Information Matrix

  • Chongfu Huang
  • Yong Shi
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 99)


This chapter introduces a novel approach, called information matrix, to illustrate a given small-sample for its information structure. There are three kinds of information matrixes, respectively, on discrete universes of discourse, crisp intervals and fuzzy intervals. This chapter is organized as follows: in section 2.1, we briefly discuss small-sample problems. In section 2.2, we describe the concept of the information matrix. In this section, we also describe the information matrix on discrete universes of discourse. In section 2.3 and 2.4, we discuss the information matrixes, respectively, on crisp intervals and fuzzy intervals. In section 2.5, we analyze three kinds of information matrixes regarding them to grab the observation’s information. In section 2.6, we review other four approaches to describe or construct relationships in systems, and then, in section 2.7, we compare the approach of the information matrix with them.


Membership Function Fuzzy Number Information Matrix Cartesian Space Fuzzy Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cross, V., and Setnes, M. (1998), A Generalized Model for Ranking Fuzzy Sets. Proceedings of FUZZ-IEEE’98, Anchorage, USA, pp. 773–778Google Scholar
  2. 2.
    Cybenko, G. (1989), Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2 (4), pp. 303–314MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Dobois, D., and Prade, H. (1978), Operations of fuzzy numbers. International Journal of Systems Science, 9 (6), pp. 613–626MathSciNetCrossRefGoogle Scholar
  4. 4.
    Feuring, T., James, J., and Hayashi, Y. (1998), A Gradient Descent Learning Algorithm for Fuzzy Neural Networks. Proceedings of FUZZ-IEEE’98, Anchorage, USA, pp. 1136–1141Google Scholar
  5. 5.
    Funahashi, K. (1989), On the approximate realization of continuous mappings by neural networks. Neural Networks, 2 (3), pp. 183–192CrossRefGoogle Scholar
  6. 6.
    Hazewinkel, M. (Ed.) (1995), Encyclopaedia of mathematics. Kluwer Academic Publishers, SingaporeMATHGoogle Scholar
  7. 7.
    Hornik, K., Stinchcombe, M., and White, H. (1989), Multilayer feedforward networks are universal approximators. Neural Networks, 2 (5), pp. 359–366CrossRefGoogle Scholar
  8. 8.
    Huber, K.-P. and Berthold, M. (1998), application of fuzzy graphs for meta-modeling. Proceedings of FUZZ-IEEE’98, Anchorage, USA, pp. 640–644Google Scholar
  9. 9.
    Kerre, E. (1999), Fuzzy Sets and Approximate Reasoning. Xian Jiaotong University Press, Xian, ChinaGoogle Scholar
  10. 10.
    Kulczycki, P. (1998), Estimating conditional distributions by neural networks. Proceedings of IJCNN’98, Anchorage, USA, pp. 1344–1349Google Scholar
  11. 11.
    Ky Van Ha (1998), Hierarchical radial basis function networks. Proceedings of IJCNN’98, Anchorage, USA, pp. 1893–1898Google Scholar
  12. 12.
    Kosko, B.(1987), Foundations of Fuzzy Estimation Theory. Ph.D. dissertation, Department of Electrical Engineering, University of California at Irvine, June 1987; Order Number 8801936, University Microfilms International, 300 N. Zeeb Road, Ann Arbor, MI 48106Google Scholar
  13. 13.
    Liu, Z.R., Huang, C.F., Kong, Q.Z., and Yin, X.F. (1987), A fuzzy quantitative study on the effect of active fault distribution on isoseismal area in Yunnan. Journal of seismology, 1, pp. 9–16Google Scholar
  14. 14.
    Marco Gori and Alberto Tesi (1992), On the problem of local minima in back-propagation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1), pp. 76–86CrossRefGoogle Scholar
  15. 15.
    Matsuyama, Y., Furukawa, S., Takeda, N., and Ikeda, T. (1998), Fast a-weighted EM learning for neural networks of module mixtures. Proceedings of IJCNN’98, Anchorage, USA, pp. 2306–2311Google Scholar
  16. 16.
    Pao, Y.H. (1989), Adaptive Pattern Recognition and Neural Networks. Addison-Wesley, Reading, MassachusettsGoogle Scholar
  17. 17.
    Patrick, K. Simpson (1990), Artificial Neural Systems Foundations, Paradigms, Applications, and Implementations. McGraw-Hill, New YorkGoogle Scholar
  18. 18.
    Ripley, B. D. (1996), Pattern Recognition and Neural Networks. Cambridge University Press, CambridgeMATHGoogle Scholar
  19. 19.
    Rosenfeld, A. (1975), Fuzzy graphs. Zadeh, L.A., Fu, K.S. Tanaka, K., and Shimura, M. (eds): Fuzzy Sets and Applications to Cognitive and Decision Processes. Academic Press, New York, pp. 77–95Google Scholar
  20. 20.
    Rumelhart, D. E., Hinton, G. E. and Williams, R. J. (1986), Learning internal representations by error propagation. Rumelhart, D.E. and McClelland, J. L. (eds): Parallel Distributed Processing: Explorations in the Microstructure of Cognitions; Vol. 1: Foundations. MIT Press, Cambridge, MA, pp. 318–362Google Scholar
  21. 21.
    Rumelhart, D.E. and McClelland J. L. (1986). Parallel Distributed Processing. MIT Press, Cambridge, MAGoogle Scholar
  22. 22.
    Simon Haykin (1994), Neural Networks: A Comprehensive Foundation. Prentice-Hall, Inc., Englewood Cliffs, New JerseyGoogle Scholar
  23. 23.
    Skiena, S. (1990), Implementing Discrete Mathematics Combinatorics and Graph Theory with Mathematics. Addison-Wesley, Reading, MAMATHGoogle Scholar
  24. 24.
    Warner, B. A. and Misra, M. (1998), Iteratively reweighted least squares based learning. Proceedings of IJCNN’98, Anchorage, USA, pp. 1327–1331Google Scholar
  25. 25.
    White, H. (1990), Connectionist nonparametric regression: multilayer feedforward network can learn arbitrary mappings. Neural Networks, 3 (5), pp. 535–549CrossRefGoogle Scholar
  26. 26.
    Wray, J. and Green, G.G.R. (1995), Neural networks, approximation theory, and finite precision computation. Neural Networks, 8 (1), pp. 31–37CrossRefGoogle Scholar
  27. 27.
    Zadeh, L.A. (1974), On the analysis of large scale system. in: Gottinger H. Ed., Systems Approaches and Environment Problems, Vandenhoeck and Ruprecht, Gottingen, pp. 23–37Google Scholar
  28. 28.
    Zadeh, L.A. (1987), Fuzzy Sets and Applications: Selected Papers by Zadeh, L.A., Yager, R. R., Ovchinnikov, S., Tong, R.M., and Nguyen, H.T. (eds), John Wiley and Sons, New YorkGoogle Scholar
  29. 29.
    Zadeh, L.A. (1994), Soft computing and fuzzy logic. IEEE Software, 11 (6), pp. 48–56CrossRefGoogle Scholar
  30. 30.
    Zadeh, L.A. (1995), Fuzzy control, fuzzy graphs, and fuzzy inference. in: Yam, Y. and Leung, K.S. Eds., Future Directions of Fuzzy Theory and Systems ( World Scientific, Singapore ), pp. 1–9CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Chongfu Huang
    • 1
  • Yong Shi
    • 2
  1. 1.Institute of Resources ScienceBeijing Normal UniversityBeijingChina
  2. 2.College of Information Science and TechnologyUniversity of Nebraska at OmahaOmahaUSA

Personalised recommendations