Statistics and Computing

, Volume 26, Issue 1–2, pp 61–71 | Cite as

Inference in finite state space non parametric Hidden Markov Models and applications

  • E. Gassiat
  • A. Cleynen
  • S. Robin


Hidden Markov models (HMMs) are intensively used in various fields to model and classify data observed along a line (e.g. time). The fit of such models strongly relies on the choice of emission distributions that are most often chosen among some parametric family. In this paper, we prove that finite state space non parametric HMMs are identifiable as soon as the transition matrix of the latent Markov chain has full rank and the emission probability distributions are linearly independent. This general result allows the use of semi- or non-parametric emission distributions. Based on this result we present a series of classification problems that can be tackled out of the strict parametric framework. We derive the corresponding inference algorithms. We also illustrate their use on few biological examples, showing that they may improve the classification performances.


Identifiability Hidden Markov Models Non-parametric 



The authors want to thank Caroline Bérard for providing the transcriptomic tiling array data. Part of this work was supported by the ABS4NGS ANR project (ANR-11-BINF-0001-06).


  1. Allman, E.S., Matias, C., Rhodes, J.: Identifiability of parameters in latent structure models with many observed variables. Ann. Stat. 37, 3099–3132 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  2. An, Y., Hu, Y., Hopkins, J., Shum, M.: Identifiability and inference of hidden Markov models. Technical report (2013)Google Scholar
  3. Baudry, J.-P., Raftery, A.E., Celeux, G., Lo, K., Gottardo, R.: Combining mixture components for clustering. J. Comput. Gr. Stat. 19(2), 332–353 (2010)MathSciNetCrossRefGoogle Scholar
  4. Benaglia, T., Chauveau, D., Hunter, D.R.: An EM-like algorithm for semi-and nonparametric estimation in multivariate mixtures. J. Comput. Gr. Stat. 18(2), 505–526 (2009)MathSciNetCrossRefGoogle Scholar
  5. Bérard, C., Martin-Magniette, M.L., Brunaud, V., Aubourg, S., Robin, S.: Unsupervised classification for tiling arrays: ChIP-chip and transcriptome. Stat. Appl. Genet. Mol. Biol. 10(1), 1–22 (2011)MathSciNetGoogle Scholar
  6. Bordes, L., Mottelet, S., Vandekerkhove, P.: Semiparametric estimation of a two components mixture model. Ann. Stat. 34, 1204–1232 (2006a)zbMATHMathSciNetCrossRefGoogle Scholar
  7. Bordes, L., Delmas, C., Vandekerkhove, P.: Semiparametric estimation of a two-component mixture model where one component is known. Scand. J. Stat. 33(4), 733–752 (2006b)zbMATHMathSciNetCrossRefGoogle Scholar
  8. Butucea, C., Vandekerkhove, P.: Semiparametric mixtures of symmetric distributions. Scand. J. Stat. 41(1), 227–239 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  9. Cappé, O., Moulines, E., Rydén, T.: Inference Hidden Markov Models. Springer, New York (2005)zbMATHGoogle Scholar
  10. Couvreur, L., Couvreur, C.: Wavelet based non-parametric HMMs: theory and methods. In: ICASSP ’00 Proceedings, pp. 604–607 (2000)Google Scholar
  11. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39, 1–38 (1977)zbMATHMathSciNetGoogle Scholar
  12. DeSantis, S.M., Bandyopadhyay, D.: Hidden Markov models for zero-inflated Poisson counts with an application to substance use. Stat. Med. 30(14), 1678–1694 (2011)MathSciNetCrossRefGoogle Scholar
  13. Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D.: Density estimation by wavelet thresholding. Ann. Stat. 24(2), 508–539 (1996)Google Scholar
  14. Du, J., Rozowsky, J.S., Korbel, J.O., Zhang, Z.D., Royce, T.E., Schultz, M.H., Snyder, M., Gerstein, M.: A supervised hidden Markov model framework for efficiently segmenting tiling array data in transcriptional and chIP-chip experiments: systematically incorporating validated biological knowledge. Bioinformatics 22(24), 3016–3024 (2006)CrossRefGoogle Scholar
  15. Dumont, T., Le Corff, S.: Nonparametric regression on hidden phi-mixing variables: identifiability and consistency of a pseudo-likelihood based estimation procedure. Technical report, arXiv:1209.0633D (Sep., 2012)
  16. Durot, C., Huet, S., Koladjo, F., Robin, S.: Least-squares estimation of a convex discrete distribution. Comput. Stat. Data Anal. 67, 282–298 (2013)Google Scholar
  17. Gassiat, E., Rousseau, J.: Non parametric finite translation hidden Markov models and extensions. Bernoulli. to appear (2014)Google Scholar
  18. Hall, P., Zhou, X.-H.: Nonparametric estimation of component distributions in a multivariate mixture. Ann. Stat. 31(1), 201–224 (2003)Google Scholar
  19. Hsu, D., Kakade, S.M., Zhang, T.: A spectral algorithm for learning hidden Markov models. J. Comput. Syst. Sci. 78, 1460–1480 (2012)Google Scholar
  20. Jin, N., Mokhtarian, F.: A non-parametric HMM learning method for shape dynamics with application to human motion recognition. In: 18th International Conference on Pattern Recognition, 2006. ICPR 2006, vol. 2, pp. 29–32. IEEE (2006)Google Scholar
  21. Lambert, M., Whiting, J., Metcalfe, A.: A non-parametric hidden Markov model for climate state identification. Hydrol. Earth Syst. Sci. 7(5), 652–667 (2003)CrossRefGoogle Scholar
  22. Lefèvre, F.: Non-parametric probability estimation for HMM-based automatic speech recognition. Comput. Speach Lang. 17, 113–136 (2003)CrossRefGoogle Scholar
  23. Levine, M., Hunter, D.R., Chauveau, D.: Maximum smoothed likelihood for multivariate mixtures. Biometrika. 98(2), 403–416 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  24. Li, H., Zhang, K., Jiang, T.: The regularized EM algorithm. In: Proceedings of the National Conference on Artificial Intelligence, vol. 20, p. 807. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999 (2005)Google Scholar
  25. Lin, T.I., Lee, J.C., Yen, S.Y.: Finite mixture modelling using the skew normal distribution. Stat. Sin. 17(3), 909–927 (2007)zbMATHMathSciNetGoogle Scholar
  26. Massart, P.: Concentration inequalities and model selection. Volume 1896 of Lecture Notes in Mathematics. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003, With a foreword by Jean Picard. Springer, Berlin (2007)Google Scholar
  27. Maugis, C., Michel, B.: A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM Probab. Stat. 15, 41–68 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  28. Olteanu, M., Ridgway, J., et al.: Hidden Markov models for time series of counts with excess zeros. Proc. ESANN 2012, 133–138 (2012)Google Scholar
  29. Petrie, T.: Probabilistic functions of finite state Markov chains. Ann. Math. Stat. 40, 97–115 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  30. Shang, L., Chan, K.: Nonparametric discriminant HMM and application to facial expression recognition. In: 2009 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pp. 2090–2096 (2009)Google Scholar
  31. Titterington, D.M., Smith, A.F.M., Makov, U.E.: Statistical Analysis of Finite Mixture Distributions. Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Wiley, Chichester (1985)Google Scholar
  32. Tune, P., Nguyen, H. X., Roughan, M.: Hidden Markov model identifiability via tensors. In: 2013 IEEE International Symposium on Information Theory Proceedings (ISIT), pp. 2299–2303. IEEE (2013)Google Scholar
  33. van de Geer, S.A.: Empirical processes in M-estimation. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2000)Google Scholar
  34. Vernet, E.: Posterior consistency for nonparametric Hidden Markov Models with finite state space. Technical report, arXiv:1311.3092V (2013)
  35. Volant, S., Bérard, C., Martin-Magniette, M.-L., Robin, S.: Hidden Markov models with mixtures as emission distributions. Stat. Comput. 1–12 (2013). doi: 10.1007/s11222-013-9383-7
  36. Yakowitz, S.J., Spragins, J.D.: On the identifiability of finite mixtures. Ann. Math. Stat. 39, 209–214 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  37. Zhai, Z., Ku, S.-Y., Luan, Y., Reinert, G., Waterman, M.S., Sun, F.: The power of detecting enriched patterns: an HMM approach. J. Comput. Biol. 17(4), 581–592 (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Laboratoire de MathématiqueUniversité Paris-SudOrsayFrance
  2. 2.Laboratoire de MathématiqueCNRSOrsayFrance
  3. 3.AgroParisTech, MIA 518ParisFrance
  4. 4.INRA, MIA 518ParisFrance

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