Bundled Causal History Interaction

Complex system arises as a result of the nonlinear interactions between components. In particular, the evolutionary dynamics of a multivariate system encodes the ways in which different variables interact with each other individually or in groups. One fundamental question that remains unanswered is: how do two non-overlapping multivariate subsets of variables interact to causally determine the outcome of a specific variable? Here we provide an information based approach to address this problem. We delineate the temporal interactions between the bundles in a probabilistic graphical model. The strength of the interactions, captured by partial information decomposition, then exposes complex behavior of dependencies and memory within the system. The proposed approach successfully illustrates complex dependence between cations and anions as determinants of \textit{pH} in an observed stream chemistry system. This example demonstrates the potentially broad applicability of the approach, establishing the foundation to study the interaction between groups of variables in a range of complex systems.