We introduce harmonic cross-correlation decomposition (HCD) as a tool to detect and visualize features in the frequency structure of multivariate time series. HCD decomposes multivariate time series into spatiotemporal harmonic modes with the leading modes representing dominant oscillatory patterns in the data. HCD is closely related to data-adaptive harmonic decomposition (DAHD) [Chekroun and Kondrashov, Chaos 27, 093110 (2017)] in that it performs an eigendecomposition of a grand matrix containing lagged cross-correlations. As for DAHD, each HCD mode is uniquely associated with a Fourier frequency, which allows for the definition of multidimensional power and phase spectra. Unlike in DAHD, however, HCD does not exhibit a systematic dependency on the ordering of the channels within the grand matrix. Further, HCD phase spectra can be related to the phase relations in the data in an intuitive way. We compare HCD with DAHD and multivariate singular spectrum analysis, a third related correlation-based decomposition, and we give illustrative applications to a simple traveling wave, as well as to simulations of three coupled Stuart-Landau oscillators and to human EEG recordings.

中文翻译：

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多元时间序列的谐波互相关分解

我们引入了谐波互相关分解 (HCD) 作为检测和可视化多元时间序列频率结构中的特征的工具。HCD 将多元时间序列分解为时空谐波模式，其中领先模式代表数据中的主要振荡模式。HCD 与数据自适应谐波分解 (DAHD) 密切相关 [Chekroun and Kondrashov, Chaos 27, 093110 (2017)] 因为它执行包含滞后互相关的大矩阵的特征分解。对于 DAHD，每个 HCD 模式都与一个傅立叶频率唯一相关联，从而可以定义多维功率和相位谱。然而，与 DAHD 不同，HCD 不表现出对大矩阵内通道排序的系统依赖性。此外，HCD 相位谱可以以直观的方式与数据中的相位关系相关联。我们将 HCD 与 DAHD 和多元奇异谱分析（第三种相关的基于相关的分解）进行了比较，并给出了对简单行波、三个耦合 Stuart-Landau 振荡器的模拟和人类 EEG 记录的说明性应用。