Dynamical Structure Function and Granger Causality: Similarities and differences

Causal networks are essential in many applications to illustrate causal relations in dynamical systems. In a view of statistics, Granger Causality (GC) gives a definition for causal ordering of time series, which implies a parametric model for stationary processes. In a systematic view, Dynamical Structure Function (DSF) is proposed to provide a general parametric representation for linear causal networks based on state space representations. It is difficult to determine which definition should be adopted for a particular application. By introducing an intermediate form of DSF, this article connects GC for stationary processes with DSF. Both GC and DSF essentially represent the same notion of causality but with important differences with respect to how they encode latent variables. This article also addresses the relations between graphs defined by GC and DSF. Furthermore, the uniqueness of parametric representations is addressed, which is essential in network inference. Results from different fields are surveyed and categorized into two categories - networks with exogeneity and networks without exogeneity. Limitations on sufficient conditions to guarantee exact identification are discussed under different assumptions on systems. In the end, a figure is used to summarize the relationships between various representations of causal dynamical networks and their identifiability conditions in LTI systems.