Residual Spectrum Applied in Brain Functional Connectivity

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.