How to monitor and mitigate stair-casing in l1 trend filtering
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In this paper we study the estimation of changing trends in time-series using ℓ_1 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 ℓ_1 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 ℓ_1 trend filtering, how to monitor their fulfillment, and how to modify the algorithm to avoid the stair-case false detection problem.
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