While many approaches have been proposed for discovering abrupt changes in piecewise constant signals, few methods are available to capture these changes in piecewise polynomial signals. In this paper, we propose a change point detection method, PRUTF, based on trend filtering. By providing a comprehensive dual solution path for trend filtering, PRUTF allows us to discover change points of the underlying signal for either a given value of the regularization parameter or a specific number of steps of the algorithm. We demonstrate that the dual solution path constitutes a Gaussian bridge process that enables us to derive an exact and efficient stopping rule for terminating the search algorithm. We also prove that the estimates produced by this algorithm are asymptotically consistent in pattern recovery. This result holds even in the case of staircases (consecutive change points of the same sign) in the signal. Finally, we investigate the performance of our proposed method for various signals and then compare its performance against some state-of-the-art methods in the context of change point detection. We apply our method to three real-world datasets including the UK House Price Index (HPI), the GISS surface Temperature Analysis (GISTEMP) and the Coronavirus disease (COVID-19) pandemic.

翻译：虽然已经提出了许多方法来发现分质恒定信号的突然变化,但很少有方法可以捕捉分质多元度信号的这些变化。在本文中,我们根据趋势过滤提出改变点检测方法,即PRUTF。通过为趋势过滤提供一个全面的双重解决方案路径,PRUTF允许我们发现趋势过滤法中基本信号的改变点,无论是对正规化参数的某个特定价值还是对算法的某个特定步骤。我们证明,双重解决方案路径是一个高斯桥路程过程,使我们能够为终止搜索算法得出准确有效的停止规则。我们还证明,这一算法产生的估计值在模式恢复中是过于同步的。即使是在信号中的斜体(同一标志的连续变化点)情况下,结果也是如此。最后,我们调查了各种信号的拟议方法的性能,然后在变化点检测中将其性能与一些最先进的方法进行比较。我们将我们的方法应用于三个真实世界数据集,包括英国房价格指数(HIPI)、GISS-19 GRIVA(GISIS) 和CRIVA-ISA性分析。