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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:11017 |
| Date | 12 May 2010 |
| Creators | Löhr, Wolfgang |
| Contributors | Jost, Jürgen, Ay, Nihat, Keller, Gerhard, Universität Leipzig |
| Publisher | Max Planck Institut für Mathematik in den Naturwissenschaften |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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