Calibration Concordance for Astronomical Instruments via Multiplicative Shrinkage

Abstract

Calibration data are often obtained by observing several well-understood objects simultaneously with multiple instruments, such as satellites for measuring astronomical sources. Analyzing such data and obtaining proper concordance among the instruments is challenging when the physical source models are not well understood, when there are uncertainties in “known” physical quantities, or when data quality varies in ways that cannot be fully quantified. Furthermore, the number of model parameters increases with both the number of instruments and the number of sources. Thus, concordance of the instruments requires careful modeling of the mean signals, the intrinsic source differences, and measurement errors. In this article, we propose a log-Normal model and a more general log-t model that respect the multiplicative nature of the mean signals via a half-variance adjustment, yet permit imperfections in the mean modeling to be absorbed by residual variances. We present analytical solutions in the form of power shrinkage in special cases and develop reliable Markov chain Monte Carlo algorithms for general cases, both of which are available in the Python module CalConcordance. We apply our method to several datasets including a combination of observations of active galactic nuclei (AGN) and spectral line emission from the supernova remnant E0102, obtained with a variety of X-ray telescopes such as Chandra, XMM- Newton, Suzaku, and Swift. The data are compiled by the International Astronomical Consortium for High Energy Calibration. We demonstrate that our method provides helpful and practical guidance for astrophysicists when adjusting for disagreements among instruments. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Keywords:

• Adjusting attributes
• Bayesian hierarchical model
• Half-variance adjustment
• Log-Normal model
• Log-t model
• Shrinkage estimator

Acknowledgments

The authors thank Matteo Guainazzi, Paul Plucinsky, Jeremy Drake, Aneta Siemiginowska, and other members of the IACHEC and CHASC collaborations for valuable discussions.

Funding

This project was conducted under the auspices of the CHASC International Astrostatistics Center. CHASC is supported by the NSF grants DMS-15-13484, DMS-15-13492, DMS-15-13546, DMS-18-11308, DMS-18-11083, and DMS-18-11661. In addition, David van Dyk’s work was supported by a Marie-Skodowska-Curie RISE (H2020-MSCA-RISE-2015-691164) Grant provided by the European Commission. Vinay Kashyap and Herman Marshall acknowledge support under NASA Contract NAS8-03060 with the Chandra X-ray Center.

Notes

Notes

1 For a Poisson random variable q with mean λ, this replacement leads to a zero-modified Poisson random variable $\tilde{q}$ with mean and variance

$\mathrm{E}(\tilde{q})=\lambda +0.5e^{-\lambda },\hspace{1em}\text{var}(\tilde{q})=\lambda (1-e^{-\lambda })+0.25e^{-\lambda }(1-e^{-\lambda }).$(2.3)

With reasonably large λ, $\tilde{q}$ approximates q extremely well.

2 See Section 4 in http://xmm2.esac.esa.int/docs/documents/CAL-TN-0052.ps.gz

Additional information

Funding

This project was conducted under the auspices of the CHASC International Astrostatistics Center. CHASC is supported by the NSF grants DMS-15-13484, DMS-15-13492, DMS-15-13546, DMS-18-11308, DMS-18-11083, and DMS-18-11661. In addition, David van Dyk’s work was supported by a Marie-Skodowska-Curie RISE (H2020-MSCA-RISE-2015-691164) Grant provided by the European Commission. Vinay Kashyap and Herman Marshall acknowledge support under NASA Contract NAS8-03060 with the Chandra X-ray Center.