$\ell_1$ Adaptive Trend Filter via Fast Coordinate Descent

Abstract

Identifying the unknown underlying trend of a given noisy signal is extremely useful for a wide range of applications. The number of potential trends might be exponential, which can be computationally exhaustive even for short signals. Another challenge, is the presence of abrupt changes and outliers at unknown times which impart resourceful information regarding the signal's characteristics. In this paper, we present the $\ell_1$ Adaptive Trend Filter, which can consistently identify the components in the underlying trend and multiple level-shifts, even in the presence of outliers. Additionally, an enhanced coordinate descent algorithm which exploit the filter design is presented. Some implementation details are discussed and a version in the Julia language is presented along with two distinct applications to illustrate the filter's potential.