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.

Types of supported datasets

Gammapy has built-in support to create and analyse the following datasets:

Dataset Type

data_type

reduced irfs

geometry

additional quantity

statistic

MapDataset

counts

background, psf, edisp, exposure,

WcsGeom or RegionGeom

cash

MapDatasetOnOff

counts

psf, edisp, exposure

WcsGeom

acceptance, acceptance_off, counts_off

wstat

SpectrumDataset

counts

background, edisp, exposure

RegionGeom

cash

SpectrumDatasetOnOff

counts

edisp, exposure

RegionGeom

acceptance, acceptance_off, counts_off

wstat

FluxPointsDataset

flux

None

None

chi2

In general, OnOff datasets should be used when the background is estimated from real off counts (eg: RingBackground or ReflectedBackground), rather than from a background model. The FluxPointDataset is used to fit pre-computed flux points when no convolution with IRFs are needed.

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 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.

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.

Using gammapy.datasets

Gammapy tutorial notebooks that show how to use this package:

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.