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Markov Additive Processes and Repeats in Sequences

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

John L. Spouge
John L. Spouge*
Affiliation:
National Library of Medicine
*
∗ Postal address: National Center for Biotechnology Information, National Library of Medicine, Bethesda, MD 20894, USA. Email address: spouge@ncbi.nlm.nih.gov
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Abstract

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Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.

Keywords

Markov additive processbiological sequence analysisrepeats

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 60G51: Processes with independent increments; L'evy processes 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 514 - 527
Copyright
Copyright © Applied Probability Trust 2007 

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