# PIG 9 - Event Sampling¶

• Author: Fabio Pintore, Andrea Giuliani, Axel Donath

• Created: May 03, 2019

• Accepted: Aug 30, 2019

• Status: accepted

• Discussion: GH 2136

## Abstract¶

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.

## Sampling methods¶

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.

## Proposal¶

We propose to include in gammapy.cube an high level interface (HLI) class, labelled as MapDatasetEventSampler or 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)
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 .sample method 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 and MC_ID. 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 from. The MapDatasetSampler should also fill the mandatory header information for event list files decribed on gadf.

### MapDatasetEventSampler¶

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 MapDataset 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. Then, 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 events.

We propose to add a Map.sample(n_events=, random_state=) method in ~gammapy.maps.Map 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: RA_TRUE, DEC_TRUE and ENERGY_TRUE .

Then, the time will be sampled independently using the temporal information stored into the MapDataset model for each source of interest. This will be done through a .sample(n_events=, random_state=) method that we propose to add to ~gammapy.time.models.LightCurveTableModel and ~gammapy.time.models.PhaseCurveTableModel. This method will take as input the GTIs (i.e. one Tstart and Tstop) in the MapDataset object. Also in this case the InverseCDFSampler 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 t_min and t_max. To define a light-curve per model component, the current SkyModel class will be extended by a SkyModel(..., temporal_model=).

The IRF correction can now be applied to sampled events. We propose to add a .sample(events=) method in both ~gammapy.cube.PSFMap and ~gammapy.cube.EdispMap. 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 ~astropy.table.Table object. The IRFs are assumed to be constant and not time-dependent. The output will be an ~astropy.table.Table with the new columns RA, DEC and ENERGY, which are the reconstructed event energies and positions.

Finally, the times and the energies/coordinates of the events will be merged into a single ~astropy.table.Table with the columns:RA, DEC and ENERGY and TIME .

The 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 sampled events.

## 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 numpy.random.poisson() based 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 ObservationTable 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 DETX and DETY position. These will be added in a future development of the simulator.

This is a proposal for a list of tasks to implement the proposed changes:

1. Implement the sample method in gammapy.maps.Map and add tests.

2. Implement the sample method in gammapy.time.models.LightCurveTableModel and gammapy.time.models.PhaseCurveTableModel and add tests.

3. Implement the sample method in gammapy.cube.PSFMap and add tests.

4. Implement the sample method in gammapy.cube.EdispMap and add tests.

5. Introduce the MapDatasetEventSampler into gammapy.cube. and add tests.

6. Add tutorials for event simulations of different kinds of sources.

## Decision¶

The PIG was discussed extensively in GH 2136, the weekly Gammapy developer calls and coding sprint in person. After the deadline for final review expired on August 20, all remaining comments were addressed and the PIG was accepted on August 30, 2019.