A Study of Periodograms Standardized Using Training Datasets and Application to Exoplanet Detection

Abstract

When the noise affecting time series is colored with unknown statistics, a difficulty for sinusoid detection is to control the true significance level of the test outcome. This paper investigates the possibility of using training data sets of the noise to improve this control. Specifically, we analyze the performances of various detectors applied to periodograms standardized using training data sets. Emphasis is put on sparse detection in the Fourier domain and on the limitation posed by the necessarily finite size of the training sets available in practice. We study the resulting false alarm and detection rates and show that standardization leads in some cases to powerful constant false alarm rate tests. The study is both analytical and numerical. Although analytical results are derived in an asymptotic regime, numerical results show that theory accurately describes the tests' behaviour for moderately large sample sizes. Throughout the paper, an application of the considered periodogram standardization is presented for exoplanet detection in radial velocity data.

3D magneto-hydrodynamical simulations of stellar convective noise for improved exoplanet detection

“…6.5). However, the analytic formulae above are quite accurate for values of N i considered in practice as shown in Sulis et al (2017a). These results allow saving a substantial amount of computation time for comparing the tests (in comparison with a MC simulation-based approach).…”

“…This has been demonstrated for the Sun as part of studies involving the comparison of velocities extracted from 3D spectra of the sodium doublet and GOLF/SoHO observations. Following the theoretical analysis described in Sulis et al (2017a), we used these synthetic time series of the granulation colored noise to design standardized periodograms. These new standardized periodograms allow for the application of tests that are both powerful and for which we can derive accurate FAP.…”

“…This section presents the considered statistical model and detection tests. Several results detailed in Sulis et al (2017a) are summarized below for the sake of completeness since Sec. 4 is aimed at validating the theoretical results from this study on real astrophysical data.…”

“…v P, k := 1 − Φ F P(ν k ), 2, 2L , with Φ F the CDF of a central F variable. Examples of adaptive tests based on the P-values are the Higher-Criticism(Donoho & Jin 2004;Sulis et al 2017a) and the Berk-Jones tests(Berk & Jones 1979;Aldor-Noiman et al 2013;Mary & Ferrari 2014;Kaplan & Goldman 2014;Gontscharuk et al 2015;Moscovich et al 2016) respectively defined as:…”

A bootstrap method for sinusoid detection in colored noise and uneven sampling. Application to exoplanet detection

“…The present study explores a bootstrap solution to this problem. As in [7], we assume that a training data set T L of the noise is available. However, this set provides only a small number of noise samples -comparable typically to the number of data samples.…”