  Isolating slower dynamics from fast fluctuations has proven remarkably
powerful, but how do we proceed from partial observations of dynamical systems
for which we lack underlying equations? Here, we construct maximally-predictive
states by concatenating measurements in time, partitioning the resulting
sequences using maximum entropy, and choosing the sequence length to maximize
short-time predictive information. Transitions between these states yield a
simple approximation of the transfer operator, which we use to reveal timescale
separation and long-lived collective modes through the operator spectrum.
Applicable to both deterministic and stochastic processes, we illustrate our
approach through partial observations of the Lorenz system and the stochastic
dynamics of a particle in a double-well potential. We use our transfer operator
approach to provide a new estimator of the Kolmogorov-Sinai entropy, which we
demonstrate in discrete and continuous-time systems, as well as the movement
behavior of the nematode worm $C. elegans$.
