Spectral Interdependency Methods

Spectral interdependency methods are a means of statistically quantifying the interrelationship between a pair of dynamic processes as a function of frequency or time period of oscillation. The measures of spectral interdependency are derived from the time series recordings of dynamic systems either by using autoregressive modeling (parametric method) or by using direct Fourier or wavelet transforms (nonparametric method). For a pair of multivariate stationary processes (1 and 2), there are three measures that characterize the spectral interdependency between these processes. They are total interdependence (M1,2), Granger causality (one-way effect or directional influence from the first process to the second process, M1! 2, or from the second to the first, M2! 1), and instantaneous causality (measure of reciprocity, M1.2). In general, the total interdependence is the sum of directional influences and instantaneous causality frequency by frequency (M1,21⁄4M1! 2 + M2! 1 + M1.2) and is related to coherence (C12) as M1,21⁄4 ln(1 C12). These measures can also be estimated for nonstationary processes by using moving time-windowed autoregressive modeling or Fourier transformations or wavelet transformations. Coherence and spectral Granger causality are well-accepted measures in neuroscience to characterize frequency-specific interdependence between multiple time series from multisite neurophysiological recordings.