Abstract
One of the most used variants of hidden Markov models (HMMs) is the standard case where the time is discrete and the state spaces (hidden and observed spaces) are finite.
In this framework, we are interested in HMMs whose emission process results from a combination of independent Markov chains. Principally, we assume that the emission process evolves as follows:
given a hidden state realization k at time t, an emission is a realization of a Markov chain
Funding statement: Supported by LIBMA Laboratory at Cadi Ayyad University, Hassan II Acad. Sc. Tech., and CNRST – Morocco, PBER, Bourse No. 9UCA2015.
References
[1] L. E. Baum, An inequality and associated maximization technique in statistical estimation for probabilistic functions of Markov processes, Inequalities III: Proceedings of the Third Symposium on Inequalities (Los Angeles 1969), Academic Press, New York (1972), 1–8. Search in Google Scholar
[2] L. E. Baum and T. Petrie, Statistical inference for probabilistic functions of finite state Markov chains, Ann. Math. Statist. 37 (1966), 1554–1563. 10.1214/aoms/1177699147Search in Google Scholar
[3] L. E. Baum, T. Petrie, G. Soules and N. Weiss, A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains, Ann. Math. Statist. 41 (1970), 164–171. 10.1214/aoms/1177697196Search in Google Scholar
[4] R. J. Boys and D. A. Henderson, A Bayesian approach to DNA sequence segmentation, Biometrics 60 (2004), no. 3, 573–588. 10.1111/j.0006-341X.2004.00206.xSearch in Google Scholar
[5] R. J. Boys, D. A. Henderson and D. J. Wilkinson, Detecting homogeneous segments in DNA sequences by using hidden Markov models, J. Roy. Statist. Soc. Ser. C 49 (2000), no. 2, 269–285. 10.1111/1467-9876.00191Search in Google Scholar
[6] J. V. Braun and H.-G. Muller, Statistical methods for DNA sequence segmentation, Statist. Sci. 13 (1998), no. 2, 142–162. 10.1214/ss/1028905933Search in Google Scholar
[7] H. Bunke and T. Caelli, Hidden Markov Models: Applications in Computer Vision, World Scientific, Hackensack, 2001. 10.1142/4648Search in Google Scholar
[8] O. Cappé, E. Moulines and T. Rydén, Inference in Hidden Markov Models, Springer Ser. Statist., Springer, New York, 2005. 10.1007/0-387-28982-8Search in Google Scholar
[9] G. A. Churchill, Stochastic models for heterogeneous DNA sequences, Bull. Math. Biol. 51 (1989), no. 1, 79–94. 10.1007/BF02458837Search in Google Scholar
[10] G. A. Churchill, Hidden Markov chains and the analysis of genome structure, Comput. Chem. 16 (1992), no. 2, 107–115. 10.1016/0097-8485(92)80037-ZSearch in Google Scholar
[11] R. E. S. Durbin, A. Krogh and G. Mitchison, Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids, Cambridge University Press, Cambridge, 1998. 10.1017/CBO9780511790492Search in Google Scholar
[12] C. Eng, Développement de méthodes de fouille de données fondées sur les modèles de Markov cachés du second ordre pour l’identification d’hétérogénéité dans les génomes bactériens, Ph.D. thesis, Université Henri Poincaré Nancy 1, Nancy, 2010. Search in Google Scholar
[13] G. D. Forney, Jr., The Viterbi algorithm, Proc. IEEE 61 (1973), 268–278. 10.1109/PROC.1973.9030Search in Google Scholar
[14] N. Friedman, K. Murphy and S. Russell, Learning the structure of dynamic probabilistic networks, Proceedings Fourteenth Conference on Uncertainty Artificial Intelligence, Morgan Kaufmann Publishers, San Francisco (1998), 139–147. Search in Google Scholar
[15] M. Gales and S. Young, The application of hidden Markov models in speech recognition, Found. Trends Signal Process. 1 (2007), no. 3, 195–304. 10.1561/2000000004Search in Google Scholar
[16] Z. Ghahramani, Learning dynamic Bayesian networks, Adaptive Processing of Sequences and Data Structures, Springer, Berlin (1998), 168–197. 10.1007/BFb0053999Search in Google Scholar
[17] Z. Ghahramani, An introduction to hidden Markov models and Bayesian networks, Int. J. Pattern Recognit. Artif. Intell. 15 (2001), no. 1, 9–42. 10.1142/S0218001401000836Search in Google Scholar
[18] J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (1989), no. 2, 357–384. 10.2307/1912559Search in Google Scholar
[19] F. Jelinek, Statistical Methods for Speech Recognition, MIT Press, Cambridge, 1997. Search in Google Scholar
[20] T. Koski, Hidden Markov Models for Bioinformatics, Comput. Biol. Ser. 2, Kluwer Academic Publishers, Dordrecht, 2001. 10.1007/978-94-010-0612-5Search in Google Scholar
[21] K. P. Murphy, Dynamic Bayesian Networks: Representation, Inference and Learning, ProQuest LLC, Ann Arbor, 2002; Ph.D. thesis, University of California, Berkeley. Search in Google Scholar
[22] T. A. Patterson, A. Parton, R. Langrock, P. G. Blackwell, L. Thomas and R. King, Statistical modelling of individual animal movement: An overview of key methods and a discussion of practical challenges, AStA Adv. Stat. Anal. 101 (2017), no. 4, 399–438. 10.1007/s10182-017-0302-7Search in Google Scholar
[23] L. R. Rabiner, A tutorial on hidden Markov models and selected applications in speech recognation, Proc. IEEE 77 (1989), no. 2, 257–286. 10.1109/5.18626Search in Google Scholar
[24] L. R. Rabiner and B.-H. Juang, Fundamentals of Speech Recognition, Prentice-Hall, Upper Saddle River, 1993. Search in Google Scholar
[25] S. Russell and P. Norvig, Artificial Intelligence: A Modern Approach, Prentice Hall, Upper Saddle River, 2002. Search in Google Scholar
[26] A. Viterbi, Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IEEE Trans. Inform. Theory 13 (1967), no. 2, 260–269. 10.1109/TIT.1967.1054010Search in Google Scholar
[27] W. Zucchini, I. L. MacDonald and R. Langrock, Hidden Markov Models for Time Series, 2nd ed., Monogr. Statist. Appl. Probab. 150, CRC Press, Boca Raton, 2016. 10.1201/b20790Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston