Analyzing time series in the frequency domain enables the development of powerful tools for investigating the second-order characteristics of multivariate processes. Parameters like the spectral density matrix and its inverse, the coherence or the partial coherence, encode comprehensively the complex linear relations between the component processes of the multivariate system. In this paper, we develop inference procedures for such parameters in a high-dimensional, time series setup. Towards this goal, we first focus on the derivation of consistent estimators of the coherence and, more importantly, of the partial coherence which possess manageable limiting distributions that are suitable for testing purposes. Statistical tests of the hypothesis that the maximum over frequencies of the coherence, respectively, of the partial coherence, do not exceed a prespecified threshold value are developed. Our approach allows for testing hypotheses for individual coherences and/or partial coherences as well as for multiple testing of large sets of such parameters. In the latter case, a consistent procedure to control the false discovery rate is developed. The finite sample performance of the inference procedures introduced is investigated by means of simulations and applications to the construction of graphical interaction models for brain connectivity based on EEG data are presented.

翻译：在频域中分析时间序列为研究多变量过程的二阶特征提供了强有力的工具。谱密度矩阵及其逆矩阵、相干性或偏相干性等参数全面编码了多变量系统中各分量过程之间的复杂线性关系。本文针对高维时间序列场景，开发了这些参数的推断方法。为此，我们首先聚焦于相干性，更重要的是偏相干性的一致估计量推导，这些估计量具有易于处理的极限分布，适用于假设检验。我们构建了如下统计检验：相干性（以及偏相干性）的频率最大值不超过预设阈值的原假设。本文方法既支持对单个相干性和/或偏相干性进行假设检验，也支持对大量此类参数进行多重检验。在后一种情况下，我们开发了控制错误发现率的一致程序。通过数值模拟和基于脑电图数据的脑连接性图形交互模型应用实例，对所提出的推断方法在有限样本下的表现进行了研究。