Advertisement

Bayesian Nonparametric Models of Circular Variables Based on Dirichlet Process Mixtures of Normal Distributions

  • Gabriel Nuñez-AntonioEmail author
  • María Concepción Ausín
  • Michael P. Wiper
Article

Abstract

This article introduces two new Bayesian nonparametric models for circular data based on Dirichlet process (DP) mixtures of normal distributions. The first model is a projected DP mixture of bivariate normals and the second approach is based on a wrapped DP mixture of normal distributions. We show how to carry out inference for these models based on a slice sampling scheme and introduce an approach to estimating a variant of the deviance information criterion which is appropriate in the context of latent variable models. Our models are then compared with both simulated and real data examples.

Keywords

Circular data Deviance information criterion Dirichlet process mixtures Projected normal distribution Wrapped normal distribution 

Notes

Acknowledgments

The work of the first author was partially supported by Sistema Nacional de Investigadores, Mexico. Support from the Department of Statistics of the University Carlos III of Madrid is also gratefully acknowledged. The second and third authors were supported by MEC grant ECO2011-25706 and MEC grant ECO2012-38442 (respectively) from the Spanish Government. The authors are grateful to an associate editor and one anonymous referee for their helpful comments and suggestions on a previous version of paper.

References

  1. Antoniak, C.E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Annals of Statistics, 6, 1152–1174.CrossRefMathSciNetGoogle Scholar
  2. Bhattacharya, S. and SenGupta, A. (2009). Bayesian Analysis of Semiparametric Linear-Circular Models. Journal of Agricultural, Biological and Environmental Statistics, 14, 33–65.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Carnicero, J.A., Ausín, M.C. and Wiper, M.P. (2013). Non-parametric copulas for circular-linear and circular-circular data: an application to wind directions. Stochastic Environmental Research and Risk Assessment, 27, 1991–2002.CrossRefGoogle Scholar
  4. ———   (2014). Density estimation of circular data with Bernstein polynomials. Hacettepe Journal of Mathematics and Statistics. In press.Google Scholar
  5. Casella, G. and Robert, C. (1996). Rao-Blackwellisation of sampling schemes. Biometrika, 83, 81–94.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Celeux, G., Forbes, F., Robert C.P. and Titterington, D.M. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 1, 651–674.CrossRefMathSciNetGoogle Scholar
  7. Coles, S. (1998). Inference for circular distributions and processes. Statistics and Computing, 8, 105–113.CrossRefGoogle Scholar
  8. Downs, T.D. (2003). Spherical regression. Biometrika, 90, 3, 655–668.CrossRefMathSciNetGoogle Scholar
  9. Escobar, M.D. and West, M. (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90, 577–588.CrossRefzbMATHMathSciNetGoogle Scholar
  10. Ferguson, T. (1973). Bayesian analysis of some nonparametric problems. Annals of Statistics, 1, 209–230.CrossRefzbMATHMathSciNetGoogle Scholar
  11. Ferrari, C. (2009). The Wrapping Approach for Circular Data Bayesian Modeling. Doctoral Thesis, Bologna University, Bologna, Italy.Google Scholar
  12. Ferreira, J.T.A.S., Juárez, M.A. and Steel, M.F.J. (2008). Directional log spline distributions. Bayesian Analysis, 3, 297–316.CrossRefMathSciNetGoogle Scholar
  13. Fisher, N.I. (1989). Smoothing a sample of circular data. Journal of Structural Geology, 11, 775-778.CrossRefGoogle Scholar
  14. ———   (1993). Statistical Analysis of Circular Data. Cambridge: University Press.Google Scholar
  15. Fisher, N.I. and Lee, A.J. (1994). Time Series Analysis of Circular Data. Journal of the Royal Statistical Society, Series B, 56, 2, 327–339.zbMATHMathSciNetGoogle Scholar
  16. Ghosh, K., Jammalamadaka, S.R. and Tiwari, R.C. (2003). Semiparametric Bayesian techniques for problems in circular data. Journal of Applied Statistics, 30, 145-161.CrossRefzbMATHMathSciNetGoogle Scholar
  17. Hall, P., Watson, G.S. and Cabrera, J. (1987). Kernel density estimation with spherical data. Biometrika, 74, 751762.MathSciNetGoogle Scholar
  18. Jammalamadaka, S.R. and SenGupta, A. (2001). Topics in Circular Statistics. Singapore: World Scientific.zbMATHGoogle Scholar
  19. Mardia, K.V. (1972). Statistics of Directional Data. Academic Press, London.zbMATHGoogle Scholar
  20. Mardia, K.V. and Jupp, P.E. (2000). Directional Statistics. Chichester: Wiley.zbMATHGoogle Scholar
  21. McVinish, R. and Mengersen, K. (2008). Semiparametric Bayesian circular statistics. Computational Statistics and Data Analysis, 52, 4722–4730.CrossRefzbMATHMathSciNetGoogle Scholar
  22. Nuñez-Antonio, G. and Gutiérrez-Peña, E. (2005). A Bayesian analysis of directional data using the projected normal distribution. Journal of Applied Statistics, 32, 995–1001.CrossRefzbMATHMathSciNetGoogle Scholar
  23. Oliveira, M., Crujeiras, R.M. and Rodríguez-Casal, A. (2012). A plug-in rule for bandwidth selection in circular density estimation. Computational Statistics and Data Analysis, 56, 3898–3908.CrossRefzbMATHMathSciNetGoogle Scholar
  24. Papaspiliopoulos, O. (2008). A note on posterior sampling from Dirichlet mixture models. Discussion paper.Google Scholar
  25. Pewsey, A., Neuhäuser, M. and Ruxton, G.D. (2013). Circular Statistics in R. Oxford: University Press.zbMATHGoogle Scholar
  26. Presnell, B., Morrison S.P. and Littell, R.C. (1998). Projected multivariate linear model for directional data. Journal of the American Statistical Association, 93, 1068–1077.CrossRefzbMATHMathSciNetGoogle Scholar
  27. Ravindran, P. and Ghosh, S.K. (2011). Bayesian Analysis of Circular Data Using Wrapped Distributions. Journal of Statistical Theory and Practice, 5, 547–561.CrossRefMathSciNetGoogle Scholar
  28. R Development Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org.
  29. Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica, 4, 639–650.zbMATHMathSciNetGoogle Scholar
  30. Shearman, L.P., Sriram, S., Weaver, D.R., Maywood, E.S. Chaves, I., Zheng, B., Kume, K., Lee, C.C., van der Horst, G.T.J., Hastings, M.H. and Reppert, S.M. (2000). Interacting molecular loops in the mammalian circadian clock. Science, 288, 1013–1018.CrossRefGoogle Scholar
  31. Spiegelhalter, D.J., Best, N.G., Carlin B.P., and van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, 64, 583–640.CrossRefzbMATHGoogle Scholar
  32. Walker, S.G. (2007). Sampling the Dirichlet Mixture Model with Slices. Communications in Statistics. Simulation and Computation, 36, 45–54.CrossRefzbMATHMathSciNetGoogle Scholar
  33. Wang, F. and Gelfand, A.E. (2013). Directional data analysis under the general projected normal distribution. Statistical Methodology, 10, 113-127.CrossRefMathSciNetGoogle Scholar

Copyright information

© International Biometric Society 2014

Authors and Affiliations

  • Gabriel Nuñez-Antonio
    • 1
    Email author
  • María Concepción Ausín
    • 2
  • Michael P. Wiper
    • 2
  1. 1.Departamento de MatemáticasUniversidad Autónoma Metropolitana – Unidad IztapalaMéxicoMexico
  2. 2.Department of StatisticsUniversidad Carlos III de MadridGetafeSpain

Personalised recommendations