Eur. Phys. J. B (2017) 90: 197
https://doi.org/10.1140/epjb/e2017-80015-x

Regular Article

Conditional Granger causality of diffusion processes

¹ Institute for Chemistry and Biology of the Marine Environment, University of Oldenburg, 26129 Oldenburg, Germany
² ForWind-Center for Wind Energy Research, Institute of Physics, University of Oldenburg, 26129 Oldenburg, Germany
³ Institute of Computer Science, Academy of Sciences of the Czech Republic, 18207 Prague, Czech Republic
⁴ National Institute of Mental Health, Klecany, Czech Republic
⁵ Research Center Neurosensory Science, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany

^a e-mail: benjamin.wahl@uni-oldenburg.de

Received: 6 January 2017
Received in final form: 14 July 2017
Published online: 16 October 2017

Abstract

The statistical concept of Granger causality is defined by prediction improvement, i.e. the causing time series contains unique information about the future of the caused one. Recently we proposed extending this concept to bivariate diffusion processes by defining Granger causality for each point of the state space as the Granger causality of a process obtained by local linearisation. This provides a Granger causality map, well-defined at least in the vicinity of stable fixed points of the deterministic part of the dynamics. This extension has convenient properties, but carries several important limitations. In the current paper we show how the Granger causality of diffusion processes can be further generalized, incorporating in particular the concept of conditional causality. Moreover, we demonstrate the application potential to systems with a more complex attractor structure such as limit cycles or bistability of fixed points.