Information Anatomy on Partition Space

Symbolic dynamics allows us to model—and design for—the effects of imperfectly measuring a time series of data, by partitioning into a finite number of possibilities. The resulting time series of discrete symbols is then made especially amenable to information-theoretic methods for understanding its temporal structure and correlation. In particular, as long as the partitioning scheme is generating, the resulting estimations for entropy rate—the rate at which the process creates information—converge to a measure of chaos in the underlying system, a dynamical invariant.

However, the entropy rate is not sensitive to what kind of generating partition: colloquially, an instrument must be at least accurate enough, but can be more fine-grained as desired. In contrast, its breakdown into a piece which affects future measurements (“bound”) and a piece which does not (“ephemeral”) depends quite dramatically on the choice of generating partition. We ask, then: is there a canonical partitioning scheme for which the full suite of information measures relate to dynamical invariants?

We find that ephemeral and bound information are extremized by the single-boundary coarsest generating partition in two-boundary sweeps of the tent and logistic maps, suggesting a canonical role for the simplest “good enough” instrument. The remaining multivariate measures, meanwhile, reveal hitherto-unseen structure in the process of imperfect measurement