Trend filtering – I. A modern statistical tool for time-domain astronomy and astronomical spectroscopy

ABSTRACT The problem of denoising a 1D signal possessing varying 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 trend filtering, a modern non-parametric statistical tool that yields significant improvements in this broad problem space of denoising spatially heterogeneous signals. When the underlying signal is spatially heterogeneous, trend filtering is superior to any statistical estimator that is a linear combination of the observed data – including kernel smoothers, LOESS, smoothing splines, Gaussian process regression, and many other popular methods. Furthermore, the trend filtering estimate can be computed with practical and scalable efficiency via a specialized convex optimization algorithm, e.g. handling sample sizes of n ≳ 107 within a few minutes. In a companion paper, we explicitly demonstrate the broad utility of trend filtering to observational astronomy by carrying out a diverse set of spectroscopic and time-domain analyses.

Nonparametric estimate remarks

Kernel smoothers belong to the most popular nonparametric functional estimates. They provide a simple way of finding structure in data. The idea of the kernel smoothing can be applied to a simple fixed design regression model. This article is focused on kernel smoothing for fixed design regresion model with three types of estimators, the Gasser-Müller estimator, the Nadaraya-Watson estimator and the local linear estimator. At the end of this article figures for ilustration of desribed estimators on simulated and real data sets are shown.