Journal of Neuroscience Methods 172 (2008) 79–93 Contents lists available at ScienceDirect Journal of Neuroscience Methods journal homepage: www.elsevier.com/locate/jneumeth Partial Granger causality—Eliminating exogenous inputs and latent variables Shuixia Guo a , Anil K. Seth d , Keith M. Kendrick e , Cong Zhou d , Jianfeng Feng b,a,c,∗ a Department of Mathematics, Hunan Normal University, Changsha 410081, PR China b Centre for Computational System Biology, Fudan University, PR China c Department of Computer Science and Mathematics, Warwick University, Coventry CV4 7AL, UK d Department of Informatics, Sussex University, Brighton BN1 9QH, UK e Laboratory of Behaviour and Cognitive Neuroscience, The Babraham Institute, Cambridge CB22 3AT, UK article info Article history: Received 4 September 2007 Received in revised form 6 April 2008 Accepted 7 April 2008 Keywords: Conditional Granger causality Partial Granger causality Latent variables abstract Attempts to identify causal interactions in multivariable biological time series (e.g., gene data, protein data, physiological data) can be undermined by the confounding influence of environmental (exogenous) inputs. Compounding this problem, we are commonly only able to record a subset of all related variables in a system. These recorded variables are likely to be influenced by unrecorded (latent) variables. To address this problem, we introduce a novel variant of a widely used statistical measure of causality – Granger causality – that is inspired by the definition of partial correlation. Our ‘partial Granger causality’ measure is extensively tested with toy models, both linear and nonlinear, and is applied to experimental data: in vivo multielectrode array (MEA) local field potentials (LFPs) recorded from the inferotemporal cortex of sheep. Our results demonstrate that partial Granger causality can reveal the underlying interactions among elements in a network in the presence of exogenous inputs and latent variables in many cases where the existing conditional Granger causality fails. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Methods for identifying intrinsic relationships among elements in a network are increasingly in demand in today’s data-rich research areas such as biology and economics. In particular, advances in experimental and computational techniques are revolutionizing the field of neuroscience. On one hand novel experimental techniques such as high-density multielectrode arrays (MEAs) have made routine the acquisition of massive amounts of empirical data. On the other, new computational techniques are increasingly in demand for interpreting this data and for generating hypotheses. A question of great interest with respect to network interactions is whether there exist causal relations among a set of measured variables (Baker et al., 2006; Datta and Siwek, 1997; Fairhall et al., 2006; Feng and Durand, 2005; Gourevitch and Eggermont, 2007; Jacobi and Moses, 2007; Knyazeva et al., 2006; Miller and White, 2007; Oswald et al., 2007). Over the last few decades several techniques such as Bayesian networks (Friedman et al., 2000) and Granger causality (Baccala et al., 1998; Gourevitch et al., 2006; Granger, 1969, 1980) have been developed to identify causal relationships in dynamic systems. Wiener (Wiener, 1956) proposed a way to measure the causal influence of one time series on another by conceiving the notion that the prediction of one time series could be improved by incorporating knowledge of the other. Granger (1969) formalized this notion in the context of linear vector autoregression (VAR) model of stochastic processes. Specifically, given two time series, if the variance of the prediction error for the second time series is reduced by including past measurements of the first time series in the linear regression model, then the first time series can be said to cause the second time series. From this definition it is clear that the flow of time plays a vital role in the inference of directed interactions from time series data and as such many applications of Granger causality remain in the time domain. Granger’s conception of causality has received a great deal of attention and has been applied widely in the econometrics literature and more recently in the biological literature (Baccala et al., 1998; Gourevitch et al., 2006; Granger, 1969, 1980). The formalism for bivariate Granger causality is described in Appendix A.1. The basic Granger causality described in Appendix A.1 is applicable only to bivariate time series. In multivariable (more than two) situations, one time series can be connected to another time series in a direct or an indirect manner, raising the important question of ∗ Corresponding author at: Department of Computer Science and Mathematics, Warwick University, Coventry CV4 7AL, UK. E-mail address: jianfeng.feng@warwick.ac.uk (J. Feng). 0165-0270/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jneumeth.2008.04.011