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The exact distribution of the k-tuple statistic for sequence homology

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  • Lou, W. Y. Wendy

Abstract

The distribution theory of runs and patterns has become increasingly useful in the field of biological sequence homology. One important application in detecting tandem duplications among DNA sequence segments is the k-tuple statistic Sn,k, the sum of matches in matching-runs of length k or longer in a sequence of n i.i.d. Bernoulli trials with success/matching probability p. Current approaches to this distribution problem are based on various approximations, due mainly to the numerical complexity of computing the exact distribution using a straightforward combinatorial approach. In this paper, we obtain a simple and efficient expression for the exact distribution of Sn,k using the principle of finite Markov chain imbedding. Our numerical results illustrate most importantly that for pattern lengths in the range n=10 to 100, a range commonly used in detecting DNA tandem repeats, the distribution, in general, is highly skewed and far from normal.

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  • Lou, W. Y. Wendy, 2003. "The exact distribution of the k-tuple statistic for sequence homology," Statistics & Probability Letters, Elsevier, vol. 61(1), pages 51-59, January.
    • Handle: RePEc:eee:stapro:v:61:y:2003:i:1:p:51-59
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    References listed on IDEAS

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    1. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
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    1. Sotirios Bersimis & Athanasios Sachlas & Pantelis G. Bagos, 2017. "Discriminating membrane proteins using the joint distribution of length sums of success and failure runs," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(2), pages 251-272, June.
    2. Frosso Makri & Zaharias Psillakis, 2011. "On runs of length exceeding a threshold: normal approximation," Statistical Papers, Springer, vol. 52(3), pages 531-551, August.
    3. Arapis, Anastasios N. & Makri, Frosso S. & Psillakis, Zaharias M., 2016. "On the length and the position of the minimum sequence containing all runs of ones in a Markovian binary sequence," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 45-54.
    4. Sáenz-de-Cabezón, Eduardo & Wynn, Henry P., 2011. "Computational algebraic algorithms for the reliability of generalized k-out-of-n and related systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(1), pages 68-78.
    5. Donald E. K. Martin, 2005. "Distribution of the Number of Successes in Success Runs of Length at Least k in Higher-Order Markovian Sequences," Methodology and Computing in Applied Probability, Springer, vol. 7(4), pages 543-554, December.
    6. Serkan Eryilmaz, 2018. "Stochastic Ordering Among Success Runs Statistics in a Sequence of Exchangeable Binary Trials," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 563-573, June.
    7. Makri, Frosso S. & Psillakis, Zaharias M. & Arapis, Anastasios N., 2015. "Length of the minimum sequence containing repeats of success runs," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 28-37.
    8. Anastasios N. Arapis & Frosso S. Makri & Zaharias M. Psillakis, 2017. "Joint distribution of k-tuple statistics in zero-one sequences of Markov-dependent trials," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-13, December.
    9. Donald E. K. Martin & Laurent Noé, 2017. "Faster exact distributions of pattern statistics through sequential elimination of states," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(1), pages 231-248, February.
    10. M. Koutras & S. Bersimis & D. Antzoulakos, 2008. "Bivariate Markov chain embeddable variables of polynomial type," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 173-191, March.

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