Nonparametric identification and maximum likelihood estimation for hidden Markov models

Alexandrovich, Grigory; Holzmann, Hajo; Leister, Anna

doi:10.1093/biomet/asw001

Cited by 32 publications

(44 citation statements)

References 21 publications

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“…He focused on the special case of two states, with one of the two state‐dependent distributions modeled parametrically, arguing that this type of model is most relevant for applications, and that computational and identifiability issues may arise in more difficult scenarios. However, it has recently been shown by Gassiat, Cleynen, and Robin () and Alexandrovich and Holzmann () that identifiability in nonparametric HMMs holds under fairly weak conditions, which in practice will usually be satisfied, namely that the transition probability matrix of the unobserved Markov chain has full rank and that the state‐dependent distributions are distinct.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric Inference in Hidden Markov Models Using P-Splines

et al. 2015

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Hidden Markov models (HMMs) are flexible time series models in which the distributions of the observations depend on unobserved serially correlated states. The statedependent distributions in HMMs are usually taken from some class of parametrically specified distributions. The choice of this class can be difficult, and an unfortunate choice can have serious consequences for example on state estimates, on forecasts and generally on the resulting model complexity and interpretation, in particular with respect to the number of states. We develop a novel approach for estimating the statedependent distributions of an HMM in a nonparametric way, which is based on the idea of representing the corresponding densities as linear combinations of a large number of standardized B-spline basis functions, imposing a penalty term on non-smoothness in order to maintain a good balance between goodness-of-fit and smoothness. We illustrate the nonparametric modeling approach in a real data application concerned with vertical speeds of a diving beaked whale, demonstrating that compared to parametric counterparts it can lead to models that are more parsimonious in terms of the number of states yet fit the data equally well.

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Section: Introductionmentioning

confidence: 99%

Nonparametric Inference in Hidden Markov Models Using P-Splines

et al. 2015

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“…The constrained parameters are then obtained by applying the transformation (4). Regarding identifiability in general, and specifically in case of the very flexible model formulation considered here, Alexandrovich et al (2016) show that for the HMM to be identifiable it is sufficient if the t.p.m. has full rank and the state-dependent distributions are distinct, conditions that can be expected to be met in most practical settings where HMMs seem natural candidate models.…”

Section: Model Fittingmentioning

confidence: 99%

Penalized estimation of flexible hidden Markov models for time series of counts

Adam

Langrock

Weiß

2019

METRON

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Hidden Markov models are versatile tools for modeling sequential observations, where it is assumed that a hidden state process selects which of finitely many distributions generates any given observation. Specifically for time series of counts, the Poisson family often provides a natural choice for the state-dependent distributions, though more flexible distributions such as the negative binomial or distributions with a bounded range can also be used. However, in practice, choosing an adequate class of (parametric) distributions is often anything but straightforward, and an inadequate choice can have severe negative consequences on the model's predictive performance, on state classification, and generally on inference related to the system considered. To address this issue, we propose an effectively nonparametric approach to fitting hidden Markov models to time series of counts, where the state-dependent distributions are estimated in a completely data-driven way without the need to select a distributional family. To avoid overfitting, we add a roughness penalty based on higher-order differences between adjacent count probabilities to the likelihood, which is demonstrated to produce smooth probability mass functions of the statedependent distributions. The feasibility of the suggested approach is assessed in * Corresponding author; email: timo.adam@uni-bielefeld.de. a simulation experiment, and illustrated in two real-data applications, where we model the distribution of i) major earthquake counts and ii) acceleration counts of an oceanic whitetip shark (Carcharhinus longimanus) over time.

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“…In , the authors prove that the transition matrix and the emission distributions of a stationary HMM are identifiable (up to state labelling) from the law of three consecutive observations, provided that the transition matrix has full rank and that the emission distributions are linearly independent. Still in the context of HMM, the authors of Alexandrovich et al (2016) use the weaker assumption that the emission distributions are all distinct to obtain identifiability up to state labelling. However, it requires to consider the law of more than three consecutive observations.…”

Section: Identifiabilitymentioning

confidence: 99%

“…In , it is proved that the transition matrix and the emission distributions of a hidden Markov model are identifiable from the joint distribution of three consecutive observations, provided that the transition matrix is non-singular and that the emission distributions are linearly independent. Alexandrovich et al (2016) proved that the identifiability can be obtained with the weaker assumption that the emission distributions are distinct. In this paper, we extend the result of by proving that the SHMM are identifiable up to state labelling, under similar assumptions.…”

Section: Introductionmentioning

confidence: 99%

Consistency of the maximum likelihood estimator in seasonal hidden Markov models

Touron

2019

Stat Comput

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In this paper, we introduce a variant of hidden Markov models in which the transition probabilities between the states, as well as the emission distributions, are not constant in time but vary in a periodic manner. This class of models, that we will call seasonal hidden Markov models (SHMM) is particularly useful in practice, as many applications involve a seasonal behaviour. However, up to now, there is no theoretical result regarding this kind of model. We show that under mild assumptions, SHMM are identifiable: we can identify the transition matrices and the emission distributions from the joint distribution of the observations on a period, up to state labelling. We also give sufficient conditions for the strong consistency of the maximum likelihood estimator (MLE). These results are applied to simulated data, using the EM algorithm to compute the MLE. Finally, we show how SHMM can be used in real world applications by applying our model to precipitation data, with mixtures of exponential distributions as emission distributions.

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Nonparametric identification and maximum likelihood estimation for hidden Markov models

Cited by 32 publications

References 21 publications

Nonparametric Inference in Hidden Markov Models Using P-Splines

Nonparametric Inference in Hidden Markov Models Using P-Splines

Penalized estimation of flexible hidden Markov models for time series of counts

Consistency of the maximum likelihood estimator in seasonal hidden Markov models

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