Abstract
It is well known that tree-structured perfect maps can be uniquely identified by computing a maximum weight spanning tree with mutual information providing the edge weights. In this paper I generalize the edge evaluation measure by stating the conditions such a measure has to satisfy in order to be able to identify tree-structured perfect maps. In addition, I show that not only mutual information, but also the well-known χ 2 measure satisfies these conditions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Borgelt, C., Kruse, R.: Graphical Models — Methods for Data Analysis and Mining. J. Wiley & Sons, Chichester (2002)
Castillo, E., Gutierrez, J.M., Hadi, A.S.: Expert Systems and Probabilistic Network Models. Springer, New York (1997)
Chow, C.K., Liu, C.N.: Approximating Discrete Probability Distributions with Dependence Trees. IEEE Trans. on Information Theory 14(3), 462–467 (1968)
Friedman, N., Goldszmidt, M.: Building Classifiers using Bayesian Networks. In: Proc. 13th Nat. Conf. on Artificial Intelligence, AAAI 1996, Portland, OR, USA, pp. 1277–1284. AAAI Press, Menlo Park (1996)
Geiger, D.: An entropy-based learning algorithm of Bayesian conditional trees. In: Proc. 8th Conf. on Uncertainty in Artificial Intelligence, UAI 1992, Stanford, CA, USA, pp. 92–97. Morgan Kaufmann, San Mateo (1992)
Jensen, F.V.: An Introduction to Bayesian Networks. UCL Press, London (1996)
Kruskal, J.B.: On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem. In: Proc. American Mathematical Society, vol. 7(1), pp. 48–50. American Mathematical Society, Providence (1956)
Kullback, S., Leibler, R.A.: On Information and Sufficiency. Annals of Mathematical Statistics 22, 79–86 (1951)
Lauritzen, S.L.: Graphical Models. Oxford University Press, Oxford (1996)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo (1988) (2nd edition 1992)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C — The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)
Prim, R.C.: Shortest Connection Networks and Some Generalizations. The Bell System Technical Journal 36, 1389–1401 (1957)
Quinlan, J.R.: C4.5: Programs for Machine Learning. Morgan Kaufmann, San Mateo (1993)
Rebane, G., Pearl, J.: The Recovery of Causal Polytrees from Statistical Data. In: Proc. 3rd Workshop on Uncertainty in Artificial Intelligence, Seattle, WA, USA, pp. 222–228 (1987)
Shannon, C.E.: The Mathematical Theory of Communication. The Bell System Technical Journal 27, 379–423 (1948)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Borgelt, C. (2003). On Identifying Tree-Structured Perfect Maps. In: Günter, A., Kruse, R., Neumann, B. (eds) KI 2003: Advances in Artificial Intelligence. KI 2003. Lecture Notes in Computer Science(), vol 2821. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39451-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-39451-8_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20059-8
Online ISBN: 978-3-540-39451-8
eBook Packages: Springer Book Archive