Bayesian nonparametric dynamic state space modeling with circular latent states

ABSTRACT

State space models are well-known for their versatility in modeling dynamic systems that arise in various scientific disciplines. Although parametric state space models are well studied, nonparametric approaches are much less explored in comparison. In this article we propose a novel Bayesian nonparametric approach to state space modeling, assuming that both the observational and evolutionary functions are unknown and are varying with time; crucially, we assume that the unknown evolutionary equation describes dynamic evolution of some latent circular random variable. Based on appropriate kernel convolution of the standard Weiner process, we model the time-varying observational and evolutionary functions as suitable Gaussian processes that take both linear and circular variables as arguments. Additionally, for the time-varying evolutionary function, we wrap the Gaussian process thus constructed around the unit circle to form an appropriate circular Gaussian process. We show that our process thus created satisfies desirable properties.

For the purpose of inference we develop a Markov-chain Monte Carlo (MCMC)-based methodology combining Gibbs sampling and Metropolis–Hastings algorithms. Applications to a simulated data set, a real wind speed data set, and a real ozone data set demonstrated quite encouraging performances of our model and methodologies.

keywords:

• Circular random variable
• kernel convolution
• Markov-chain Monte Carlo
• state space model
• Weiner process
• wrapped Gaussian process

AMS Subject Classification:

• 62M10
• 60G15

Acknowledgments

The authors are thankful to Moumita Das for very useful discussions. The authors are also thankful to the anonymous referees and an associate editor for their valuable comments and suggestions, which helped improve our article significantly.