Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Batschelet, E.: Circular Statistics in Biology. Academic Press, London (1981)
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)
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)
Bowman, K., Shenton, L.: Methods of moments. Encycl. Stat. Sci. 5, 467–473 (1985)
Chow, C., Liu, C.: Approximating discrete probability distributions with dependence trees. IEEE Trans. Inf. Theory 14(3), 462–467 (1968)
Fisher, N.I.: Statistical Analysis of Circular Data. Cambridge University, Cambridge (1995)
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)
Jammalamadaka, S.R., Sengupta, A.: Topics in Circular Statistics. World Scientific, River Edge (2001)
Kato, S.: A distribution for a pair of unit vectors generated by Brownian motion. Bernoulli 15(3), 898–921 (2009)
Kato, S., Pewsey, A.: A Möbius transformation-induced distribution on the torus. Biometrika 102(2), 359–370 (2015)
Kent, J.T.: The Fisher-Bingham distribution on the sphere. J. Roy. Stat. Soc. Ser. B (Methodol.) 44(1), 71–80 (1982)
Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge (2009)
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)
Lévy, P.: L’addition des variables aléatoires définies sur une circonférence. Bulletin de la Société Mathématique de France 67, 1–41 (1939)
Mardia, K.V.: Statistics of directional data. J. Roy. Stat. Soc. Ser. B (Methodol.) 37, 349–393 (1975)
Mardia, K.V.: Bayesian analysis for bivariate von Mises distributions. J. Appl. Stat. 37(3), 515–528 (2010)
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)
Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, Hoboken (2009)
McCullagh, P.: Möbius transformation and Cauchy parameter estimation. Ann. Stat. 24(2), 787–808 (1996)
von Mises, R.: Über die Ganzzahligkeit der Atomgewichte und verwandte Fragen. Zeitschrift für Physik 19, 490–500 (1918)
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, http://www.R-project.org
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)
Spirtes, P., Glymour, C.N., Scheines, R.: Causation, Prediction, and Search. MIT Press, Cambridge (2000)
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)
Wintner, A.: On the shape of the angular case of Cauchy’s distribution curves. Ann. Math. Stat. 18(4), 589–593 (1947)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Leguey, I., Bielza, C., Larrañaga, P. (2016). Tree-Structured Bayesian Networks for Wrapped Cauchy Directional Distributions. In: Luaces , O., et al. Advances in Artificial Intelligence. CAEPIA 2016. Lecture Notes in Computer Science(), vol 9868. Springer, Cham. https://doi.org/10.1007/978-3-319-44636-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-44636-3_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-44635-6
Online ISBN: 978-3-319-44636-3
eBook Packages: Computer ScienceComputer Science (R0)