$L_1$ Trend Filtering: A Modern Statistical Tool for Time-Domain Astronomy and Astronomical Spectroscopy

The problem of estimating a one-dimensional signal possessing mixed degrees of smoothness is ubiquitous in time-domain astronomy and astronomical spectroscopy. For example, in the time domain, an astronomical object may exhibit a smoothly varying intensity that is occasionally interrupted by abrupt dips or spikes. Likewise, in the spectroscopic setting, a noiseless spectrum typically contains intervals of relative smoothness mixed with localized higher frequency components such as emission peaks and absorption lines. In this work, we present $L_1$ trend filtering (Steidl et al.; Kim et al.), a modern nonparametric statistical tool that yields significant improvements in this broad problem space of estimating spatially heterogeneous signals. When the underlying signal is spatially heterogeneous, the $L_1$ trend filter has been shown to be strictly superior to any estimator that is a linear combination of the observed data, including smoothing splines, kernels, and Gaussian processes. Moreover, the $L_1$ trend filter does not require the restrictive setup of wavelets --- the definitive classical approach for modeling spatially heterogeneous signals. In the spirit of illustrating the wide applicability of $L_1$ trend filtering, we briefly demonstrate its utility on several relevant astrophysical data sets: two Kepler light curves (an exoplanet transit and an eclipsing binary system), a Palomar Transient Factory supernova light curve, and an SDSS galaxy spectrum. Furthermore, we present a more rigorous analysis of the Lyman-alpha forest of SDSS quasar spectra --- a standard cosmological tool for probing the large-scale structure of the high redshift intergalactic medium.