Some applications of nonlinear and non-Gaussian state–space modelling by means of hidden Markov models

Abstract

Nonlinear and non-Gaussian state–space models (SSMs) are fitted to different types of time series. The applications include homogeneous and seasonal time series, in particular earthquake counts, polio counts, rainfall occurrence data, glacial varve data and daily returns on a share. The considered SSMs comprise Poisson, Bernoulli, gamma and Student-t distributions at the observation level. Parameter estimations for the SSMs are carried out using a likelihood approximation that is obtained after discretization of the state space. The approximation can be made arbitrarily accurate, and the approximated likelihood is precisely that of a finite-state hidden Markov model (HMM). The proposed method enables us to apply standard HMM techniques. It is easy to implement and can be extended to all kinds of SSMs in a straightforward manner.

Keywords:

• time series
• numerical integration
• count data
• binary data
• stochastic volatility
• pseudo-residuals
• Viterbi algorithm

Acknowledgements

The author would like to thank Walter Zucchini, Iain L. MacDonald and two anonymous referees for helpful comments.