# Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction

## TL;DR

This paper develops a spectral analysis framework for complex stochastic processes using meromorphic functional calculus to handle nonnormal, nondiagonalizable operators, enabling new insights into system organization and complexity measures.

## Contribution

It introduces the application of meromorphic functional calculus to analyze nonnormal operators in stochastic processes, providing a foundation for explicit complexity calculations.

## Key findings

- Spectral decomposition of nonnormal operators is achieved using meromorphic calculus.
- Special properties of projection operators reveal subprocess organization.
- Circumvents infinities in traditional spectral analysis methods.

## Abstract

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 an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.