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Pattern Markov Chains: Optimal Markov Chain Embedding Through Deterministic Finite Automata

Part of: Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Grégory Nuel
Grégory Nuel*
Affiliation:
University of Evry
*
∗Current address: MAP5, UMR CNRS 8145, Université Paris Descartes, 45 rue des Saints-Pères, F-75006 Paris, France. Email address: gregory.nuel@math-info.univ-paris5.fr
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Abstract

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In the framework of patterns in random texts, the Markov chain embedding techniques consist of turning the occurrences of a pattern over an order-m Markov sequence into those of a subset of states into an order-1 Markov chain. In this paper we use the theory of language and automata to provide space-optimal Markov chain embedding using the new notion of pattern Markov chains (PMCs), and we give explicit constructive algorithms to build the PMC associated to any given pattern problem. The interest of PMCs is then illustrated through the exact computation of P-values whose complexity is discussed and compared to other classical asymptotic approximations. Finally, we consider two illustrative examples of highly degenerated pattern problems (structured motifs and PROSITE signatures), which further illustrate the usefulness of our approach.

Keywords

Languageregular expressionexact distributionstructured motifPROSITE signature

MSC classification

Secondary: 65C40: Computational Markov chains
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 226 - 243
Copyright
Copyright © Applied Probability Trust 2008 

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