  Most information dynamics and statistical causal analysis frameworks rely on
the common intuition that causal interactions are intrinsically pairwise --
every 'cause' variable has an associated 'effect' variable, so that a 'causal
arrow' can be drawn between them. However, analyses that depict
interdependencies as directed graphs fail to discriminate the rich variety of
modes of information flow that can coexist within a system. This, in turn,
creates problems with attempts to operationalise the concepts of 'dynamical
complexity' or `integrated information.' To address this shortcoming, we
combine concepts of partial information decomposition and integrated
information, and obtain what we call Integrated Information Decomposition, or
$\Phi$ID. We show how $\Phi$ID paves the way for more detailed analyses of
interdependencies in multivariate time series, and sheds light on collective
modes of information dynamics that have not been reported before. Additionally,
$\Phi$ID reveals that what is typically referred to as 'integration' is
actually an aggregate of several heterogeneous phenomena. Furthermore, $\Phi$ID
can be used to formulate new, tailored measures of integrated information, as
well as to understand and alleviate the limitations of existing measures.
