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

## TL;DR

This paper develops a spectral decomposition framework for analyzing complex stochastic processes, providing exact formulas for various information-theoretic measures and revealing mathematical similarities among them.

## Contribution

It introduces a novel spectral analysis method for hidden Markov models, enabling exact computation of complexity measures and exploring their interrelations.

## Key findings

- Derived closed-form expressions for entropy-related measures.
- Revealed mathematical similarities between different complexity measures.
- Introduced coronal spectrograms for frequency-dependent analysis.

## 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.