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Duality Between the Local Score of One Sequence and Constrained Hidden Markov Model

Author

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  • Sabine Mercier

    (Université de Toulouse 2 Jean Jaurès)

  • Grégory Nuel

    (Laboratoire de Probabilités Statistique Modélisation (LPSM), CNRS 8001Sorbonne Université)

Abstract

We are interested here in a theoretical and practical approach for detecting atypical segments in a multi-state sequence. We prove in this article that the segmentation approach through an underlying constrained Hidden Markov Model (HMM) is equivalent to using the maximum scoring subsequence (also called local score), when the latter uses an appropriate rescaled scoring function. This equivalence allows results from both HMM or local score to be transposed into each other. We propose an adaptation of the standard forward-backward algorithm which provides exact estimates of posterior probabilities in a linear time. Additionally it can provide posterior probabilities on the segment length and starting/ending indexes. We explain how this equivalence allows one to manage ambiguous or uncertain sequence letters and to construct relevant scoring functions. We illustrate our approach by considering the TM-tendency scoring function.

Suggested Citation

  • Sabine Mercier & Grégory Nuel, 2022. "Duality Between the Local Score of One Sequence and Constrained Hidden Markov Model," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1411-1438, September.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09856-8
    DOI: 10.1007/s11009-021-09856-8
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    References listed on IDEAS

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    1. Bo Zhao & Joseph Glaz, 2017. "Scan statistics for detecting a local change in variance for two-dimensional normal data," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(11), pages 5517-5530, June.
    2. Daudin, Jean-Jacques & Etienne, Marie Pierre & Vallois, Pierre, 2003. "Asymptotic behavior of the local score of independent and identically distributed random sequences," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 1-28, September.
    3. Wolfsheimer Stefan & Hartmann Alexander & Rabus Ralf & Nuel Gregory, 2012. "Computing Posterior Probabilities for Score-based Alignments Using ppALIGN," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(4), pages 1-37, May.
    4. Lagnoux, Agnès & Mercier, Sabine & Vallois, Pierre, 2019. "Probability density function of the local score position," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3664-3689.
    5. Chabriac, Claudie & Lagnoux, Agnès & Mercier, Sabine & Vallois, Pierre, 2014. "Elements related to the largest complete excursion of a reflected BM stopped at a fixed time. Application to local score," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4202-4223.
    6. Jie Chen & Joseph Glaz, 2016. "Scan statistics for monitoring data modeled by a negative binomial distribution," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(6), pages 1632-1642, March.
    7. Claudie Hassenforder & Sabine Mercier, 2007. "Exact Distribution of the Local Score for Markovian Sequences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(4), pages 741-755, December.
    8. Arribas-Gil Ana & Matias Catherine, 2012. "A Context Dependent Pair Hidden Markov Model for Statistical Alignment," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(1), pages 1-29, January.
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