Authors: Nathan Jackson, Ryan James, James Crutchfield
Mentors: Mikhael Semaan
Insitution: University of Utah
Symbolic dynamics allows for modeling---and designing 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?
To tackle this, we simulate the well-known tent and logistic maps, sweeping over all two-boundary partitions, calculating the full information anatomies for each. We find that ephemeral and bound information are extremized by the single-boundary coarsest generating partition, 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.