$\ell_1$ Adaptive Trend Filter via Fast Coordinate Descent
This addresses a computational challenge in signal processing for applications requiring trend analysis, though it appears incremental as it builds on existing filtering methods.
The paper tackles the problem of identifying underlying trends and level-shifts in noisy signals with outliers, presenting the ℓ₁ Adaptive Trend Filter that consistently identifies these components, along with an enhanced coordinate descent algorithm for efficient computation.
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.