Causalized convergent cross mapping and its approximate equivalence with directed information in causality analysis

Abstract

Convergent cross mapping (CCM) has attracted increased attention recently due to its capability in detecting causality in non-separable systems under deterministic settings, which may not be covered by the traditional Granger causality. From an information-theoretic perspective, causality is often characterized as the directed information (DI) flowing from one side to the other. As information is essentially nondeterministic, a natural question is: does CCM measure directed information flow? Here we first causalize CCM so that it aligns with the presumption in causality analysis—the future values of one process cannot influence the past of the other, and then establish and validate the approximate equivalence of causalized CCM (cCCM) and directed information (DI) under Gaussian variables through both theoretical derivations and fMRI-based brain network causality analysis. Our simulation result indicates that, in general, cCCM tends to be more robust than DI in causality detection. The underlying argument is that DI relies heavily on probability estimation, which is sensitive to data size as well as the digitization procedures; cCCM, on the other hand, gets around this problem through geometric cross-mapping between the manifolds involved. Overall, our analysis demonstrates that cross-mapping provides an alternative way to evaluate directed information and is potentially an effective technique for identifying both linear and nonlinear causal coupling in brain neural network and other settings, either random or deterministic, or both.