Many Roads to Synchrony: Natural Time Scales and Their Algorithms
- Author(s): James, Ryan G.
- Mahoney, John R.
- Ellison, Christopher J.
- Crutchfield, James P.
- et al.
Published Web Locationhttps://arxiv.org/pdf/1010.5545.pdf
We consider two important time scales---the Markov and cryptic orders---that monitor how an observer synchronizes to a finitary stochastic process. We show how to compute these orders exactly and that they are most efficiently calculated from the epsilon-machine, a process's minimal unifilar model. Surprisingly, though the Markov order is a basic concept from stochastic process theory, it is not a probabilistic property of a process. Rather, it is a topological property and, moreover, it is not computable from any finite-state model other than the epsilon-machine. Via an exhaustive survey, we close by demonstrating that infinite Markov and infinite cryptic orders are a dominant feature in the space of finite-memory processes. We draw out the roles played in statistical mechanical spin systems by these two complementary length scales.