Granger Causality: Theory and Applications

Abstract

A question of great interest in systems biology is how to uncover complex network structures from experimental data[1, 3, 18, 38, 55]. With the rapid progress of experimental techniques, a crucial task is to develop methodologies that are both statistically sound and computationally feasible for analysing increasingly large datasets and reliably inferring biological interactions from them [16, 17, 22, 37, 40, 42]. The building block of such enterprise is to being able to detect relations (causal, statistical or functional) between nodes of the network. Over the past two decades, a number of approaches have been developed: information theory ([4]), control theory ([17]) or Bayesian statistics ([35]). Here we will be focusing on another successful alternative approach: Granger causality. In recent Cell papers [7, 12], the authors have come to the conclusion that the ordinary differential equation approach outperforms the other reverse engineering approaches (Bayesian network and information theory) in building causal networks. We have demonstrated that the Granger causality achieves better results than the ordinary differential approach [34]. The basic idea of Granger causality can be traced back to Wiener[47] who conceived the notion that, if the prediction of one time series is improved by incorporating the knowledge of a second time series, then the latter is said to have a causal influence on the first. Granger[23, 24] later formalized Wiener’s idea in the context of linear regression models. Specifically, two auto-regressive models are fitted to the first time series – with and without including the second time series – and the improvement of the prediction is measured by the ratio of the variance of the error terms. A ratio larger than one signifies an improvement, hence a causal connection. At worst, the ratio is 1 and signifies causal independence from the second time series to the first. Geweke’s decomposition of a vector autoregressive process ([20, 21]) led to a set of causality measures which have a spectral representation and make the interpretation more informative and useful by extending Granger causality to the frequency domain. In this chapter, we aim to present Granger causality and how its original formalism has been extended to address biological and computational issues, as summarized in Fig. 5.1.