Journal of Statistical Theory and Practice

, Volume 11, Issue 4, pp 693–718 | Cite as

Nonparametric dynamic state space modeling of observed circular time series with circular latent states: A Bayesian perspective

  • Satyaki Mazumder
  • Sourabh BhattacharyaEmail author


Circular time series have received relatively little attention in statistics, and modeling complex circular time series using the state space approach is nonexistent in the literature. In this article we introduce a flexible Bayesian nonparametric approach to state-space modeling of observed circular time series where even the latent states are circular random variables. Crucially, we assume that the forms of the observational and evolutionary functions, both of which are circular in nature, are unknown and time-varying. We model these unknown circular functions by appropriate wrapped Gaussian processes having desirable properties. We develop an effective Markov-chain Monte Carlo strategy for implementing our Bayesian model by judiciously combining Gibbs sampling and Metropolis–Hastings methods. Validation of our ideas with a simulation study and two real bivariate circular time-series data sets, where we assume one of the variables to be unobserved, revealed very encouraging performance of our model and methods. We finally analyze a data set consisting of directions of whale migration, considering the unobserved ocean current direction asthe latent circular process of interest. The results thatweobtain are encouraging, and the posterior predictive distribution of the observed process correctly predicts the observed whale movement.


Circular time series latent circular process look-up table Markov-chain Monte Carlo state-space model wrapped Gaussian process 

AMS Subject Classification

62M10 65C05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

42519_2017_1305922_MOESM1_ESM.pdf (224 kb)
Supplement to “Nonparametric Dynamic State Space Modeling of Observed Circular Time Series with Circular Latent States: A Bayesian Perspective”


  1. Bhattacharya, S. 2007. A simulation approach to Bayesian emulation of complex dynamic computer models. Bayesian Analysis 2:783–816. doi:10.1214/07-BA232.MathSciNetCrossRefGoogle Scholar
  2. Breckling, J. 1989. The analysis of directional time series: Applications to wind speed and direction. Number 61 in Lecture Notes in Statistics. Berlin, Germany: Springer-Verlag.CrossRefGoogle Scholar
  3. Carlin, B. P., N. G. Polson, and D. S. Stoffer. 1992. A Monte Carlo approach to nonnormal and nonlinear state-space modeling. Journal of the American Statistical Association 87:493–500. doi:10.1080/01621459.1992.10475231.CrossRefGoogle Scholar
  4. Di Marzio, M., A. Panzera, and C. C. Taylor. 2012. Non-parametric smoothing and prediction for nonlinear circular time series. Journal of Time Series Analysis 33:620–30. doi:10.1111/jtsa.2012.33.issue-4.MathSciNetCrossRefGoogle Scholar
  5. Fisher, N. I. 1993. Statistical analysis of circular data. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  6. Fisher, N. I., and A. J. Lee. 1994. Time series analysis of circular data. Journal of the Royal Statistical Society. Series B 56:327–39.MathSciNetzbMATHGoogle Scholar
  7. Ghosh, A., S. Mukhopadhyay, S. Roy, and S. Bhattacharya. 2014. Bayesian inference in nonpaametric dynamic state space models. Statistical Methodology 21:35–48. doi:10.1016/j.stamet.2014.02.004.MathSciNetCrossRefGoogle Scholar
  8. Holzmann, H., A. Munk, M. Suster, and W. Zucchini. 2006. Hidden Markov models for circular and linear-circular time series. Environmental and Ecological Statistics 13:325–47. doi:10.1007/s10651-006-0015-7.MathSciNetCrossRefGoogle Scholar
  9. Hughes, G. 2007. Multivariate and time series models for circular data with applications to protein conformational angles. Doctoral thesis, University of Leeds, Leeds, UK.Google Scholar
  10. Liu, J. 2001. Monte Carlo strategies in scientific computing. New York, NY: Springer-Verlag.zbMATHGoogle Scholar
  11. Lund, U. 1999. Least circular distance regression for directional data. Journal of Applied Statistics 26:723–33. doi:10.1080/02664769922160.MathSciNetCrossRefGoogle Scholar
  12. Mazumder, S., and S. Bhattacharya. 2016. Bayesian nonparametric dynamic state-space modeling with circular latent states. Journal of Statistical Theory and Practice 10:154–78. doi:10.1080/15598608.2015.1100562.MathSciNetCrossRefGoogle Scholar
  13. Ravindran, P., and S. Ghosh. 2011. Bayesian analysis of circular data using wrapped distributions. Journal of Statistical Theory and Practice 4:1–20.MathSciNetGoogle Scholar
  14. Robert, C. P., and G. Casella. 2004. Monte Carlo statistical methods. New York, NY: Springer-Verlag.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIndian Institute of Science Education and Research KolkataKolkataIndia
  2. 2.Interdisciplinary Statistical Research UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations