Detecting periodicities with Gaussian processes

We consider the problem of detecting and quantifying the periodic component of a function given noise-corrupted observations of a limited number of input/output tuples. Our approach is based on Gaussian process regression, which provides a flexible non-parametric framework for modelling periodic data. We introduce a novel decomposition of the covariance function as the sum of periodic and aperiodic kernels. This decomposition allows for the creation of sub-models which capture the periodic nature of the signal and its complement. To quantify the periodicity of the signal, we derive a periodicity ratio which reflects the uncertainty in the fitted sub-models. Although the method can be applied to many kernels, we give a special emphasis to the Matérn family, from the expression of the reproducing kernel Hilbert space inner product to the implementation of the associated periodic kernels in a Gaussian process toolkit. The proposed method is illustrated by considering the detection of periodically expressed genes in the arabidopsis genome

Global testing against sparse alternatives in time-frequency analysis

In this paper, an over-sampled periodogram higher criticism (OPHC) test is proposed for the global detection of sparse periodic effects in a complex-valued time series. An explicit minimax detection boundary is established between the rareness and weakness of the complex sinusoids hidden in the series. The OPHC test is shown to be asymptotically powerful in the detectable region. Numerical simulations illustrate and verify the effectiveness of the proposed test. Furthermore, the periodogram over-sampled by $O(\log N)$ is proven universally optimal in global testing for periodicities under a mild minimum separation condition.Comment: Published at http://dx.doi.org/10.1214/15-AOS1412 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evidence for 9 planets in the HD 10180 system

We re-analyse the HARPS radial velocities of HD 10180 and calculate the probabilities of models with differing numbers of periodic signals in the data. We test the significance of the seven signals, corresponding to seven exoplanets orbiting the star, in the Bayesian framework and perform comparisons of models with up to nine periodicities. We use posterior samplings and Bayesian model probabilities in our analyses together with suitable prior probability densities and prior model probabilities to extract all the significant signals from the data and to receive reliable uncertainties for the orbital parameters of the six, possibly seven, known exoplanets in the system. According to our results, there is evidence for up to nine planets orbiting HD 10180, which would make this this star a record holder in having more planets in its orbits than there are in the Solar system. We revise the uncertainties of the previously reported six planets in the system, verify the existence of the seventh signal, and announce the detection of two additional statistically significant signals in the data. If of planetary origin, these two additional signals would correspond to planets with minimum masses of 5.1$^{+3.1}_{-3.2}$ and 1.9$^{+1.6}_{-1.8}$ M$_{\oplus}$ on orbits with 67.55$^{+0.68}_{-0.88}$ and 9.655$^{+0.022}_{-0.072}$ days periods (denoted using the 99% credibility intervals), respectively.Comment: 12 pages, 7 figures, accepted for publication in the Astronomy and Astrophysic

Searching data for periodic signals

We present two statistical tests for periodicities in the time series. We apply the two tests to the data taken from Glasgow prototype interferometer in March 1996. We find that the data contain several very narrow spectral features. We investigate whether these features can be confused with gravitational wave signals from pulsars.Comment: 7 pages, 2 figure

The evidence for and against astronomical impacts on climate change and mass extinctions: A review

Numerous studies over the past 30 years have suggested there is a causal connection between the motion of the Sun through the Galaxy and terrestrial mass extinctions or climate change. Proposed mechanisms include comet impacts (via perturbation of the Oort cloud), cosmic rays and supernovae, the effects of which are modulated by the passage of the Sun through the Galactic midplane or spiral arms. Supposed periodicities in the fossil record, impact cratering dates or climate proxies over the Phanerozoic (past 545 Myr) are frequently cited as evidence in support of these hypotheses. This remains a controversial subject, with many refutations and replies having been published. Here I review both the mechanisms and the evidence for and against the relevance of astronomical phenomena to climate change and evolution. This necessarily includes a critical assessment of time series analysis techniques and hypothesis testing. Some of the studies have suffered from flaws in methodology, in particular drawing incorrect conclusions based on ruling out a null hypothesis. I conclude that there is little evidence for intrinsic periodicities in biodiversity, impact cratering or climate on timescales of tens to hundreds of Myr. Furthermore, Galactic midplane and spiral arm crossings seem to have little or no impact on biological or climate variation above background level. (truncated)Comment: 51 pages, 7 figures, 140 references. To appear in the International Journal of Astrobiology. For hyperref version with full resolution figures see http://www.mpia-hd.mpg.de/homes/calj/astimpact_ija.pd