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Dependence tree structure estimation via copula

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Abstract

We propose an approach for dependence tree structure learning via copula. A nonparametric algorithm for copula estimation is presented. Then a Chow-Liu like method based on dependence measure via copula is proposed to estimate maximum spanning bivariate copula associated with bivariate dependence relations. The main advantage of the approach is that learning with empirical copula focuses on dependence relations among random variables, without the need to know the properties of individual variables as well as without the requirement to specify parametric family of entire underlying distribution for individual variables. Experiments on two real-application data sets show the effectiveness of the proposed method.

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References

  1. D. Heckerman, D. Geiger, D. M. Chickering. Learning Bayesian networks: The combination of knowledge and statistical data. Machine Learning, vol. 20, no. 3, pp. 197–243, 1995.

    MATH  Google Scholar 

  2. W. Buntine. A guide to the literature on learning probabilistic networks from data. IEEE Transactions on Knowledge and Data Engineering, vol. 8, no. 2, pp. 195–210, 1996.

    Article  Google Scholar 

  3. M. I. Jordan. Learning in Graphical Models, Holland: Kluwer Academic Publishers, 1998.

    MATH  Google Scholar 

  4. C. M. Bishhop. A new framework for machine learning. Computational Intelligence: Research Frontiers, J. M. Zu rada, G. G. Yen, J. Wang, Eds., Heidelberg, Germany: Springer, pp. 1–24, 2008.

    Chapter  Google Scholar 

  5. H. Joe. Multivariate Models and Dependence Concepts, London, UK: Chapmann & Hall, 1997.

    MATH  Google Scholar 

  6. R. B. Nelsen. An Introduction to Copulas. New York, USA: Springer, 1999.

    MATH  Google Scholar 

  7. E. Bouyé, V. Durrleman, A. Nikeghbali, G. Riboulet, T. Roncalli. Copulas for finance — A reading guide and some applications, [Online], Available: http://ssrn.com/abstract=1032533, October 27, 2011.

  8. S. X. Chen, T. M. Huang. Nonparametric estimation of copula functions for dependence modelling. Canadian Journal of Statistics, vol. 35, no. 2, pp. 265–282, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Ma, Z. Sun. Copula component analysis. In Proceedings of the 7th International Conference on Independent Component Analysis and Signal Separation, ACM, London, UK, pp. 73–80, 2007.

  10. K. Abayomi, U. Lall, V. de la Pena. Copula based independent component analysis, [Online], Available: http://ssrn.com/abstract=1028822, October 27, 2011.

  11. S. Kirshner. Learning with tree-averaged densities and distributions. Advances in Neural Information Processing Systems, J. C. Platt, D. Koller, Y. Singer, S. Roweis, Eds., Cambridge, USA: MIT Press, pp. 761–768, 2000.

    Google Scholar 

  12. X. H. Chen, W. B. Wu, Y. P. Yi. Efficient estimation of copula-based semiparametric Markov models. The Annals of Statistics, vol. 37, no. 6B, pp. 4214–4253, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Sklar. Fonctions de repartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris, vol. 8, pp. 229–231, 1959. (In French)

    MathSciNet  Google Scholar 

  14. L. Rüschendorf. On the distributional transform, Sklar’s theorem, and the empirical copula process. Journal of Statistical Planning and Inference, vol. 139, no. 11, pp. 3921–3927, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. K. Chow, C. N. Liu. Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory, vol. 14, no. 3, pp. 462–467, 1968.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. D. Fermanian. Goodness-of-fit tests for copulas. Journal of Multivariate Analysis, vol. 95, no. 1, pp. 119–152, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Yan. Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, [Online], Available: http://www.jstatsoft.org/v21/i04/paper, October 27, 2011.

  18. X. Shen, Y. Zhu, L. Song. Linear B-spline copulas with applications to nonparametric estimation of copulas. Computational Statistics and Data Analysis, vol. 52, no. 7, pp. 3806–3819, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Deheuvels. La fonction de dépendance empirique et ses propriétés — Un test non paramétrique d’indépendance. Académie Royale de Belgique — Bulletin de la Classe des Sciences — 5e Série, vol. 65, pp. 274–292, 1979. (In French)

    MathSciNet  MATH  Google Scholar 

  20. P. Deheuvels. A non parametric test for independence. Publications de l’Institut de Statistique de l’Université de Paris, vol. 26, pp. 29–50, 1981.

    MathSciNet  MATH  Google Scholar 

  21. H. A. David, H. N. Nagaraja. Order Statistics, the 3rd Edition, New York, USA: John Wiley & Sons, 2003.

    Book  MATH  Google Scholar 

  22. T. M. Cover, J. A. Thomas. Elements of Information Theory, New York, USA: John Wiley & Sons, 1991.

    Book  MATH  Google Scholar 

  23. B. W. Silverman. Density Estimation for Statistics and Data Analysis. London, UK: Chapmann & Hall, 1986.

    MATH  Google Scholar 

  24. E. Parzen. On estimation of a probability density function and mode. Annals of Mathematical Statistics, vol. 33, no. 3, pp. 1065–1076, 1962.

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Chen, X. Hong, C. J. Harris. An orthogonal forward regression technique for sparse kernel density estimation. Neurocomputing, vol. 71, no. 4–6, pp. 931–943, 2008.

    Article  Google Scholar 

  26. F. Topsøe. On the Glivenko-Cantelli theorem. Probability Theory and Related Fields, vol. 14, no. 3, pp. 239–250, 1970.

    MATH  Google Scholar 

  27. J. B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. In Proceedings of the American Mathematical Society, vol. 7, no. 1, pp. 48–50, 1956.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. C. Prim. Shortest connection networks and some generalizations. Bell System Technical Journal, vol. 36, pp. 1389–1401, 1957.

    Google Scholar 

  29. A. Asuncion, D. J. Newman. UCI Machine Learning Repository, University of California, Irvine, School of Information and Computer Sciences, [Online], Available: http://archive.ics.uci.edu/ml/datasets.html, October 28, 2011.

  30. D. Harrison, D. L. Rubinfeld. Hedonic housing prices and the demand for clean air. Journal of Environmental Economics and Management, vol. 5, no. 1, pp. 81–102, 1978.

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Tsinghua University, Beijing, 100084, PRC

    Jian Ma & Zeng-Qi Sun

  2. Electronics and Computer Science, Faculty of Physical and Applied Sciences, University of Southampton, Southampton, SO17 1BJ, UK

    Sheng Chen

  3. Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Sheng Chen

  4. Intelligent Systems and Robotics Research Group, School of Creative Technologies, University of Portsmouth, Portsmouth, PO1 2DJ, UK

    Hong-Hai Liu

Authors
  1. Jian Ma
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  2. Zeng-Qi Sun
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  3. Sheng Chen
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  4. Hong-Hai Liu
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Corresponding author

Correspondence to Sheng Chen.

Additional information

Jian Ma received his B. Sc. and M. Sc. degrees in computer science from Hangzhou Dianzi University, Hangzhou, PRC in 2000 and 2003, respectively, and his Ph.D. degree in computer science and technology from Tsinghua University, Beijing, PRC in 2009. He is currently a post-doctoral researcher with the Department of Automation, Tsinghua University.

His research interests include machine learning, data analysis, and information theory.

Zeng-Qi Sun received his B. Sc. degree from the Department of Automatic Control, Tsinghua University, Beijing, PRC in 1966, and his Ph.D. degree in control engineering from Chalmers University of Technology, Gothenburg, Sweden in 1981. He is currently a full professor in the Department of Computer Science and Technology, Tsinghua University. He is the author and coauthor of over 200 research papers and eight books on computer control theory, intelligent control, and robotics.

His research interests include intelligent control, robotics, fuzzy systems, neural networks, and evolution computing.

Sheng Chen received his B. Eng. degree from Huadong Petroleum Institute, Dongying, PRC in January 1982, and his Ph.D. degree from the City University, London, UK in September 1986, both in control engineering. He was awarded the Doctor of Sciences (D. Sc.) degree by the University of Southampton, Southampton, UK in 2004. From October 1986 to August 1999, he held research and academic appointments at the University of Sheffield, the University of Edinburgh and the University of Portsmouth, all in UK. Since September 1999, he has been with the Electronics and Computer Science, the University of Southampton, UK, where he currently holds the post of professor of intelligent systems and signal processing. He is a Distinguished Adjunct Professor at King Abdulaziz University, Jeddah, Saudi Arabia. He has published over 450 research papers. He is a Chartered Engineer (CEng), a fellow of IET and a fellow of IEEE. In the database of the world’s most highly cited researchers, compiled by Institute for Scientific Information (ISI) of the USA, he is on the list of the highly cited researchers in the engineering category.

His research interests include wireless communications, adaptive signal processing for communications, machine learning, evolutionary computation methods, and intelligent control systems.

Hong-Hai Liu received his Ph.D. degree in robotics from Kings College, University of London, UK in 2003. He joined the University of Portsmouth, UK in September 2005, where he currently holds a post of professor of intelligent systems. He previously held research appointments at Universities of London and Aberdeen, UK, and project leader appointments in the industrial control and system integration industries. He has published over 200 research papers including three Best Paper Awards. He is a senior member of IEEE.

His research interests include computational intelligence methods and applications with a focus on those approaches which could make contributions to the intelligent connection of perception to action.

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Ma, J., Sun, ZQ., Chen, S. et al. Dependence tree structure estimation via copula. Int. J. Autom. Comput. 9, 113–121 (2012). https://doi.org/10.1007/s11633-012-0624-6

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  • DOI: https://doi.org/10.1007/s11633-012-0624-6

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