  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.
