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

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.

27 pages, 12 figures, 2 tables; most recent version at http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt2.htm

References in corpus (4)

• Nearly Maximally Predictive Features and Their Dimensions
• Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
• Diffraction Patterns of Layered Close-packed Structures from Hidden Markov Models
• Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators

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• Shannon Entropy Rate of Hidden Markov Processes
• Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes
• Local Causal States and Discrete Coherent Structures
• Fraudulent White Noise: Flat power spectra belie arbitrarily complex processes
• Discovering Causal Structure with Reproducing-Kernel Hilbert Space $\epsilon$-Machines
• Algebraic Theory of Patterns as Generalized Symmetries