In this paper we revisit the problem of decomposing a signal into a tendency and a residual. The tendency describes an executive summary of a signal that encapsulates its notable characteristics while disregarding seemingly random, less interesting aspects. Building upon the Intrinsic Time Decomposition (ITD) and information-theoretical analysis, we introduce two alternative procedures for selecting the tendency from the ITD baselines. The first is based on the maximum extrema prominence, namely the maximum difference between extrema within each baseline. Specifically this method selects the tendency as the baseline from which an ITD step would produce the largest decline of the maximum prominence. The second method uses the rotations from the ITD and selects the tendency as the last baseline for which the associated rotation is statistically stationary. We delve into a comparative analysis of the information content and interpretability of the tendencies obtained by our proposed methods and those obtained through conventional low-pass filtering schemes, particularly the Hodrik-Prescott (HP) filter. Our findings underscore a fundamental distinction in the nature and interpretability of these tendencies, highlighting their context-dependent utility with emphasis in multi-scale signals. Through a series of real-world applications, we demonstrate the computational robustness and practical utility of our proposed tendencies, emphasizing their adaptability and relevance in diverse time series contexts.

翻译：本文重新审视了将信号分解为趋势与残差的问题。趋势描述了信号的执行摘要，它概括了信号的显著特征，同时忽略了看似随机、不太重要的方面。基于内在时间分解（ITD）和信息论分析，我们提出了两种从ITD基线中选择趋势的替代方法。第一种方法基于最大极值显著性，即每个基线内极值之间的最大差异。具体而言，该方法将趋势选择为ITD步骤中会导致最大显著性下降幅度最大的基线。第二种方法利用ITD的旋转特性，将趋势选择为相应旋转在统计上平稳的最后一个基线。我们深入分析了通过所提方法获得的趋势与通过传统低通滤波方案（特别是霍德里克-普雷斯科特（HP）滤波器）获得的趋势在信息内容和可解释性上的差异。研究结果揭示了这些趋势在性质和可解释性上的根本区别，强调了它们在不同情境下的实用性，尤其关注多尺度信号。通过一系列实际应用，我们证明了所提趋势的计算鲁棒性和实际效用，突出了它们在多样化时间序列情境中的适应性和相关性。