Tree-Structured Bayesian Networks for Wrapped Cauchy Directional Distributions

  • Ignacio Leguey
  • Concha Bielza
  • Pedro Larrañaga
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9868)


Modelling the relationship between directional variables is a nearly unexplored field. The bivariate wrapped Cauchy distribution has recently emerged as the first closed family of bivariate directional distributions (marginals and conditionals belong to the same family). In this paper, we introduce a tree-structured Bayesian network suitable for modelling directional data with bivariate wrapped Cauchy distributions. We describe the structure learning algorithm used to learn the Bayesian network. We also report some simulation studies to illustrate the algorithms including a comparison with the Gaussian structure learning algorithm and an empirical experiment on real morphological data from juvenile rat somatosensory cortex cells.


Directional statistics Wrapped Cauchy distribution Tree-structure Bayesian networks 



This work has been partially supported by the Spanish Ministry of Economy and Competitiveness through the TIN2013-41592-P and Cajal Blue Brain (C080020-09), by the Regional Government of Madrid through the S2013/ICE-2845-CASI-CAM-CM project. I.L. is supported by the Spanish Ministry of Education, Culture and Sport Fellowship (FPU13/01941). The authors thankfully acknowledge the Cortical Circuits Laboratory (CSIC-UPM) for the neurons dataset.


  1. 1.
    Batschelet, E.: Circular Statistics in Biology. Academic Press, London (1981)zbMATHGoogle Scholar
  2. 2.
    Berntsen, J., Espelid, T.O., Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw. 17(4), 437–451 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boomsma, W., Kent, J.T., Mardia, K.V., Taylor, C.C., Hamelryck, T.: Graphical models and directional statistics capture protein structure. Interdisc. Stat. Bioinform. 25, 91–94 (2006)Google Scholar
  4. 4.
    Bowman, K., Shenton, L.: Methods of moments. Encycl. Stat. Sci. 5, 467–473 (1985)Google Scholar
  5. 5.
    Chow, C., Liu, C.: Approximating discrete probability distributions with dependence trees. IEEE Trans. Inf. Theory 14(3), 462–467 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fisher, N.I.: Statistical Analysis of Circular Data. Cambridge University, Cambridge (1995)Google Scholar
  7. 7.
    Geiger, D., Heckerman, D.: Learning gaussian networks. In: Proceedings of the Tenth International Conference on Uncertainty in Artificial Intelligence, pp. 235–243. Morgan Kaufmann Publishers Inc. (1994)Google Scholar
  8. 8.
    Jammalamadaka, S.R., Sengupta, A.: Topics in Circular Statistics. World Scientific, River Edge (2001)CrossRefGoogle Scholar
  9. 9.
    Kato, S.: A distribution for a pair of unit vectors generated by Brownian motion. Bernoulli 15(3), 898–921 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kato, S., Pewsey, A.: A Möbius transformation-induced distribution on the torus. Biometrika 102(2), 359–370 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kent, J.T.: The Fisher-Bingham distribution on the sphere. J. Roy. Stat. Soc. Ser. B (Methodol.) 44(1), 71–80 (1982)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge (2009)zbMATHGoogle Scholar
  13. 13.
    Leguey, I., Bielza, C., Larrañaga, P., Kastanauskaite, A., Rojo, C., Benavides-Piccione, R., DeFelipe, J.: Dendritic branching angles of pyramidal cells across layers of the juvenile rat somatosensory cortex. J. Comp. Neurol. 524(13), 2567–2576 (2016)CrossRefGoogle Scholar
  14. 14.
    Lévy, P.: L’addition des variables aléatoires définies sur une circonférence. Bulletin de la Société Mathématique de France 67, 1–41 (1939)zbMATHGoogle Scholar
  15. 15.
    Mardia, K.V.: Statistics of directional data. J. Roy. Stat. Soc. Ser. B (Methodol.) 37, 349–393 (1975)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mardia, K.V.: Bayesian analysis for bivariate von Mises distributions. J. Appl. Stat. 37(3), 515–528 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mardia, K.V., Hughes, G., Taylor, C.C., Singh, H.: A multivariate von Mises distribution with applications to bioinformatics. Can. J. Stat. 36(1), 99–109 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, Hoboken (2009)zbMATHGoogle Scholar
  19. 19.
    McCullagh, P.: Möbius transformation and Cauchy parameter estimation. Ann. Stat. 24(2), 787–808 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    von Mises, R.: Über die Ganzzahligkeit der Atomgewichte und verwandte Fragen. Zeitschrift für Physik 19, 490–500 (1918)zbMATHGoogle Scholar
  21. 21.
    R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2008). ISBN 3-900051-07-0,
  22. 22.
    Razavian, N., Kamisetty, H., Langmead, C.J.: The von Mises graphical model: regularized structure and parameter learning. Technical report CMU-CS-11-108. Carnegie Mellon University, Department of Computer Science (2011)Google Scholar
  23. 23.
    Spirtes, P., Glymour, C.N., Scheines, R.: Causation, Prediction, and Search. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  24. 24.
    Van Dooren, P., de Ridder, L.: An adaptive algorithm for numerical integration over an \(N\)-dimensional cube. J. Comput. Appl. Math. 2(3), 207–217 (1976)CrossRefzbMATHGoogle Scholar
  25. 25.
    Wintner, A.: On the shape of the angular case of Cauchy’s distribution curves. Ann. Math. Stat. 18(4), 589–593 (1947)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ignacio Leguey
    • 1
  • Concha Bielza
    • 1
  • Pedro Larrañaga
    • 1
  1. 1.Departamento de Inteligencia ArtificialUniversidad Politécnica de MadridBoadilla del Monte, MadridSpain

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