Skip to main content
eScholarship
Open Access Publications from the University of California

UC Davis

UC Davis Previously Published Works bannerUC Davis

Spectral Simplicity of Apparent Complexity, Part II: Exact Complexities and Complexity Spectra

  • Author(s): Riechers, Paul M.
  • Crutchfield, James P.
  • et al.

Published Web Location

https://arxiv.org/pdf/1706.00883.pdf
No data is associated with this publication.
Abstract

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 generic to the metadynamics of predicting their behavior. Using the resulting spectral decomposition, we derive closed-form expressions for correlation functions, finite-length Shannon entropy-rate approximates, asymptotic entropy rate, excess entropy, transient information, transient and asymptotic state uncertainty, and synchronization information of stochastic processes generated by finite-state hidden Markov models. This introduces analytical tractability to investigating information processing in discrete-event stochastic processes, symbolic dynamics, and chaotic dynamical systems. Comparisons reveal mathematical similarities between complexity measures originally thought to capture distinct informational and computational properties. We also introduce a new kind of spectral analysis via coronal spectrograms and the frequency-dependent spectra of past-future mutual information. We analyze a number of examples to illustrate the methods, emphasizing processes with multivariate dependencies beyond pairwise correlation. An appendix presents spectral decomposition calculations for one example in full detail.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Item not freely available? Link broken?
Report a problem accessing this item