The \(\ell _{1}\) trend filtering enables us to estimate a continuous piecewise linear trend of univariate time series. This filter and its variants have subsequently been applied in various fields, including astronomy, climatology, economics, electronics, environmental science, finance, and geophysics. Although the \(\ell _{1}\) trend filtering can estimate a continuous piecewise linear trend of univariate time series, it cannot estimate a common continuous piecewise linear trend of multiple time series. This paper develops a statistical procedure that enables us to estimate it, which is a multivariate extension of the \(\ell _{1}\) trend filtering. We provide an algorithm for estimating it and a clue to specify the tuning parameter of the procedure, both required for its application. We also (i) numerically illustrate how well the algorithm works, (ii) provide an empirical illustration, and (iii) introduce a generalization of our novel method.

的\（\ ELL _ {1} \）趋势滤波使我们能够估计连续的分段线性单变量的时间序列的趋势。此过滤器及其变体随后已应用于各个领域，包括天文学，气候学，经济学，电子学，环境科学，金融和地球物理学。尽管\（\ ell _ {1} \）趋势过滤可以估计单变量时间序列的连续分段线性趋势，但无法估计多个时间序列的公共连续分段线性趋势。本文开发了一种统计程序，使我们能够对其进行估计，它是\（\ ell _ {1} \）的多元扩展。趋势过滤。我们提供了一种估计算法，并提供了指定过程调整参数的线索，这都是其应用所必需的。我们还（i）以数字方式说明了该算法的工作效果，（ii）提供了经验说明，并且（iii）介绍了我们的新方法的概括。