Computational and Mathematical Methods in Medicine
Volume 2012 (2012), Article ID 140513, 18 pages
Research Article

Measuring Connectivity in Linear Multivariate Processes: Definitions, Interpretation, and Practical Analysis

Laboratorio Biosegnali, Dipartimento di Fisica & BIOtech, Università di Trento, via delle Regole 101, 38123 Mattarello, Trento, Italy

Received 28 October 2011; Revised 22 February 2012; Accepted 3 March 2012

Academic Editor: Dimitris Kugiumtzis

Copyright © 2012 Luca Faes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This tutorial paper introduces a common framework for the evaluation of widely used frequency-domain measures of coupling (coherence, partial coherence) and causality (directed coherence, partial directed coherence) from the parametric representation of linear multivariate (MV) processes. After providing a comprehensive time-domain definition of the various forms of connectivity observed in MV processes, we particularize them to MV autoregressive (MVAR) processes and derive the corresponding frequency-domain measures. Then, we discuss the theoretical interpretation of these MVAR-based connectivity measures, showing that each of them reflects a specific time-domain connectivity definition and how this results in the description of peculiar aspects of the information transfer in MV processes. Furthermore, issues related to the practical utilization of these measures on real-time series are pointed out, including MVAR model estimation and significance assessment. Finally, limitations and pitfalls arising from model mis-specification are discussed, indicating possible solutions and providing practical recommendations for a safe computation of the connectivity measures. An example of estimation of the presented measures from multiple EEG signals recorded during a combined visuomotor task is also reported, showing how evaluation of coupling and causality in the frequency domain may help describing specific neurophysiological mechanisms.