Linear models for systematics and nuisances

The target of many astronomical studies is the recovery of tiny astrophysical signals living in a sea of uninteresting (but usually dominant) noise. In many contexts (i.e., stellar time-series, or high-contrast imaging, or stellar spectroscopy), there are structured components in this noise caused by systematic effects in the astronomical source, the atmosphere, the telescope, or the detector. More often than not, evaluation of the true physical model for these nuisances is computationally intractable and dependent on too many (unknown) parameters to allow rigorous probabilistic inference. Sometimes, housekeeping data---and often the science data themselves---can be used as predictors of the systematic noise. Linear combinations of simple functions of these predictors are often used as computationally tractable models that can capture the nuisances. These models can be used to fit and subtract systematics prior to investigation of the signals of interest, or they can be used in a simultaneous fit of the systematics and the signals. In this Note, we show that if a Gaussian prior is placed on the weights of the linear components, the weights can be marginalized out with an operation in pure linear algebra, which can (often) be made fast. We illustrate this model by demonstrating the applicability of a linear model for the non-linear systematics in K2 time-series data, where the dominant noise source for many stars is spacecraft motion and variability.Comment: 5 pages, 1 figure. Accepted to Research Notes of the AA

Systematics-insensitive periodic signal search with K2

From pulsating stars to transiting exoplanets, the search for periodic signals in K2 data, Kepler's 2-wheeled extension, is relevant to a long list of scientific goals. Systematics affecting K2 light curves due to the decreased spacecraft pointing precision inhibit the easy extraction of periodic signals from the data. We here develop a method for producing periodograms of K2 light curves that are insensitive to pointing-induced systematics; the Systematics-Insensitive Periodogram (SIP). Traditional sine-fitting periodograms use a generative model to find the frequency of a sinusoid that best describes the data. We extend this principle by including systematic trends, based on a set of 'Eigen light curves', following Foreman-Mackey et al. (2015), in our generative model as well as a sum of sine and cosine functions over a grid of frequencies. Using this method we are able to produce periodograms with vastly reduced systematic features. The quality of the resulting periodograms are such that we can recover acoustic oscillations in giant stars and measure stellar rotation periods without the need for any detrending. The algorithm is also applicable to the detection of other periodic phenomena such as variable stars, eclipsing binaries and short-period exoplanet candidates. The SIP code is available at https://github.com/RuthAngus/SIPK2

Fast and scalable Gaussian process modeling with applications to astronomical time series

The growing field of large-scale time domain astronomy requires methods for probabilistic data analysis that are computationally tractable, even with large datasets. Gaussian Processes are a popular class of models used for this purpose but, since the computational cost scales, in general, as the cube of the number of data points, their application has been limited to small datasets. In this paper, we present a novel method for Gaussian Process modeling in one-dimension where the computational requirements scale linearly with the size of the dataset. We demonstrate the method by applying it to simulated and real astronomical time series datasets. These demonstrations are examples of probabilistic inference of stellar rotation periods, asteroseismic oscillation spectra, and transiting planet parameters. The method exploits structure in the problem when the covariance function is expressed as a mixture of complex exponentials, without requiring evenly spaced observations or uniform noise. This form of covariance arises naturally when the process is a mixture of stochastically-driven damped harmonic oscillators -- providing a physical motivation for and interpretation of this choice -- but we also demonstrate that it can be a useful effective model in some other cases. We present a mathematical description of the method and compare it to existing scalable Gaussian Process methods. The method is fast and interpretable, with a range of potential applications within astronomical data analysis and beyond. We provide well-tested and documented open-source implementations of this method in C++, Python, and Julia.Comment: Updated in response to referee. Submitted to the AAS Journals. Comments (still) welcome. Code available: https://github.com/dfm/celerit

A Causal, Data-Driven Approach to Modeling the Kepler Data

Astronomical observations are affected by several kinds of noise, each with its own causal source; there is photon noise, stochastic source variability, and residuals coming from imperfect calibration of the detector or telescope. The precision of NASA Kepler photometry for exoplanet science---the most precise photometric measurements of stars ever made---appears to be limited by unknown or untracked variations in spacecraft pointing and temperature, and unmodeled stellar variability. Here we present the Causal Pixel Model (CPM) for Kepler data, a data-driven model intended to capture variability but preserve transit signals. The CPM works at the pixel level so that it can capture very fine-grained information about the variation of the spacecraft. The CPM predicts each target pixel value from a large number of pixels of other stars sharing the instrument variabilities while not containing any information on possible transits in the target star. In addition, we use the target star's future and past (auto-regression). By appropriately separating, for each data point, the data into training and test sets, we ensure that information about any transit will be perfectly isolated from the model. The method has four hyper-parameters (the number of predictor stars, the auto-regressive window size, and two L2-regularization amplitudes for model components), which we set by cross-validation. We determine a generic set of hyper-parameters that works well for most of the stars and apply the method to a corresponding set of target stars. We find that we can consistently outperform (for the purposes of exoplanet detection) the Kepler Pre-search Data Conditioning (PDC) method for exoplanet discovery.Comment: Accepted for publication in the PAS