  We introduce a notion of emergence for coarse-grained macroscopic variables
associated with highly-multivariate microscopic dynamical processes, in the
context of a coupled dynamical environment. Dynamical independence instantiates
the intuition of an emergent macroscopic process as one possessing the
characteristics of a dynamical system "in its own right", with its own
dynamical laws distinct from those of the underlying microscopic dynamics. We
quantify (departure from) dynamical independence by a transformation-invariant
Shannon information-based measure of dynamical dependence. We emphasise the
data-driven discovery of dynamically-independent macroscopic variables, and
introduce the idea of a multiscale "emergence portrait" for complex systems. We
show how dynamical dependence may be computed explicitly for linear systems via
state-space modelling, in both time and frequency domains, facilitating
discovery of emergent phenomena at all spatiotemporal scales. We discuss
application of the state-space operationalisation to inference of the emergence
portrait for neural systems from neurophysiological time-series data. We also
examine dynamical independence for discrete- and continuous-time deterministic
dynamics, with potential application to Hamiltonian mechanics and classical
complex systems such as flocking and cellular automata.
