Autoregressive Times Series Methods for Time Domain Astronomy

Abstract

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-are 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 α "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.Keywords: time domain astronomy, irregularly sampled time series, variable stars, quasars, statistical methods, time series analysis, autoregressive modeling, ARIMA THE VARIABILITY OF COSMIC POPULATIONSExcept for five roving planets and an occasional comet or nova, the nighttime sky seems immutable to the human eye. The pattern and brightness of stars appears unchanging as from our childhood to old age. Myths from ancient Egyptian, Greek, and Australian Aboriginal cultures suggest that a few stars (such as Algol, Mira, and Aldeberan) were recognized as variables [1,2]. As telescopic studies proliferated from the seventeenth through twenty first centuries, more variable stars were found with a wide range of characteristics. Some are periodic due to pulsations, rotationally modulated spots, or eclipses of binary companions. Others vary in irregular ways from magnetic flares, eruptions, pulsations, accretion of gas from companions, and most spectacularly, nova and supernova explosions. Ten thousand stars in two dozen categories were cataloged by Kukarkin and Parenago [3]; this catalog now has over 50,000 stars with >100 classes [4]. NASA's Kepler mission has recently shown that most ordinary stars are variable when observed with ∼0.001% accuracy and dense cadences [5].The study of celestial objects with variable brightness has broadened hugely in recent decades, emerging as a recognized discipline called "time domain astronomy" [6]. The brightest sources in the X-ray and gamma-ray sky are highly variable, typically from accretion of gas onto neutron stars and black holes. Timescales range from milliseconds to decades with a bewildering range of

Autoregressive Planet Search: Methodology

“…Equally-spaced data: The mathematics of autoregessive modeling discussed in this paper has been developed for evenly-spaced data (Hamilton 1994), but most astronomical time series are irregularly spaced. Two approaches exist to extend ARMA-type methodology for irregularly sampled observations: resample the data into an equally-spaced grid since ARIMA models can treat missing data, or use extensions of the methodology to handle continuous processes (Jones 1985;Feigelson et al 2018). The first approach is examined for the planetary transit problem by Stuhr et al (2019).…”

“…The model is multiscale in the sense that the AR and MA components treat small-scale variations while the I and FI components treat long-timescale variations. ARIMA-type models have shown to be very effective on a variety of applications and fields, and these successes motivated us to explore their application in astronomy where they have not seen widespread use (Feigelson et al 2018).…”

“…Though not as pervasive as in other fields, simple ARMA-type models are increasingly used in time domain astronomy, currently around 25 studies annually (Feigelson et al 2018). The richer classes of ARIMA and ARFIMA models that treat non-stationarity and long-memory 'red' noise as well as short-memory processes only rarely appear in astronomical studies.…”

Autoregressive Planet Search: Feasibility Study for Irregular Time Series

“…It is therefore satisfying that, in most cases, the ARIMA models significantly reduce the structure in irregularly spaced stellar lightcurves examined here with a few parameters, and that the TCF periodogram captures the known planet transit signal in the ARIMA residuals (Figures 1 and 7). ARIMA modeling and the TCF periodogram are basically successful in detecting planetary signals continuous-time autoregressive modeling of astronomical irregular time series is given by Feigelson et al (2018); see also the astronomical discussion byHanif & Protopapas (2015). under these circumstances.…”

Stingray: A Modern Python Library for Spectral Timing

“…CARMA (see for example Kelly et al 2014;Foreman-Mackey et al 2017), and ARIMA (e.g. Feigelson et al 2018) processes. Developing spectral timing methods for irregularly sampled light curves is a major future challenge for the field, and a high-priority long-term goal for stingray.…”