Testing frequency-domain causality in multivariate time series

We introduce a new hypothesis-testing framework, based on surrogate data generation, to assess in the frequency domain, the concept of causality among multivariate (MV) time series. The approach extends the traditional Fourier transform (FT) method for generating surrogate data in a MV process and adapts it to the specific issue of causality. It generates causal FT (CFT) surrogates with FT modulus taken from the original series, and FT phase taken from a set of series with causal interactions set to zero over the direction of interest and preserved over all other directions. Two different zero-setting procedures, acting on the parameters of a MV autoregressive (MVAR) model fitted on the original series, were used to test the null hypotheses of absence of direct causal influence (CFTd surrogates) and of full (direct and indirect) causal influence (CFTf surrogates), respectively. CFTf and CFTd surrogates were utilized in combination with the directed coherence (DC) and the partial DC (PDC) spectral causality estimators, respectively. Simulations reproducing different causality patterns in linear MVAR processes demonstrated the better accuracy of CFTf and CFTd surrogates with respect to traditional FT surrogates. Application on real MV biological data measured from healthy humans, i.e., heart period, arterial pressure, and respiration variability, as well as multichannel EEG signals, showed that CFT surrogates disclose causal patterns in accordance with expected cardiorespiratory and neurophysiological mechanisms.