$\ell_1$ Adaptive Trend Filter via Fast Coordinate Descent

TL;DR

This paper introduces an $$ Adaptive Trend Filter that effectively detects underlying trends and level-shifts in noisy signals with outliers, using a fast coordinate descent algorithm for efficient computation.

Contribution

The paper proposes a novel $$ Adaptive Trend Filter and an optimized coordinate descent algorithm tailored for trend detection in noisy, outlier-contaminated signals.

Findings

Successfully identifies underlying trends and level-shifts.

Efficient implementation with a Julia version.

Demonstrated effectiveness through two applications.

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.