Coherence is a widely used measure to assess linear relationships between time series. However, it fails to capture nonlinear dependencies. To overcome this limitation, this paper introduces the notion of residual spectral density as a higher-order extension of the squared coherence. The method is based on an orthogonal decomposition of time series regression models. We propose a test for testing the existence of the residual spectrum and derive its fundamental properties. A numerical study illustrates finite sample performance of the proposed method. An application of the method shows that the residual spectrum can effectively detect brain connectivity.

翻译：相干性是一种广泛用于评估时间序列线性关系的度量，但无法捕捉非线性依赖关系。为克服这一局限，本文提出将残差谱密度作为平方相干性的高阶拓展。该方法基于时间序列回归模型的正交分解。我们提出了一种检验残差谱存在性的统计检验方法，并推导出其基本性质。数值研究展示了所提方法在有限样本下的性能表现。方法应用表明，残差谱能有效检测脑连接性。