetSampler should also fill the mandatory header information for event list
files decribed on gamma-astro-data-formats.
The general design of the
sample method is as follows:
def sample(dataset, random_state) """Sample events from a ``MapDataset``""" events_list =  for evaluator in dataset.evaluators: npred = evaluator.compute_npred() n_events = random_state.poisson(npred.data.sum()) events = npred.sample(n_events, random_state) time = LightCurveTableModel.sample(n_events=, lc=, random_state=) events = hstack(events,time) events_list.append(events) event_list["MC_ID"] = evaluator.model.name events_src = vstack(events_list) events_src = dataset.psf.sample(events_src, random_state) events_src = dataset.edisp.sample(events_src, random_state) n_events_bkg = random_state.poisson(dataset.background_model.map.data.sum()) events_bkg = dataset.background_model.sample(n_events, random_state) events_total = vstack([events_src, events_bkg]) events_total.meta = get_events_meta_data(dataset) return EventList(events_total)
In more detail,
sample starts a loop over the sources stored into the
model. Then, for each source, the
src.compute_npred method will calculate the predicted
number of source counts
npred. In particular, it is important to note that
npred = exposure * flux, where
exposure is defined as
effective_area * exposure_time.
npred is therefore calculated irrespective of the energy dispersion and of PSF.
npred will be the input of the
npred.sample method. The latter uses a Poisson
distribution, with mean equal to the predicted counts, to estimate the random number of sampled
We propose to add a
Map.sample(n_events=, random_state=) method in
that will be the core of the sampling process. The
sample is based on the
~gammapy.utils.random.InverseCDFSampler class described in GH_2229 . The output
will be an
~astropy.table.Table with columns:
Then, the time will be sampled independently using the temporal information stored into
MapDataset model for each source of interest. This will be done through a
.sample(n_events=, random_state=) method that we propose to add
This method will take as input the GTIs (i.e. one Tstart and Tstop) in the
object. Also in this case the
class is the machine used to sample the time of the events. In the case the temporal
model is not provided, the time is uniformly sampled in the time range
To define a light-curve per model component, the current
SkyModel class will be
extended by a
The IRF correction can now be applied to sampled events. We propose to add a
.sample(events=) method in both
The method interpolates the “correct” IRF at the position of a given event and
applies it. In more detail, the method calculates the psf and the energy dispersion at
the events true positions and true energies, which are given in input as an
The IRFs are assumed to be constant and not time-dependent. The output will be an
with the new columns
ENERGY, which are the reconstructed event energies and
Finally, the times and the energies/coordinates of the events will be merged into a
~astropy.table.Table with the columns:
MapDatasetEventSampler can be used to sample background events using the
Map.sample(n_events=, random_state=) as well. The time of the events is sampled
assuming a constant event rate. Finally, the IRF corrections are not applied to background
Performance and precision evaluation¶
To evaluate the precision and performance of the described framework we propose to implement a prototype for a simulation / fitting pipeline. Starting from a selection of spatial, spectral and temporal models, data are simulated and fitted multiple times to derive distributions and pull-distributions of the reconstructed model parameters. This pipeline should also monitor the required cpu and memory usage. This first prototype can be used to evaluate the optimal bin-size (with the best compromise between performance and precision) for the simulations and to verify the over-all correctness of the simulated data. This will be valid for a set of input maps and IRFs. Later this prototype can be developed further into a full simulation / fitting continuous integration system.
Alternatives / Outlook¶
So far Gammapy only supports binned likelihood analysis and technically most use-
cases for the event sampling could be solved with binned simulations. A binned
simulation can be basically achieved by a call to
on the predicted number of counts map. This is conceptionally simpler as well as
computationally more efficient than a sampling of event lists. In
Gammapy a similar
dataset simulation is already implemented in
Dataset.fake(), although this has a
limited number of use cases than an event sampler.
However, to support the full data access and data reduction process for simulations,
event lists are required. In future Gammapy possibly also supports event based analysis methods
(unbinned likelihood, but also e.g. clustering algorithms), that also require event
lists. For this reason binned simulations cannot present a full equivalent
solution to event sampling.
The question of the API to simulate multiple observations from e.g. an
or a list of
GTIs as it is needed for simulating data for the CTA data challenge
is not addressed in this PIG. For the scope of this PIG, the fundamental class
MapDatasetEventSampler to simulate events corresponding to a given observation
and/or single GTI is in place.
The proposed Event Sampler will not provide, for each event, the corresponding
DETY position. These will be added in a future development of the
This is a proposal for a list of tasks to implement the proposed changes:
- Implement the
gammapy.maps.Mapand add tests.
- Implement the
gammapy.time.models.PhaseCurveTableModeland add tests.
- Implement the
gammapy.cube.PSFMapand add tests.
- Implement the
gammapy.cube.EdispMapand add tests.
- Introduce the
gammapy.cube.and add tests.
- Add tutorials for event simulations of different kinds of sources.