This dissertation is an information-theoretic analysis of discrete-time, discrete-valued, stationary, stochastic dynamical systems that are modeled by edge-emitting, finite-state hidden Markov models. The approach follows that of computational mechanics, a discipline which uses the optimal predictor of a dynamical system, known as the ε-machine, to uniquely characterize its structural properties. One of the primary contributions of this thesis is a closed-form expression for the excess entropy, a measure of structure which describes the amount of information that the entire past of the system shares with the entire future of the system. Generalizing away from ε-machines, we then define a number of structural measures for generic hidden Markov models through the decomposition of their state entropy. This decomposition breaks the state entropy into four components: excess entropy, crypticity, oracular information, and gauge information. Finally, this thesis utilizes optimal predictors and retrodictors in a detailed study of irreversibility. The bidirectional machine is introduced as an encompassing model of dynamical systems from which one can calculate, in closed-form, a number of well-known, and also new, measures of structural complexity.
|Advisor:||Crutchfield, James P.|
|Commitee:||D'Souza, Raissa M., Rundle, John B.|
|School:||University of California, Davis|
|School Location:||United States -- California|
|Source:||DAI-B 73/03, Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Theoretical physics, Computer science|
|Keywords:||Automata, Computational mechanics, Dynamical systems, Hidden Markov models, Information theory, Stochastic processes|
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