A framework for assessing frequency domain causality in physiological time series with instantaneous effects

Luca Faes, Silvia Erla, Alberto Porta, Giandomenico Nollo

Research output: Contribution to journalArticlepeer-review


We present an approach for the quantification of directional relations in multiple time series exhibiting significant zero-lag interactions. To overcome the limitations of the traditional multivariate autoregressive (MVAR) modelling of multiple series, we introduce an extended MVAR (eMVAR) framework allowing either exclusive consideration of time-lagged effects according to the classic notion of Granger causality, or consideration of combined instantaneous and lagged effects according to an extended causality definition. The spectral representation of the eMVAR model is exploited to derive novel frequency domain causality measures that generalize to the case of instantaneous effects the known directed coherence (DC) and partial DC measures. The new measures are illustrated in theoretical examples showing that they reduce to the known measures in the absence of instantaneous causality, and describe peculiar aspects of directional interaction among multiple series when instantaneous causality is non-negligible. Then, the issue of estimating eMVAR models from time-series data is faced, proposing two approaches for model identification and discussing problems related to the underlying model assumptions. Finally, applications of the framework on cardiovascular variability series and multichannel EEG recordings are presented, showing how it allows one to highlight patterns of frequency domain causality consistent with wellinterpretable physiological interaction mechanisms.

Original languageEnglish
Article number20110618
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number1997
Publication statusPublished - Aug 28 2013


  • Directed coherence
  • Granger causality
  • Multivariate autoregressive models
  • Partial directed coherence

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Engineering(all)


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