Deux modèles de Markov caché pour processus multiples et leur contribution à l’élaboration d’une notion de style postural
Journal de la société française de statistique, Tome 150 (2009) no. 1, pp. 73-100.

Nous décrivons dans cet article deux modèles de Markov caché (hmms) pour processus multiples que nous mettons au point à des fins d’application à un problème biomédical d’étude du maintien postural. Informellement, ces modèles sont des hmms standard considérant simultanément plusieurs processus individuels. Ceux-ci partagent les mêmes distributions de chaîne de Markov cachée et d’émission, avec l’ajout éventuel d’un effet de type mixte dans la partie conditionnelle du modèle. Nous discutons des aspects théoriques de l’estimation de l’ordre des deux modèles, ainsi que des aspects numériques. Nous contribuons enfin à l’élaboration d’une notion de “style postural”.

Two hidden Markov models (hmms) for multiple processes are introduced and applied to the biomedical study of how people maintain posture. Informally, they model simultaneously multiple processes by making them share the same hidden Markov and observed conditional distributions. One of them incorporates covariates and random effects in the conditional part. We discuss the theoretical issue of order estimation for both models as well as computational issues. Finally, we contribute to the definition of a notion of “postural style”.

Classification : 62M02, 62M05, 62P10
Mot clés : BIC, estimation de l’ordre, modèle de Markov caché (HMM), posture
Keywords: BIC, hidden Markov models (HMM), order estimation, posture 
@article{JSFS_2009__150_1_73_0,
     author = {Chambaz, Antoine and Bonan, Isabelle and Vidal, Pierre-Paul},
     title = {Deux mod\`eles de {Markov} cach\'e pour processus multiples et leur contribution \`a l{\textquoteright}\'elaboration d{\textquoteright}une notion de style postural},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {73--100},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {150},
     number = {1},
     year = {2009},
     mrnumber = {2609698},
     zbl = {1311.62126},
     language = {fr},
     url = {http://www.numdam.org/item/JSFS_2009__150_1_73_0/}
}
TY  - JOUR
AU  - Chambaz, Antoine
AU  - Bonan, Isabelle
AU  - Vidal, Pierre-Paul
TI  - Deux modèles de Markov caché pour processus multiples et leur contribution à l’élaboration d’une notion de style postural
JO  - Journal de la société française de statistique
PY  - 2009
SP  - 73
EP  - 100
VL  - 150
IS  - 1
PB  - Société française de statistique
UR  - http://www.numdam.org/item/JSFS_2009__150_1_73_0/
LA  - fr
ID  - JSFS_2009__150_1_73_0
ER  - 
%0 Journal Article
%A Chambaz, Antoine
%A Bonan, Isabelle
%A Vidal, Pierre-Paul
%T Deux modèles de Markov caché pour processus multiples et leur contribution à l’élaboration d’une notion de style postural
%J Journal de la société française de statistique
%D 2009
%P 73-100
%V 150
%N 1
%I Société française de statistique
%U http://www.numdam.org/item/JSFS_2009__150_1_73_0/
%G fr
%F JSFS_2009__150_1_73_0
Chambaz, Antoine; Bonan, Isabelle; Vidal, Pierre-Paul. Deux modèles de Markov caché pour processus multiples et leur contribution à l’élaboration d’une notion de style postural. Journal de la société française de statistique, Tome 150 (2009) no. 1, pp. 73-100. http://www.numdam.org/item/JSFS_2009__150_1_73_0/

[1] Altman, R. MacKay Mixed hidden Markov models : an extension of the hidden Markov model to the longitudinal data setting, J. Amer. Statist. Assoc., Volume 102 (2007) no. 477, pp. 201-210 | MR | Zbl

[2] Albert, P. S.; McFarland, H. F.; Smith, M. E.; A., Frank . Time series for modelling counts from a relapsed-remitting disease : application to modelling disease activity in multiple sclerosis, Statistics in Medicine (1994) no. 24, pp. 453-466

[3] Bardet, J-M.; Bertrand, P. Identification of the multiscale fractional Brownian motion with biomechanical applications, J. Time Ser. Anal., Volume 28 (2007) no. 1, pp. 1-52 | MR | Zbl

[4] Bertrand, P.; Bardet, J-M.; Dabonneville, M.; Mouzat, A.; Vaslin, P. Automatic determination of the different control mechanisms in upright position by a wavelet method, IEEE Engineering in Medicine and Biology Society, Volume 2 (2001), pp. 1163-1166

[5] Biernacki, C.; Celeux, G.; Govaert, G. Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood, IEEE Trans. Pattern Anal. Mach. Intell., Volume 22 (2000) no. 7, pp. 719-725 | DOI

[6] Chiari, L.; Cappello, A.; Lenzi, D.; Della Croce, U. An improved technique for the extraction of stochastic parameters from stabilograms, Gait Posture, Volume 12 (2000) no. 3, pp. 225-234

[7] Collins, J.-J.; De Lucas, C. J. Open-loop and closed-loop control of posture : a random-walk analysis of center-of-pressure trajectories, Exp. Brain Res. (1993)

[8] Chambaz, A.; Garivier, A.; Gassiat, E. A minimum description length approach to hidden Markov models with Poisson and Gaussian emissions. Application to order identification, J. Statist. Plann. Inference (2008), pp. 1-16 | DOI | MR | Zbl

[9] Cappé, O.; Moulines, E.; Rydén, T. Inference in hidden Markov models, Springer Series in Statistics, Springer, New York, 2005, xviii+652 pages | MR | Zbl

[10] Davisson, Lee D.; McEliece, Robert J.; Pursley, Michael B.; Wallace, Mark S. Efficient universal noiseless source codes, IEEE Trans. Inform. Theory, Volume 27 (1981) no. 3, pp. 269-279 | MR | Zbl

[11] Frank, T. D.; Daffertshofer, A.; Beek, P. J. Multivariate Ornstein-Uhlenbeck processes with mean-field dependent coefficients : Application to postural sway, Phys. Rev. E, Volume 63 (2000) no. 1 | DOI

[12] Fransson, P. A.; Hafström, A.; Karlberg, M.; Magnusson, M.; Tjäder, A.; Johansson, R. Postural control adaptation during galvanic vestibular and vibratory proprioceptive stimulation, IEEE Trans. Biomed. Eng., Volume 50 (2003) no. 12, pp. 1310-1319

[13] Finesso, L. Consistent estimation of the order for Markov and hidden Markov chains, PhD Thesis, University of Maryland (1991) | MR

[14] Fransson, P. A.; Johansson, R.; Hafström, A.; Magnusson, M. Methods for evaluation of postural control adaptation, Gait Posture, Volume 12 (2000) no. 1, pp. 14-24

[15] Fransson, P. A.; Kristinsdottir, E. K.; Hafström, A.; Magnusson, M.; Johansson, R. Balance control and adaptation during vibratory perturbations in middle-aged and elderly humans, Eur. J. Appl. Physiol., Volume 91 (2004) no. 5-6, pp. 595-603

[16] Gassiat, E. Likelihood ratio inequalities with applications to various mixtures, Ann. Inst. H. Poincaré Probab. Statist., Volume 38 (2002) no. 6, pp. 897-906 (En l’honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov) | Numdam | MR | Zbl

[17] Koski, T. Hidden Markov models for bioinformatics, Computational Biology Series, 2, Kluwer Academic Publishers, Dordrecht, 2001, xviii+391 pages | MR | Zbl

[18] Laurens, J.; Droulez, J. Bayesian processing of vestibular information, Biol. Cybern., Volume 96 (2007) no. 4, pp. 389-404 | DOI | MR | Zbl

[19] Lepecq, J. C.; De Waele, C.; Mertz-Josse, S.; Teyssèdre, C.; Huy, P. T.; Baudonnière, P. M.; Vidal, P-P. Galvanic vestibular stimulation modifies vection paths in healthy subjects, J. Neurophysiol., Volume 95 (2006) no. 5, pp. 3199-3207

[20] Leroux, B. G. Maximum-likelihood estimation for hidden Markov models, Stochastic Process. Appl., Volume 40 (1992) no. 1, pp. 127-143 | MR | Zbl

[21] Liu, C. C.; Narayan, P. Order estimation and sequential universal data compression of a hidden Markov source by the method of mixtures, Canad. J. Statist., Volume 30 (1994) no. 4, pp. 573-589 | Zbl

[22] Levison, S. E.; Rabiner, L. R.; Sondhi, M. M. An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition, Bell System Tech. J., Volume 62 (1983), pp. 1035-1074 | MR | Zbl

[23] Mergner, T.; Schweigart, G.; Maurer, C.; Blümle, A. Human postural responses to motion of real and virtual visual environments under different support base conditions, Exp. Brain Res., Volume 167 (2005) no. 4, pp. 535-556

[24] Nash, J. C. Compact numerical methods for computers : linear algebra and function minimisation, John Wiley & Sons Inc., New York, 1979, ix+227 pages (A Halsted Press Book) | MR | Zbl

[25] Newell, K. M.; Slobounov, S. M.; Slobounova, E. S.; Molenaar, P. C. Stochastic processes in postural center-of-pressure profiles, Exp. Brain Res., Volume 113 (1997) no. 1, pp. 158-164

[26] Olshen, R. A.; Biden, E. N.; Wyatt, Marilynn P.; Sutherland, D. H. Gait analysis and the bootstrap, Ann. Statist., Volume 17 (1989) no. 4, pp. 1419-1440 | MR | Zbl

[27] R Development Core Team R : A Language and Environment for Statistical Computing (2008) http://www.R-project.org (ISBN 3-900051-07-0)

[28] Sabatini, A.M. A statistical mechanical analysis of postural sway using non-Gaussian FARIMA stochastic models, IEEE Trans. Biomed. Eng., Volume 47 (2000) no. 9, pp. 1219-1227