datasets - Reduced datasets

Introduction

The gammapy.datasets sub-package contains classes to handle reduced gamma-ray data for modeling and fitting.

The Dataset class bundles reduced data, IRFs and model to perform likelihood fitting and joint-likelihood fitting. All datasets contain a Models container with one or more SkyModel objects that represent additive emission components.

To model and fit data in Gammapy, you have to create a Datasets container object with one or multiple Dataset objects. Gammapy has built-in support to create and analyse the following datasets: MapDataset, MapDatasetOnOff, SpectrumDataset, SpectrumDatasetOnOff and FluxPointsDataset.

The map datasets represent 3D cubes (WcsNDMap objects) with two spatial and one energy axis. For 2D images the same map objects and map datasets are used, an energy axis is present but only has one energy bin. The MapDataset contains a counzts map, background is modeled with a BackgroundModel, and the fit statistic used is cash. The MapDatasetOnOff contains on and off count maps, background is implicitly modeled via the off counts map, and the wstat fit statistic.

The spectrum datasets represent 1D spectra (RegionNDMap objects) with an energy axis. There are no spatial axes or information, the 1D spectra are obtained for a given on region. The SpectrumDataset contains a counts spectrum, background is modeled with a RegionNDMap, and the fit statistic used is cash. The SpectrumDatasetOnOff contains on on and off count spectra, background is implicitly modeled via the off counts spectrum, and the wstat fit statistic. The FluxPointsDataset contains estimatorsFluxPoints and a spectral model, the fit statistic used is chi2.

Note that in Gammapy, 2D image analyses are done with 3D cubes with a single energy bin, e.g. for modeling and fitting, see the 2D map analysis tutorial.

To analyse multiple runs, you can either stack the datasets together, or perform a joint fit across multiple datasets.

Predicted counts

The total number of predicted counts from a MapDataset are computed per bin like:

\[N_{Pred} = N_{Bkg} + \sum_{Src} N_{Src}\]

Where \(N_{Bkg}\) is the expected counts from the residual hadronic background model and \(N_{Src}\) the predicted counts from a given source model component. The predicted counts from the hadronic background are computed directly from the model in reconstructed energy and spatial coordinates, while the predicted counts from a source are obtained by forward folding with the instrument response:

\[N_{Src} = \mathrm{PSF_{Src}} \circledast \mathrm{EDISP_{Src}}(\mathcal{E} \cdot F_{Src}(l, b, E_{True}))\]

Where \(F_{Src}\) is the integrated flux of the source model, \(\mathcal{E}\) the exposure, \(\mathrm{EDISP}\) the energy dispersion matrix and \(\mathrm{PSF}\) the PSF convolution kernel. The corresponding IRFs are extracted at the current position of the model component defined by \((l, b)\) and assumed to be constant across the size of the source. The detailed expressions to compute the predicted number of counts from a source and corresponding IRFs are given in IRF Theory.

Stacking Multiple Datasets

Stacking datasets implies that the counts, background and reduced IRFs from all the runs are binned together to get one final dataset for which a likelihood is computed during the fit. Stacking is often useful to reduce the computation effort while analysing multiple runs.

The following table lists how the individual quantities are handled during stacking. Here, \(k\) denotes a bin in reconstructed energy, \(l\) a bin in true energy and \(j\) is the dataset number

Dataset attribute

Behaviour

Implementation

livetime

Sum of individual livetimes

\(\overline{t} = \sum_j t_j\)

mask_safe

True if the pixel is included in the safe data range.

\(\overline{\epsilon_k} = \sum_{j} \epsilon_{jk}\)

mask_fit

Dropped

counts

Summed in the data range defined by mask_safe

\(\overline{\mathrm{counts}_k} = \sum_j \mathrm{counts}_{jk} \cdot \epsilon_{jk}\)

background

Summed in the data range defined by mask_safe

\(\overline{\mathrm{bkg}_k} = \sum_j \mathrm{bkg}_{jk} \cdot \epsilon_{jk}\)

exposure

Summed in the data range defined by mask_safe

\(\overline{\mathrm{exposure}_l} = \sum_{j} \mathrm{exposure}_{jl} \cdot \sum_k \epsilon_{jk}\)

psf

Exposure weighted average

\(\overline{\mathrm{psf}_l} = \frac{\sum_{j} \mathrm{psf}_{jl} \cdot \mathrm{exposure}_{jl}} {\sum_{j} \mathrm{exposure}_{jl}}\)

edisp

Exposure weighted average, with mask on reconstructed energy

\(\overline{\mathrm{edisp}_{kl}} = \frac{\sum_{j}\mathrm{edisp}_{jkl} \cdot \epsilon_{jk} \cdot \mathrm{exposure}_{jl}} {\sum_{j} \mathrm{exposure}_{jl}}\)

gti

Union of individual gti

For the model evaluation, an important factor that needs to be accounted for is that the energy threshold changes between obseravtions. With the above implementation using a EDispersionMap, the npred is conserved, ie, the predicted number of counts on the stacked dataset is the sum expected by stacking the npred of the individual runs,

The following plot shows the individual and stacked energy dispersion kernel and npred for two SpectrumDataset

(png, hires.png, pdf)

../_images/plot_stack.png

Note

  • A stacked analysis is reasonable only when adding runs taken by the same instrument.

  • Stacking happens in-place, ie, dataset1.stack(dataset2) will overwrite dataset1

  • To properly handle masks, it is necessary to stack onto an empty dataset.

  • Stacking only works for maps with equivalent geometry. Two geometries are called equivalent if one is exactly the same as, or can be obtained from a cutout of, the other.

Joint Analysis

An alternative to stacking datasets is a joint fit across all the datasets. For a definition, see Glossary.

The totat fit statistic of datasets is the sum of the fit statistic of each dataset. Note that this is not equal to the stacked fit statistic.

A joint fit usually allows a better modeling of the background because the background model parameters can be fit for each dataset simultaneously with the source models. However, a joint fit is, performance wise, very computationally intensive. The fit convergence time increases non-linearly with the number of datasets to be fit. Moreover, depending upon the number of parameters in the background model, even fit convergence might be an issue for a large number of datasets.

To strike a balance, what might be a practical solution for analysis of many runs is to stack runs taken under similar conditions and then do a joint fit on the stacked datasets.

Reference/API

gammapy.datasets Package

Classes

Dataset

Dataset abstract base class.

Datasets([datasets])

Dataset collection.

MapDatasetOnOff([models, counts, …])

Map dataset for on-off likelihood fitting.

SpectrumDataset([models, counts, exposure, …])

MapDatasetEventSampler([random_state])

Sample events from a map dataset

MapDataset([models, counts, exposure, …])

Perform sky model likelihood fit on maps.

SpectrumDatasetOnOff([models, counts, …])

FluxPointsDataset([models, data, mask_fit, …])

Fit a set of flux points with a parametric model.

Variables

DATASET_REGISTRY

Registry of dataset classes in Gammapy.