How to monitor and mitigate stair-casing in L1 trend filtering
In this paper we study the estimation of changing trends in time-series using ℓ1 trend filtering. This method generalizes 1D Total Variation (TV) denoising for detection of step changes in means to detecting changes in trends, and it relies on a convex optimization problem for which there are very efficient numerical algorithms. It is known that TV denoising suffers from the so-called stair-case effect, which leads to detecting false change points. The objective of this paper is to show that ℓ1 trend filtering also suffers from a certain stair-case problem. The analysis is based on an interpretation of the dual variables of the optimization problem in the method as integrated random walk. We discuss consistency conditions for ℓ1 trend filtering, how to monitor their fulfillment, and how to modify the algorithm to avoid the stair-case false detection problem.