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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

Models of Discrete-Time Stochastic Processes and Associated Complexity Measures / Modelle stochastischer Prozesse in diskreter Zeit und zugehörige Komplexitätsmaße

Löhr, Wolfgang 24 June 2010 (has links) (PDF)
Many complexity measures are defined as the size of a minimal representation in a specific model class. One such complexity measure, which is important because it is widely applied, is statistical complexity. It is defined for discrete-time, stationary stochastic processes within a theory called computational mechanics. Here, a mathematically rigorous, more general version of this theory is presented, and abstract properties of statistical complexity as a function on the space of processes are investigated. In particular, weak-* lower semi-continuity and concavity are shown, and it is argued that these properties should be shared by all sensible complexity measures. Furthermore, a formula for the ergodic decomposition is obtained. The same results are also proven for two other complexity measures that are defined by different model classes, namely process dimension and generative complexity. These two quantities, and also the information theoretic complexity measure called excess entropy, are related to statistical complexity, and this relation is discussed here. It is also shown that computational mechanics can be reformulated in terms of Frank Knight's prediction process, which is of both conceptual and technical interest. In particular, it allows for a unified treatment of different processes and facilitates topological considerations. Continuity of the Markov transition kernel of a discrete version of the prediction process is obtained as a new result.
2

Models of Discrete-Time Stochastic Processes and Associated Complexity Measures

Löhr, Wolfgang 12 May 2010 (has links)
Many complexity measures are defined as the size of a minimal representation in a specific model class. One such complexity measure, which is important because it is widely applied, is statistical complexity. It is defined for discrete-time, stationary stochastic processes within a theory called computational mechanics. Here, a mathematically rigorous, more general version of this theory is presented, and abstract properties of statistical complexity as a function on the space of processes are investigated. In particular, weak-* lower semi-continuity and concavity are shown, and it is argued that these properties should be shared by all sensible complexity measures. Furthermore, a formula for the ergodic decomposition is obtained. The same results are also proven for two other complexity measures that are defined by different model classes, namely process dimension and generative complexity. These two quantities, and also the information theoretic complexity measure called excess entropy, are related to statistical complexity, and this relation is discussed here. It is also shown that computational mechanics can be reformulated in terms of Frank Knight''s prediction process, which is of both conceptual and technical interest. In particular, it allows for a unified treatment of different processes and facilitates topological considerations. Continuity of the Markov transition kernel of a discrete version of the prediction process is obtained as a new result.

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