Abstract
Hidden Markov models are a class of probabilistic graphical models used to describe the evolution of a sequence of unknown variables from a set of observed variables. They are statistical models introduced by Baum and Petrie in Baum (JMA 101:789–810) and belong to the class of latent variable models. Initially developed and applied in the context of speech recognition, they have attracted much attention in many fields of application. The central objective of this research work is upon an extension of these models. More accurately, we define multiparameter hidden Markov models, using multiple observation processes and the Riesz distribution on the space of symmetric matrices as a natural extension of the gamma one. Some basic related properties are discussed and marginal and posterior distributions are derived. We conduct the Forward-Backward dynamic programming algorithm and the classical Expectation Maximization algorithm to estimate the global set of parameters. Using simulated data, the performance of these estimators is conveniently achieved by the Matlab program. This allows us to assess the quality of the proposed estimators by means of the mean square errors between the true and the estimated values.
Similar content being viewed by others
References
Andersson SA, Klein T (2010) On Riesz and Wishart distributions associated with decomposable undirected graphs. J Multivar Anal 101(4):789–810
Baum LE, Petrie T (1966) Statistical inference for probabilistic functions of finite state Markov chains. Ann math Stat 37(6):1554–1563
Brunel N, Lapuyade-Lahorgue J, Pieczynski W (2010) Modeling and unsupervised classification of multivariate hidden Markov chains with copulas. IEEE Trans on Automatic Control 55(2):338–349
Cappé O, Moulines E, Rydén T (2005) Inference in hidden Markov models. Springer-VerlagBerlin, Heidelberg
Cho W, Na MH, Kim S (2016a) Human action recognition using variational Bayesian HMM with Dirichlet process mixture of Gaussian Wishart emission model. World Acad Sci Eng Technol Int J Comput Elect Autom Control Information Eng 10:2025–2031
Cho W, Kim S, Park S (2016b) Human action recognition using variational Bayesian hidden Markov model with Gaussian-Wishart emission mixture model. In: international conference on machine learning and cybernetics (ICMLC) 201–206
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Statist 39(1):1–38
Devijver PA, Dekesel M (1988) Champs aléatoires de Pickard et modélisation d’images digitales. Trait Signal 5(5):131–150
Díaz-García JA (2014) On Riesz distribution. Metrika 77:469–481
Douc R, Fort G, Moulines E, Priouret P (2009) Forgetting of the initial distribution for hidden Markov models. Stoch Process Their Appl 119(4):1235–1256
Dunn JC (1973) A Fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. J Cybern 3:32–57
El-Emary IMM, Fezari M, Atoui H (2011) Hidden Markov model/Gaussian mixture models (HMM/GMM) based voice command system: A way to improve the control of remotely operated robot arm TR45. Scientific Research and Essays 6(2):341–350
Fan W, Yang L, Bouguila N, Chen Y (2020) Sequentially spherical data modeling with hidden Markov models and its application to fMRI data analysis. Knowl Based Syst 206:106341
Faraut J, Korányi F (1994) Analysis on symmetric cones. Oxford University Press
Ghahramani Z (2001) An Introduction to hidden Markov models and Bayesian networks. Int J Pattern Recognit Artif Intell 15:9–42
Ghorbel E, Louati M (2019) The multiparameter t’distribution. Filomat 33(13):4137–4150
Ghorbel E, Kammoun K, Louati M (2020) Bayesian estimation of the precision matrix with monotone missing data. Lith Math J 60(4):470–481
Ghorbel E, Kammoun K, Louati M, Sallem A (2022) Estimation of the parameters of a Wishart extension on symmetric matrices. J Korean Statist Soc 51:1071–1089
Gindikin SG (1964) Analysis in homogeneous domains. Russian Math Surveys 19(4):1–89
Haff LR (1982) Identities for the inverse Wishart distribution with computational results in linear and quadratic discrimination. Sankhyā Ser B 44(3):245–258
Hassairi A, Louati M (2009) Multivariate stable exponential families and Tweedie scale. J Stat Plan Inference 139:143–158
Hidot S, Saint-Jean C (2010) An Expectation-Maximization algorithm for the Wishart mixture model: Application to movement clustering. Patt Recognit Lett 31:2318–2324
Higgins C, Vidaurre D, Kolling N, Liu Y, Behrens T, Woolrich M (2022) Spatiotemporally resolved multivariate pattern analysis for M/EEG. Hum Brain Mapp 43(10): 3062–3085
Ishi H (2000) Positive Riesz distributions on homogeneous cones. J Math Soc Japan 52(1), 161–186
Kaleh GK, Vallet R (1994) Joint parameter estimation and symbol detection for linear or nonlinear uknown channels. IEEE Trans Commun 42(7), 406–413
Kammoun K, Louati M, Masmoudi A (2017) Maximum likelihood estimator of the scale parameter for the Riesz distribution. Stat Probabil Lett 126:127–131
Li J, Lee JY, Liao L (2021) A new algorithm to train hidden Markov models for biological sequences with partial labels. BMC Bioinform 22:162
Lin P (1972) Some characterizations of the multivariate t distribution. J Multivar Anal 2:339–344
Louati M (2013) Mixture of the Riesz distribution with respect to the generalized multivariate gamma distribution. J Korean Statist Soc 42:83–93
Louati M, Masmoudi A (2015) Moment for the inverse Riesz distributions. Stat Probabil lett 102:30–37
Mesa A, Basterrech A, Guerberoff G, Alveraz-Valin F (2015) Hidden Markov models for gene sequence classification. Pattern Anal Appl 19:793–805
Osatohanmwen P, Omotayo-Tomo MS, Oyegue FO, Mazona V, Ewere F, Bilesanmi A, Osawe NL, (2023) A note on hidden Markov models with application to criminal intelligence. J Appl Sci Environ Manag 27(2):277–282
Petropoulos A, Chatzis SP, Xanthopoulos S (2016) A novel corporate credit rating system based on Student’s-t hidden Markov models. Expert Syst Appl 53:87–105
Qian W, Titterington DM (1989) On the use of Gibbs Markov chain models in the analysis of images based on second-order pairwise interactive distributions. J Appl Stat 6(2), 267–282
Rösler M (2020) Riesz distributions and Laplace transform in the Dunkl setting of type A. J Funct Anal 278(12):108506
Stigler J, Ziegler F, Gieseke A, Gebhardt JCM, Rief M (2011) The complex folding network of single calmodulin molecule. Science 334(6055): 512–516
Suleiman D, Awajan A, Al Etaiwi W (2017) The use of hidden Markov model in natural ARABIC language processing: a survey. Procedia Comput Sci 113:240–247
Tian F, Zhou Q, Yang C (2020) Gaussian mixture model-hidden Markov model based nonlinear equalizer for optical fiber transmission. Opt Express 28:9728–9737
Veleva E (2009) Testing a Normal covariance matrix for small samples with monotone missing data. Appl math sci 3(54):2671–2679
Viterbi A (1967) Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans Inform Theory 13:260–269
von Rosen D (1988) Moments for the inverted Wishart distribution. Scand J Stat 15:97–109
Xuan G, Wei Zhang W, Chai P (2001) EM algorithms of Gaussian mixture model and hidden Markov model. Int Conf Image Process 1:145–148
Zhang H, Zhang W, Palazoglu A, Sun W (2012) Prediction of ozone levels using a hidden Markov model (HMM) with gamma distribution. Atmosph Environ 62:64–73
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ghorbel, E., Louati, M. An expectation maximization algorithm for the hidden markov models with multiparameter student-t observations. Comput Stat (2023). https://doi.org/10.1007/s00180-023-01432-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00180-023-01432-7