# Consistency of the maximum likelihood estimator in seasonal hidden Markov models

arXiv: https://arxiv.org/abs/1802.08161 · HTML: https://mlsanity.com/paper/1802.08161/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2018.

## Built on by (most cited first, 2 of 2)

- [HERMES: Hybrid Error-corrector Model with inclusion of External Signals for nonstationary fashion time series](https://mlsanity.com/paper/2202.03224/) (2022)
- [Variational quantization for state space models](https://mlsanity.com/paper/2404.11117/) (2024)

## Abstract (arXiv)

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.