title: The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference.
abstract: BACKGROUND
  Wiener-Granger causality ("G-causality") is a statistical notion of causality applicable to time series data, whereby cause precedes, and helps predict, effect. It is defined in both time and frequency domains, and allows for the conditioning out of common causal influences. Originally developed in the context of econometric theory, it has since achieved broad application in the neurosciences and beyond. Prediction in the G-causality formalism is based on VAR (vector autoregressive) modelling.
  
  
  METHODS
  The MVGC Matlab© Toolbox approach to G-causal inference is based on multiple equivalent representations of a VAR model by (i) regression parameters, (ii) the autocovariance sequence and (iii) the cross-power spectral density of the underlying process. It features a variety of algorithms for moving between these representations, enabling selection of the most suitable algorithms with regard to computational efficiency and numerical accuracy.
  
  
  RESULTS
  In this paper we explain the theoretical basis, computational strategy and application to empirical G-causal inference of the MVGC Toolbox. We also show via numerical simulations the advantages of our Toolbox over previous methods in terms of computational accuracy and statistical inference.
  
  
  CONCLUSIONS
  The standard method of computing G-causality involves estimation of parameters for both a full and a nested (reduced) VAR model. The MVGC approach, by contrast, avoids explicit estimation of the reduced model, thus eliminating a source of estimation error and improving statistical power, and in addition facilitates fast and accurate estimation of the computationally awkward case of conditional G-causality in the frequency domain.
  
  
  CONCLUSIONS
  The MVGC Toolbox implements a flexible, powerful and efficient approach to G-causal inference.
authors.affiliation: Sackler Centre for Consciousness Science and School of Informatics, University of Sussex, Brighton BN1 9QJ, UK. Electronic address: l.c.barnett@sussex.ac.uk.
authors.forename: Lionel
authors.initials: L
authors.lastname: Barnett
authors.affiliation: Sackler Centre for Consciousness Science and School of Informatics, University of Sussex, Brighton BN1 9QJ, UK. Electronic address: a.k.seth@sussex.ac.uk.
authors.forename: Anil K
authors.initials: AK
authors.lastname: Seth
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D006801:Humans; D008962:Models, Theoretical; D012815:Signal Processing, Computer-Assisted; D012984:Software; D013997:Time Factors
keywords: Granger causality; Time series analysis; Vector autoregressive modelling