Asymmetric separation for local independence graphs
This work addresses the need for asymmetric graphical representations in continuous-time stochastic processes, offering a theoretical advancement for researchers in causality and time-series analysis, though it appears incremental as it builds on prior generalizations of Granger-causality.
The paper tackles the problem of representing dynamic dependencies among stochastic processes using directed possibly cyclic graphs, developing an asymmetric notion of separation called delta-separation to capture local independence based on generalized Granger-causality. It investigates the properties of this separation and local independence within an asymmetric (semi)graphoid framework to enhance interpretability of such graphs.
Directed possibly cyclic graphs have been proposed by Didelez (2000) and Nodelmann et al. (2002) in order to represent the dynamic dependencies among stochastic processes. These dependencies are based on a generalization of Granger-causality to continuous time, first developed by Schweder (1970) for Markov processes, who called them local dependencies. They deserve special attention as they are asymmetric unlike stochastic (in)dependence. In this paper we focus on their graphical representation and develop a suitable, i.e. asymmetric notion of separation, called delta-separation. The properties of this graph separation as well as of local independence are investigated in detail within a framework of asymmetric (semi)graphoids allowing a deeper insight into what information can be read off these graphs.