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Hidden Markov Models with mixtures as emission distributions

Abstract

In unsupervised classification, Hidden Markov Models (HMM) are used to account for a neighborhood structure between observations. The emission distributions are often supposed to belong to some parametric family. In this paper, a semiparametric model where the emission distributions are a mixture of parametric distributions is proposed to get a higher flexibility. We show that the standard EM algorithm can be adapted to infer the model parameters. For the initialization step, starting from a large number of components, a hierarchical method to combine them into the hidden states is proposed. Three likelihood-based criteria to select the components to be combined are discussed. To estimate the number of hidden states, BIC-like criteria are derived. A simulation study is carried out both to determine the best combination between the combining criteria and the model selection criteria and to evaluate the accuracy of classification. The proposed method is also illustrated using a biological dataset from the model plant Arabidopsis thaliana. A R package HMMmix is freely available on the CRAN.

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Correspondence to Stevenn Volant.

Appendix

Appendix

A.1 Algorithm

We present in this appendix the algorithm we proposed for combining components of an HMM. This algorithm has been written with respect to the results we obtained in Sect. 4.1.2. However, this algorithm can easily be written for other criteria.

  1. 1.

    Fit an HMM with K components.

  2. 2.

    From G=K,K−1,…,1

    • Select the clusters k and l to be combined as:

    • Update the parameters with a few steps of the EM algorithm to get closer to a local optimum.

  3. 3.

    Selection of the number of groups \(\widehat{D}\):

A.2 Mean and variance of the Gaussian distributions for the simulation study (Sect. 4.1)

and,

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Volant, S., Bérard, C., Martin-Magniette, ML. et al. Hidden Markov Models with mixtures as emission distributions. Stat Comput 24, 493–504 (2014). https://doi.org/10.1007/s11222-013-9383-7

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  • DOI: https://doi.org/10.1007/s11222-013-9383-7

Keywords

  • Hidden Markov models
  • Model-based clustering
  • Mixture model
  • Hierarchical algorithm