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Shannon Entropy Rate of Hidden Markov Processes
Alexandra M. Jurgens, James P. Crutchfield
Hidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidd…
Spectral Simplicity of Apparent Complexity, Part II: Exact Complexities and Complexity Spectra
Paul M. Riechers, James P. Crutchfield
The meromorphic functional calculus developed in Part I overcomes the nondiagonalizability of linear operators that arises often in the temporal evolution of complex systems and is…
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Paul M. Riechers, James P. Crutchfield
Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by…
Chaos Forgets and Remembers: Measuring Information Creation, Destruction, and Storage
Ryan G. James, Korana Burke, James P. Crutchfield
The hallmark of deterministic chaos is that it creates information---the rate being given by the Kolmogorov-Sinai metric entropy. Since its introduction half a century ago, the met…
The Past and the Future in the Present
James P. Crutchfield, Christopher J. Ellison
We show how the shared information between the past and future---the excess entropy---derives from the components of directional information stored in the present---the predictive…
Asymptotic Synchronization for Finite-State Sources
Nicholas F. Travers, James P. Crutchfield
We extend a recent synchronization analysis of exact finite-state sources to nonexact sources for which synchronization occurs only asymptotically. Although the proof methods are q…