A Gaussian process framework for modelling instrumental systematics: application to transmission spectroscopy

Transmission spectroscopy, which consists of measuring the wavelength-dependent absorption of starlight by a planet’s atmosphere during a transit, is a powerful probe of atmospheric composition. However, the expected signal is typically orders of magnitude smaller than instrumental systematics and the results are crucially dependent on the treatment of the latter. In this paper, we propose a new method to infer transit parameters in the presence of systematic noise using Gaussian processes, a technique widely used in the machine learning community for Bayesian regression and classification problems. Our method makes use of auxiliary information about the state of the instrument, but does so in a non-parametric manner, without imposing a specific dependence of the systematics on the instrumental parameters, and naturally allows for the correlated nature of the noise. We give an example application of the method to archival NICMOS transmission spectroscopy of the hot Jupiter HD 189733, which goes some way towards reconciling the controversy surrounding this data set in the literature. Finally, we provide an appendix giving a general introduction to Gaussian processes for regression, in order to encourage their application to a wider range of problems.

[1]  C. Moutou,et al.  Detection of atmospheric haze on an extrasolar planet: the 0.55–1.05 μm transmission spectrum of HD 189733b with the Hubble Space Telescope , 2007, 0712.1374.

[2]  Gautam Vasisht,et al.  The presence of methane in the atmosphere of an extrasolar planet , 2008, Nature.

[3]  David Charbonneau,et al.  Detection of Thermal Emission from an Extrasolar Planet , 2005 .

[4]  C. Q. Lee,et al.  The Computer Journal , 1958, Nature.

[5]  David Charbonneau,et al.  Hubble Space Telescope Time-Series Photometry of the Transiting Planet of HD?209458 , 2001 .

[6]  William H. Press,et al.  Numerical recipes in C , 2002 .

[7]  M. Mayor,et al.  An extended upper atmosphere around the extrasolar planet HD209458b , 2003, Nature.

[8]  Ashok Srivastava,et al.  Stable and Efficient Gaussian Process Calculations , 2009, J. Mach. Learn. Res..

[9]  Princeton,et al.  Theoretical Transmission Spectra during Extrasolar Giant Planet Transits , 1999, astro-ph/9912241.

[10]  James Demmel,et al.  LU, QR and Cholesky Factorizations using Vector Capabilities of GPUs , 2008 .

[11]  Frederic Pont,et al.  The effect of red noise on planetary transit detection , 2006, astro-ph/0608597.

[12]  M. Way,et al.  NEW APPROACHES TO PHOTOMETRIC REDSHIFT PREDICTION VIA GAUSSIAN PROCESS REGRESSION IN THE SLOAN DIGITAL SKY SURVEY , 2009, 0905.4081.

[13]  M. Holman,et al.  Transit infrared spectroscopy of the hot Neptune around GJ 436 with the Hubble Space Telescope , 2008, 0810.5731.

[14]  T. Brown.  Transmission Spectra as Diagnostics of Extrasolar Giant Planet Atmospheres , 2001, astro-ph/0101307.

[15]  Geoffrey E. Hinton,et al.  Bayesian Learning for Neural Networks , 1995 .

[16]  E. Agol,et al.  Analytic Light Curves for Planetary Transit Searches , 2002, astro-ph/0210099.

[17]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[18]  R. Gilliland,et al.  Detection of an Extrasolar Planet Atmosphere , 2001, astro-ph/0111544.

[19]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[20]  Joshua N. Winn,et al.  PARAMETER ESTIMATION FROM TIME-SERIES DATA WITH CORRELATED ERRORS: A WAVELET-BASED METHOD AND ITS APPLICATION TO TRANSIT LIGHT CURVES , 2009, 0909.0747.

[21]  I. A. Steele,et al.  A transit timing analysis of seven RISE light curves of the exoplanet system HAT-P-3 , 2009, 0909.4170.

[22]  David K. Sing,et al.  Stellar limb-darkening coefficients for CoRot and Kepler , 2009, 0912.2274.

[23]  C. Moutou,et al.  Hubble Space Telescope time-series photometry of the planetary transit of HD 189733: no moon, no rings, starspots , 2007, 0707.1940.

[24]  David Charbonneau,et al.  The transit light curve project. I. Four consecutive transits of the exoplanet XO-1b , 2006 .

[25]  R. G. West,et al.  Efficient identification of exoplanetary transit candidates from SuperWASP light curves , 2007, 0707.0417.

[26]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[27]  Steven Reece,et al.  Sequential Bayesian Prediction in the Presence of Changepoints and Faults , 2010, Comput. J..

[28]  S. Zucker,et al.  Reassessing the radial-velocity evidence for planets around CoRoT-7 , 2010, 1008.3859.

[29]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[30]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[31]  Bernhard Schölkopf,et al.  Sparse multiscale gaussian process regression , 2008, ICML '08.

[32]  Drake Deming,et al.  Infrared radiation from an extrasolar planet , 2005, Nature.

[33]  University of Exeter,et al.  A new look at NICMOS transmission spectroscopy of HD 189733, GJ-436 and XO-1: no conclusive evidence for molecular features , 2010, 1010.1753.

[34]  E. K. Simpson,et al.  A TRANSIT TIMING ANALYSIS OF NINE RISE LIGHT CURVES OF THE EXOPLANET SYSTEM TrES-3 , 2009, 0905.4680.

[35]  Carl E. Rasmussen,et al.  A Unifying View of Sparse Approximate Gaussian Process Regression , 2005, J. Mach. Learn. Res..

[36]  M. Way,et al.  Novel Methods for Predicting Photometric Redshifts from Broadband Photometry Using Virtual Sensors , 2006 .

[37]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .