## Journal of Applied Probability

### Pattern Markov chains: optimal Markov chain embedding through deterministic finite automata

Grégory Nuel

#### Abstract

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.

#### Article information

Source
J. Appl. Probab. Volume 45, Number 1 (2008), 226-243.

Dates
First available in Project Euclid: 16 April 2008

Permanent link to this document
http://projecteuclid.org/euclid.jap/1208358964

Digital Object Identifier
doi:10.1239/jap/1208358964

Mathematical Reviews number (MathSciNet)
MR2409323

Zentralblatt MATH identifier
1142.65010

Subjects
Primary: 65C40: Computational Markov chains

#### Citation

Nuel, Grégory. Pattern Markov chains: optimal Markov chain embedding through deterministic finite automata. J. Appl. Probab. 45 (2008), no. 1, 226--243. doi:10.1239/jap/1208358964. http://projecteuclid.org/euclid.jap/1208358964.

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