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Celestial objects exhibit a wide range of variability in brightness at different wavebands. Surprisingly, the most common methods for characterizing time series in statistics -- parametric autoregressive modeling -- is rarely used to interpret astronomical light curves. We review standard ARMA, ARIMA and ARFIMA (autoregressive moving average fractionally integrated) models that treat short-memory autocorrelation, long-memory $1/f^alpha$ `red noise, and nonstationary trends. Though designed for evenly spaced time series, moderately irregular cadences can be treated as evenly-spaced time series with missing data. Fitting algorithms are efficient and software implementations are widely available. We apply ARIMA models to light curves of four variable stars, discussing their effectiveness for different temporal characteristics. A variety of extensions to ARIMA are outlined, with emphasis on recently developed continuous-time models like CARMA and CARFIMA designed for irregularly spaced time series. Strengths and weakness of ARIMA-type modeling for astronomical data analysis and astrophysical insights are reviewed.
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Progress in astronomy comes from interpreting the signals encoded in the light received from distant objects: the distribution of light over the sky (images), over photon wavelength (spectrum), over polarization angle, and over time (usually called light curves by astronomers). In the time domain we see transient events such as supernovae, gamma-ray bursts, and other powerful explosions; we see periodic phenomena such as the orbits of planets around nearby stars, radio pulsars, and pulsations of stars in nearby galaxies; and persistent aperiodic variations (`noise) from powerful systems like accreting black holes. I review just a few of the recent and future challenges in the burgeoning area of Time Domain Astrophysics, with particular attention to persistently variable sources, the recovery of reliable noise power spectra from sparsely sampled time series, higher-order properties of accreting black holes, and time delays and correlations in multivariate time series.
Advection-dominated dynamical systems, characterized by partial differential equations, are found in applications ranging from weather forecasting to engineering design where accuracy and robustness are crucial. There has been significant interest in the use of techniques borrowed from machine learning to reduce the computational expense and/or improve the accuracy of predictions for these systems. These rely on the identification of a basis that reduces the dimensionality of the problem and the subsequent use of time series and sequential learning methods to forecast the evolution of the reduced state. Often, however, machine-learned predictions after reduced-basis projection are plagued by issues of stability stemming from incomplete capture of multiscale processes as well as due to error growth for long forecast durations. To address these issues, we have developed a emph{non-autoregressive} time series approach for predicting linear reduced-basis time histories of forward models. In particular, we demonstrate that non-autoregressive counterparts of sequential learning methods such as long short-term memory (LSTM) considerably improve the stability of machine-learned reduced-order models. We evaluate our approach on the inviscid shallow water equations and show that a non-autoregressive variant of the standard LSTM approach that is bidirectional in the PCA components obtains the best accuracy for recreating the nonlinear dynamics of partial observations. Moreover---and critical for many applications of these surrogates---inference times are reduced by three orders of magnitude using our approach, compared with both the equation-based Galerkin projection method and the standard LSTM approach.
The large-scale surveys such as PTF, CRTS and Pan-STARRS-1 that have emerged within the past 5 years or so employ digital databases and modern analysis tools to accentuate research into Time Domain Astronomy (TDA). Preparations are underway for LSST which, in another 6 years, will usher in the second decade of modern TDA. By that time the Digital Access to a Sky Century @ Harvard (DASCH) project will have made available to the community the full sky Historical TDA database and digitized images for a century (1890--1990) of coverage. We describe the current DASCH development and some initial results, and outline plans for the production scanning phase and data distribution which is to begin in 2012. That will open a 100-year window into temporal astrophysics, revealing rare transients and (especially) astrophysical phenomena that vary on time-scales of a decade. It will also provide context and archival comparisons for the deeper modern surveys
This review outlines concepts of mathematical statistics, elements of probability theory, hypothesis tests and point estimation for use in the analysis of modern astronomical data. Least squares, maximum likelihood, and Bayesian approaches to statistical inference are treated. Resampling methods, particularly the bootstrap, provide valuable procedures when distributions functions of statistics are not known. Several approaches to model selection and good- ness of fit are considered. Applied statistics relevant to astronomical research are briefly discussed: nonparametric methods for use when little is known about the behavior of the astronomical populations or processes; data smoothing with kernel density estimation and nonparametric regression; unsupervised clustering and supervised classification procedures for multivariate problems; survival analysis for astronomical datasets with nondetections; time- and frequency-domain times series analysis for light curves; and spatial statistics to interpret the spatial distributions of points in low dimensions. Two types of resources are presented: about 40 recommended texts and monographs in various fields of statistics, and the public domain R software system for statistical analysis. Together with its sim 3500 (and growing) add-on CRAN packages, R implements a vast range of statistical procedures in a coherent high-level language with advanced graphics.
THESEUS is a medium size space mission of the European Space Agency, currently under evaluation for a possible launch in 2032. Its main objectives are to investigate the early Universe through the observation of gamma-ray bursts and to study the gravitational waves electromagnetic counterparts and neutrino events. On the other hand, its instruments, which include a wide field of view X-ray (0.3-5 keV) telescope based on lobster-eye focusing optics and a gamma-ray spectrometer with imaging capabilities in the 2-150 keV range, are also ideal for carrying out unprecedented studies in time domain astrophysics. In addition, the presence onboard of a 70 cm near infrared telescope will allow simultaneous multi-wavelegth studies. Here we present the THESEUS capabilities for studying the time variability of different classes of sources in parallel to, and without affecting, the gamma-ray bursts hunt.
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