Journal of Statistical Theory and Practice

, Volume 5, Issue 4, pp 547–561 | Cite as

Bayesian Analysis of Circular Data Using Wrapped Distributions

  • Palanikumar RavindranEmail author
  • Sujit K. Ghosh


Circular data arise in a number of different areas such as geological, meteorological, biological and industrial sciences. Standard statistical techniques can not be used to model circular data due to the circular geometry of the sample space. One of the common methods to analyze circular data is known as the wrapping approach. This approach is based on a simple fact that a probability distribution on a circle can be obtained by wrapping a probability distribution defined on the real line. A large class of probability distributions that are flexible to account for different features of circular data can be obtained by the aforementioned approach. However, the likelihood-based inference for wrapped distributions can be very complicated and computationally intensive. The EM algorithm to compute the MLE is feasible, but is computationally unsatisfactory. A data augmentation method using slice sampling is proposed to overcome such computational difficulties. The proposed method turns out to be flexible and computationally efficient to fit a wide class of wrapped distributions. In addition, a new model selection criteria for circular data is developed. Results from an extensive simulation study are presented to validate the performance of the proposed estimation method and the model selection criteria. Application to a real data set is also presented and parameter estimates are compared to those that are available in the literature.

AMS Subject Classification

62F15 62F10 62H11 65C05 


Bayesian inference Markov chain Monte Carlo (MCMC) Wrapped Cauchy Wrapped Double Exponential Wrapped Normal 


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  1. Best, D.J., Fisher, N.I., 1978. Efficient simulation of the von Mises distribution. Appl.Statist., 28, 152–157.CrossRefGoogle Scholar
  2. Coles, S., 1998. Inference for circular distributions and processes. Statistics and Computing, 8, 105–113.CrossRefGoogle Scholar
  3. Damien, P., Wakefield, J., Walker, S., 1999. Gibbs sampling for Bayesian non-conjugate hierarchical models by using auxiliary variables. J. R. Statist. Soc. B, 61, 331–344.MathSciNetCrossRefGoogle Scholar
  4. Damien, P., Walker, S., 1999. A full Bayesian analysis of circular data using von Mises distribution. The Canadian Journal of Statistics, 27, 291–298.CrossRefGoogle Scholar
  5. Fisher, N.I., 1993. Statistical Analysis of Circular Data. Cambridge Univ. Press, Cambridge.CrossRefGoogle Scholar
  6. Fisher, N.I., Lee, A.J., 1994. Time series analysis of circular data. J. R. Statist. Soc. B, 56, 327–339.MathSciNetzbMATHGoogle Scholar
  7. Gelfand, A.E., Ghosh, S.K., 1998. Model choice: A minimum posterior predictive loss function. Biometrika, 85, 1–11.MathSciNetCrossRefGoogle Scholar
  8. Higdon, D.M., 1998. Auxiliary variable methods for Markov Chain Monte Carlo with applications. J. Am. Statist. Assoc., 93, 585–595.CrossRefGoogle Scholar
  9. Jammalamadaka, S.R., Kozubowski, T.J., 2004. New families of wrapped distributions for modeling skew circular data. Communications in Statistics — Theory and Methods, 33, 2059–2074.MathSciNetCrossRefGoogle Scholar
  10. Jammalamadaka, S.R., SenGupta, A., 2001. Topics in Circular Statistics. Wiley & Sons, New York.CrossRefGoogle Scholar
  11. Jander, R., 1957. Die optische Richtungsorientierung der roten Waldameise (Formica rufa L.). Zeitschrift Fur Vergleichende Physiologie, 40, 162–238.CrossRefGoogle Scholar
  12. Kent, J.T., Tyler, D.E., 1988. Maximum likelihood estimation for the wrapped Cauchy distribution. Journal of Applied Statistics, 15, 247–254.CrossRefGoogle Scholar
  13. Mardia, K.V., 1972. Statistics of Directional Data. Academic Press, London.zbMATHGoogle Scholar
  14. Mardia, K.V., Jupp, P.E., 2000. Directional Statistics. Wiley, Chichester.zbMATHGoogle Scholar
  15. McCullagh, P., Nelder, J.A., 1989. Generalized Linear Models, 2nd Edition. Chapman and Hall, New York.CrossRefGoogle Scholar
  16. Neal, R., 2003. Slice sampling (with discussion). Annals of Statistics, 31, 705–767.MathSciNetCrossRefGoogle Scholar
  17. Ravindran, P., 2003. Bayesian Analysis of Circular Data Using Wrapped Distributions. Doctoral Thesis, North Carolina State University, Raleigh, North Carolina. URL: Scholar
  18. Sengupta, A., Pal, C., 2001. On optimal tests for isotropy against the symmetric wrapped stable-circular uniform mixture family. Journal of Applied Statistics, 28, 129–143.MathSciNetCrossRefGoogle Scholar
  19. Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A., 2002. Bayesian measures of model complexity and fit, (with discussion and rejoinder). Journal of the Royal Statistical Society, Series B, 64, 583–639.MathSciNetCrossRefGoogle Scholar
  20. Stephens, M.A., 1969. Techniques for directional data. Technical Report 150, Department of Statistics, Stanford University.CrossRefGoogle Scholar
  21. Tanner, M.A., Wong W.H., 1987. The calculation of posterior distributions by data augmentation. J. Am. Statist. Assoc., 82, 528–540.MathSciNetCrossRefGoogle Scholar
  22. van Dyk, D.A., Meng, X.L., 2001. The art of data augmentation. Journal of Computational and Graphical Statistics, 10, 1–50.MathSciNetCrossRefGoogle Scholar
  23. von Mises, R., 1918. Über die “Ganzzahligkeit” der Atomgewicht und verwandte Fragen. Physikalische Zeitschrift. 19, 490–500.zbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2011

Authors and Affiliations

  1. 1.Hoffmann-La Roche IncNutleyUSA
  2. 2.Department of StatisticsNC State UniversityRaleighUSA

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