标题： 如何监控和缓解L1趋势过滤中的阶梯效应 显示英文标题

标题： How to monitor and mitigate stair-casing in l1 trend filtering

摘要： 本文研究了利用$\ell_1$趋势过滤来估计时间序列中变化趋势的方法。该方法将一维总变差（TV）去噪推广到检测均值的跃变点，用于检测趋势的变化，并且依赖于一个凸优化问题，对此已有非常高效的数值算法。众所周知，TV去噪存在所谓的阶梯效应，这会导致检测到虚假的变点。本文的目标是表明$\ell_1$趋势过滤也存在某种形式的阶梯问题。分析基于将该方法中的优化问题对偶变量解释为积分随机游走。我们讨论了$\ell_1$趋势过滤的一致性条件，如何监控这些条件的满足情况，以及如何修改算法以避免阶梯效应导致的错误检测问题。 显示英文摘要

摘要： In this paper we study the estimation of changing trends in time-series using $\ell_1$ 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 $\ell_1$ 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 $\ell_1$ trend filtering, how to monitor their fulfillment, and how to modify the algorithm to avoid the stair-case false detection problem.