Kepler Data Analysis: Non-Gaussian Noise and Fourier Gaussian Process Analysis of Stellar Variability

Abstract

We develop a statistical analysis model of Kepler star flux data in the presence of planet transits, non-Gaussian noise, and star variability. We first develop a model for Kepler noise probability distribution in the presence of outliers, which make the noise probability distribution non-Gaussian. We develop a signal likelihood analysis based on this probability distribution, in which we model the signal as a sum of the star variability and planetary transits. We argue these components need to be modeled together if optimal signal is to be extracted from the data. For the star variability model we develop an optimal Gaussian process analysis using a Fourier based Wiener filter approach, where the power spectrum is non-parametric and learned from the data. We develop high dimensional optimization of the objective function, where we jointly optimize all the model parameters, including thousands of star variability modes, and planet transit parameters. We apply the method to Kepler-90 data and show that it gives a better match to the star variability than the standard spline method, and robustly handles noise outliers. As a consequence, the planet radii have a higher value than the standard spline method.

fBLS -- a fast-folding BLS algorithm

“…For longer periods, the effect of correlated noise may significantly suppress the detection of small planets (see Pont et al 2006;Hartman & Bakos 2016;Cubillos et al 2017), and affect the reliability of our validation scheme. Robnik & Seljak (2020) recently demonstrated that correlated noise induced by stellar variability becomes significant for frequencies 0.25 d -1 , therefore expected to affect the detection of long-duration transit signals. In a follow-up study, Robnik & Seljak (2021) also pointed out that the noise properties may vary significantly between different Kepler stars.…”

Kepler-90: Giant transit-timing variations reveal a super-puff

“…The final result are the values of the TTV-s, TDV-s and their errors, as well as the latent parameters. The procedure is near optimal, and has been shown to give smaller and more reliable errors relative to the other existing pipelines (Robnik & Seljak 2020). Posterior distribution of the parameters around their optimal values is somewhat non-Gaussian, but we summarize it by a single symmetrized one sigma error to make further analysis tractable.…”

“…We first describe how the TTV and TDV data and its errors are extracted from the Kepler lightcurves. We apply the method developed in Robnik & Seljak (2020). We use PDCSAP flux of the Kepler data 2 processed through the Pre-search data conditioning module (Jenkins et al 2017), meaning that long term trends and most of the systematics have already been removed.…”

“…We analyzed the TTVs using a likelihood analysis pipeline developed by Robnik & Seljak (2020): this analysis models the non-Gaussian noise of Kepler data, and performs a Fourier based Gaussian process analysis, fitting star variability parameters with optimal prior power spectrum jointly with the planet parameters. The resulting outputs are TTVs and their associated uncertainties.…”

The look-elsewhere effect from a unified Bayesian and frequentist perspective

“…The look-elsewhere effect is also prevalent in other areas of physics and beyond, for example: in astronomy it occurs when detecting exoplanets via stellar photometry, where the period and phase of the planets' transits are unknown (e.g. [16]); in biology it occurs when considering large DNA sequences to study genetic association [17,18]; in medicine it occurs when testing the effectiveness of drugs in clinical trials [19]; and in theology it occurs when attempting to find hidden prophecies in religious texts [20]. Therefore, given the apparent ubiquity of the look-elsewhere effect, there is much motivation for a fast method to account for it.…”