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Learning and Decision-Making in the Framework of Fuzzy Lattices

  • Chapter
New Learning Paradigms in Soft Computing

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 84))

Abstract

A novel theoretical framework is delineated for supervised and unsupervised learning. It is called framework of fuzzy lattices, or FL-framework for short, and it suggests mathematically sound tools for dealing separately and/or jointly with disparate types of data including vectors of numbers, fuzzy sets, symbols, etc. Specific schemes are proposed for clustering and classification having the capacity to deal with both missing and don’t care data values; the schemes in question can be implemented as neural networks. The proposed learning schemes are employed here for pattern recognition on seven data sets including benchmark data sets, and the results are compared with those ones by various learning techniques from the literature. Finally, aiming at a mutual cross-fertilization, the FL-framework is associated with established theories for learning and/or decision-making including probability theory, fuzzy set theory, Bayesian decision-making, theory of evidence, and adaptive resonance theory.

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References

  1. Aha, D.W. (1998), “The omnipresence of case-based reasoning in science and application,” Knowledge-Based Systems, vol. 11, no. 5–6, pp. 261–273.

    Article  Google Scholar 

  2. Amari, S. and Wu, S. (1999), “Improving Support Vector Machine Classifiers by Modifying Kernel Functions,” Neural Networks, vol. 12, no. 6, pp. 783–789.

    Article  Google Scholar 

  3. Baker, D., Brett, P.N., Griffiths, M.V., and Reyes, L. (1996), “A Mechatronic Tool for Ear Surgery: A Case Study of Some Design Characteristics,” Mechatronics, vol. 6, no. 4, pp. 461–477.

    Article  Google Scholar 

  4. Baralis, E., Ceri, S., and Paraboschi, S. (1998), “Compile-Time and Runtime Analysis of Active Behaviors,” IEEE Trans. Knowledge Data Engineering, vol. 10, no. 3, pp. 353–370.

    Article  Google Scholar 

  5. Bianchini, M., Frasconi, P., and Gori, M. (1995), “Learning Without Local Minima in Radial Basis Function Networks,” IEEE Trans. Neural Networks, vol. 6, no. 3, pp. 749–756.

    Article  Google Scholar 

  6. Birkhoff, G. (1967), Lattice Theory, American Mathematical Society, Colloquium Publications, vol. 25, Providence, RI.

    MATH  Google Scholar 

  7. Bishop, C. (1991), “Improving the generalization properties of radial basis function neural networks,” Neural Computation, vol. 3, no. 4, pp. 579–588.

    Article  Google Scholar 

  8. Blumer, A., Ehrenfeucht, A., Haussier, D. and Warmuth, M.K. (1989), “Learnability and the Vapnik-Chervonenkis dimension,” Journal of the ACM, vol. 36, no. 4, pp. 929–965.

    Article  MATH  Google Scholar 

  9. Bogler, P.L. (1987) “Shafer-Dempster reasoning with applications to multisensor target identification systems,” IEEE Trans. Systems, Man, and Cybernetics, vol. 17, no. 6, pp. 968–977.

    Google Scholar 

  10. Bryson, N. and Mobolurin, A. (1998), “A qualitative discriminant approach for generating quantitative belief functions,” IEEE Trans. Knowledge Data Engineering, vol. 10, no. 2, pp. 345–348.

    Article  Google Scholar 

  11. Buede, D.M. and Girardi, P. (1997), “A target identification comparison of Bayesian and Dempster-Shafer multisensor fusion,” IEEE Trans. Sys., Man, and Cybernetics–A, vol. 27, no. 5, pp. 569–577.

    Article  Google Scholar 

  12. Card, H.C., Rosendahl, G.K., McNeill, D.K., and McLeod, R.D. (1998), “Competitive learning algorithms and neurocomputer architecture,” IEEE Trans. Computers, vol. 47, no. 8, pp. 847–858.

    Article  Google Scholar 

  13. Carpenter, G.A. and Grossberg, S. (1987), “A massively parallel architecture for self-organizing neural pattern recognition machine,” Computer Vision, Graphics and Image Understanding, vol. 37, pp. 54–115.

    Article  MATH  Google Scholar 

  14. Carpenter, G.A., Grossberg, S., and Rosen, D.B. (1991), “Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system,” Neural Networks, vol. 4, pp. 759–771.

    Article  Google Scholar 

  15. Cercone, N., An, A., and Chan, C. (1999), “Rule-induction and case-based reasoning: hybrid architectures appear advantageous,” IEEE Trans. Knowledge and Data Engineering, vol. 11, no. 1, pp. 166–174.

    Article  Google Scholar 

  16. Chen, J-Q. and Xi, Y-G. (1998), “Nonlinear system modeling by competitive learning and adaptive fuzzy inference system,” IEEE Trans. Systems, Man, and Cybernetics–C, vol. 28, no. 2, pp. 231–238.

    Article  Google Scholar 

  17. Chen, S., Cowan, C.F.N., and Grant, P.M. (1991), “Orthogonal least squares learning for radial basis function networks,” IEEE Trans. Neural Networks, vol. 2, no. 2, pp. 302–309.

    Article  Google Scholar 

  18. Davey, B.A. and Priestley, H.A. (1990), Introduction to Lattices and Order, Cambridge University Press, Cambridge, Great Britain.

    Google Scholar 

  19. Dempster, A.P. (1967), “Upper and lower probabilities induced by multivalued mappings,” Annals of Math. Stat., vol. 38, pp. 325–329.

    Article  MathSciNet  MATH  Google Scholar 

  20. Dunyak, J., Saad, I.W., and Wunsch, D. (1999), “A theory of independent fuzzy probability for system reliability,” IEEE Trans. Fuzzy Systems, vol. 7, no. 2, pp. 286–294.

    Article  Google Scholar 

  21. Evans, B. and Fisher, D. (1994), “Overcoming process delays with decision tree induction,” IEEE Expert, pp. 60–66, February.

    Google Scholar 

  22. Fahlman, S.E. and White, M., Carnegie Mellon University (CMU) repository of neural net benchmarks. Available at http://www.boltz.cs.cmu.edu/.

  23. Feller, W. (1968), An Introduction to Probability Theory and Its Applications, third edition, John Wiley and Sons, New York, NY.

    MATH  Google Scholar 

  24. Fixsen, D. and Mahler, R.P.S. (1997), “The modified Dempster-Shafer approach to classification,” IEEE Trans. Systems, Man, and Cybernetics–A, vol. 27, no. 1, pp. 96–104.

    Article  Google Scholar 

  25. Fritzke, B. (1994), “Growing cell structures–A self-organizing network for unsupervised learning,” Neural Networks, vol. 7, no. 9, pp. 1441–1460.

    Article  Google Scholar 

  26. Gaines, B.R. (1978), “Fuzzy and probability uncertainty logics,” Information and Control, vol. 38, pp. 154–169.

    Article  MathSciNet  MATH  Google Scholar 

  27. Georgiopoulos, M., Dagher, I., Heileman, G.L., and Bebis, G. (1999), “Properties of learning of a fuzzy ART variant,” Neural Networks, vol. 12, no. 6, pp. 837–850.

    Article  Google Scholar 

  28. Giarratano, J. and Riley G. (1994), Expert Systems - Principles and Programming, second edition, PWS Publishing Company, Boston, MA.

    Google Scholar 

  29. Giles, C.L., Miller, C.B., Chen, D., Chen, H.H., Sun, G.Z., and Lee. Y.C. (1992), “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Computation, vol. 4, no. 3, pp. 393–405.

    Article  Google Scholar 

  30. Grätzer, G. (1971), Lattice theory, W.H. Freeman, San Francisco, CA.

    MATH  Google Scholar 

  31. Healy, M.J. (1999), “A topological semantics for rule extraction with neural networks,” Connection Science, vol. 11, no. 1, pp. 91–113.

    Article  Google Scholar 

  32. Healy, M.J. and Caudell, T.P. (1997), “Acquiring rule sets as a product of learning in a logical neural architecture,” IEEE Trans. Neural Networks, vol. 8, no. 3, pp. 461–474.

    Article  Google Scholar 

  33. Hong, L. and Jain, A. (1998), “Integrating faces and fingerprints for personal identification,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 12, pp. 1295–1307.

    Article  Google Scholar 

  34. Horiuchi, T. (1998), “Decision rule for pattern classification by integrating interval feature values,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 4, pp. 440–448.

    Article  MathSciNet  Google Scholar 

  35. Hummel, R. and Manevitz, L. (1996), “A statistical approach to the representation of uncertainty in beliefs using spread of opinion,” IEEE Trans. Sys., Man, and Cybernetics–A, vol. 26, no. 3, pp. 378–384.

    Article  Google Scholar 

  36. Ishibuchi, H., Fujioka, R., and Tanaka, H. (1993), “Neural networks that learn from fuzzy if-then rules,” IEEE Trans. on Fuzzy Systems, vol. 1, no. 2, pp. 8597.

    Article  Google Scholar 

  37. John, M.F.St. and McClelland, J.L. (1990), “Learning and applying contextual constraints in sentence comprehension,” Artificial Intelligence, vol. 46, pp. 5–46.

    Article  Google Scholar 

  38. Joshi, A., Ramakrishman, N., Houstis, E.N., and Rice, J.R. (1997), “On neurobiological, neuro-fuzzy, machine learning, and statistical pattern recognition techniques,” IEEE Trans. Neural Networks, vol. 8, no. 1, pp. 18–31.

    Article  Google Scholar 

  39. Kaburlasos, V.G. (1992), Adaptive Resonance Theory with Supervised Learning and Large Database Applications, Ph.D. Thesis, Dept. Electrical Engineering, University of Nevada, Reno.

    Google Scholar 

  40. Kaburlasos, V.G. and Petridis, V. (1997), “Fuzzy lattice neurocomputing (FLN): A novel connectionist scheme for versatile learning and decision making by clustering,” International Journal of Computers and Their Applications, vol. 4, no. 2, pp. 31–43.

    Google Scholar 

  41. Kaburlasos, V.G. and Petridis, V. (1998), “A unifying framework for hybrid information processing,” Proceedings of the ISCA 7th Intl. Conf. on Intelligent Systems (ICIS’98), Paris, France, pp. 68–71.

    Google Scholar 

  42. Kaburlasos, V.G., Petridis, V., Brett, P., and Baker, D. (1998), “Learning a linear association of drilling profiles in stapedotomy surgery,” Proceedings of the IEEE 1998 Intl. Conf. on Robotics and Automation (ICRA’98), Leuven, Belgium, pp. 705–710.

    Google Scholar 

  43. Kearns, M.J. and Vazirani, U.V. (1994), An Introduction to Computational Learning Theory, The MIT Press, Cambridge, MA.

    Google Scholar 

  44. Kim, I. and Vachtsevanos, G. (1998), “Overlapping object recognition: a paradigm for multiple sensor fusion,” IEEE Robotics and Automation Magazine, pp. 37–44, September.

    Google Scholar 

  45. Klir, G.J. and Folger, T.A. (1988), Fuzzy Sets, Uncertainty, and Information, Prentice-Hall International Editions, London, UK.

    Google Scholar 

  46. Klir, G.J. and Yuan, B. (1997), Fuzzy Sets, and Fuzzy Logic, Prentice-Hall of India, New Delhi, India.

    Google Scholar 

  47. Kohonen, T. (1995), Self-Organizing Maps, Springer-Verlag, Berl in Heidelberg.

    Book  Google Scholar 

  48. Kosko, B. (1998), “Global stability of generalized additive fuzzy systems,” IEEE Trans. Systems, Man, and Cybernetics–C, vol. 28, no. 3, pp. 441–452.

    Article  Google Scholar 

  49. Kuo, Y-H., Hsu, J-P., and Wang, C-W. (1998), “A parallel fuzzy inference model with distributed prediction scheme for reinforcement learning,” IEEE Trans. Systems, Man, and Cybernetics–B, vol. 28, no. 2, pp. 160–172.

    Article  Google Scholar 

  50. Lin, C-T., Lin, C-J., and Lee, C.S.G. (1995), “Fuzzy adaptive learning control network with on-line neural learning,” Fuzzy Sets and Systems, vol. 71, pp. 2545.

    Article  MathSciNet  Google Scholar 

  51. Lucas, C. and Araabi, B.N. (1999), “Generalization of the Dempster-Shafer theory: A fuzzy-valued measure,” IEEE Trans. Fuzzy Systems, vol. 7, no. 3, pp. 255–270.

    Article  MathSciNet  Google Scholar 

  52. Luo, R.C. and Kay, M.G. (1989), “Multisensor integration and fusion in intelligent systems,” IEEE Trans. Systems, Man, and Cybernetics, vol. 19, no. 5, pp. 901–931.

    Article  Google Scholar 

  53. Mamdani, E.H. and Assilian, S. (1975), “An experiment in linguistic synthesis with a fuzzy logic controller,” Intl. J. of Man-Machine Studies, vol. 7, pp. 1–13.

    Article  MATH  Google Scholar 

  54. Mardia, K.V., Kent, J.T., and Bibby, J.M. (1979), Multivariate Analysis, Academic Press, London, UK.

    Google Scholar 

  55. Margaris, A. (1990), First Order Mathematical Logic, Dover Publications, Inc., New York, NY.

    Google Scholar 

  56. McLachlan, G.J. (1992), Discriminant Analysis and Statistical Pattern Recognition, John Wiley and Sons, New York, NY.

    Book  Google Scholar 

  57. Merz, C.J. and Murphy, P.M. (1996), UCI Repository of Machine Learning Databases, Dept. Inform. Comput. Sci., University of California, Irvine.

    Google Scholar 

  58. Murphy, R.R. (1998), “Dempster-Shafer theory for sensor fusion in autonomous mobile robots,” IEEE Trans. Robotics and Automation, vol. 14, no. 2, pp. 197206.

    Google Scholar 

  59. Nguyen, H.T. and Walker, E.A. (1997), Fuzzy Logic, CRC Press Inc., New York, NY.

    Google Scholar 

  60. Papoulis, A. (1991), Probability, Random Variables, and Stochastic Processes, third edition, McGraw-Hill, Inc., New York, NY.

    Google Scholar 

  61. Park, J.S., Chen, M-S., and Yu, P.S. (1997), “Using a hash-based method with transaction trimming for mining association rules,” IEEE Trans. Knowledge and Data Engring, vol. 9, no. 5, pp. 813–825.

    Article  Google Scholar 

  62. Petridis, V., Kaburlasos, V.G., Brett, P., Parker, T., and Day, J.C.C. (1996), “Two level fuzzy lattice (2L-FL) supervised clustering: A new method for soft tissue identification in surgery,” Proceedings of the CESA’96 IMACS Multiconference, Lille, France, pp. 232–237.

    Google Scholar 

  63. Petridis, V. and Kaburlasos, V.G. (1998), “Fuzzy lattice neural network (FLNN): A hybrid model for learning,” IEEE Trans. Neural Networks, vol. 9, no. 5, pp. 877–890.

    Article  Google Scholar 

  64. Petridis, V. and Kaburlasos, V.G. (1999), “Learning in the framework of fuzzy lattices,” IEEE Trans. Fuzzy Systems, vol. 7, no. 4, pp. 422–440.

    Article  Google Scholar 

  65. Pontil, M. and Verri, A. (1998), “Support vector machines for 3D object recognition,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 6, pp. 637–646.

    Article  Google Scholar 

  66. Pindyck, R.S. and Rubinfeld, D.L. (1991), Econometric Models and Economic Forecasts, McGraw-Hill, Inc., New York, NY.

    Google Scholar 

  67. Rutherford, D.E. (1965), Introduction to Lattice Theory, Oliver and Boyd Ltd., Edinburgh, Great Britain.

    Google Scholar 

  68. Saridis, G.N. (1983), “Intelligent robotic control,” IEEE Trans. Automatic Control, vol. 28, no. 5, pp. 547–557.

    Article  MathSciNet  MATH  Google Scholar 

  69. Sarkar, S., Chakrabarti, P.P., and Ghose, S. (1998), “A framework for learning in search-based systems,” IEEE Trans. Knowledge and Data Engineering, vol. 10, no. 4, pp. 563–575.

    Article  Google Scholar 

  70. Setnes, M., Babuska, R., Kaymak, U., and van Nauta Lemke, H.R. (1998), “Similarity measures in fuzzy rule base simplification,” IEEE Trans. Systems, Man, and Cybernetics–B, vol. 28, no. 3, pp. 376–386.

    Article  Google Scholar 

  71. Shafer, G. (1976), A Mathematical Theory of Evidence, Princeton University Press.

    Google Scholar 

  72. Simpson, P.K. (1992), “Fuzzy min-max neural networks - partl: classification-, IEEE Trans. Neural Networks,vol. 3, no. 5, pp. 776–786.

    Google Scholar 

  73. Simpson, P.K. (1993), “Fuzzy min-max neural networks–part2: clustering,” IEEE Trans. Fuzzy Systems, vol. 1, no. 1, pp. 32–45.

    Article  Google Scholar 

  74. Smets, P. (1990), “The combination of evidence in the transferable belief model,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 5, pp. 447–458.

    Article  Google Scholar 

  75. Tan, A-H. (1995), “Adaptive resonance associative map,” Neural Networks, vol. 8, no. 3, pp. 437–446.

    Article  Google Scholar 

  76. Valiant, L.G. (1984), “A theory of the learnable,” Communications of the ACM, vol. 27, no. 11, pp. 1134–1142.

    Article  MATH  Google Scholar 

  77. Vapnik, V.N. and Chervonenkis, Y.A. (1971), “On the uniform convergence of relative frequencies of events to their probabilities,” Theory of Probability and its Applications, vol. 16, no. 2, pp. 264–280.

    Article  MathSciNet  MATH  Google Scholar 

  78. Vidyasagar, M. (1997), A Theory of Learning and Generalization, Springer-Verlag, London, England.

    Google Scholar 

  79. Waugh, S. (1995), “Extending and benchmarking Cascade-Correlation,” Ph.D. Thesis, Dept. Computer Science, University of Tasmania.

    Google Scholar 

  80. Williamson, J.R. (1996), “Gaussian ARTMAP: A neural network for fast incremental learning of noisy multidimensional maps,” Neural Networks, vol. 9, no. 5, pp. 881–897.

    Article  Google Scholar 

  81. Xu, H., Hsia, Y-T., and Smets, P. (1996), “Transferable belief model for decision making in the valuation-based systems,” IEEE Trans. Systems, Man, and Cybernetics–A, vol. 26, no. 6, pp. 698–707.

    Article  Google Scholar 

  82. Yager, R.R. (1996), “On the aggregation of prioritized belief structured belief structures,” IEEE Trans. Systems, Man, and Cybernetics–A, vol. 26, no. 6, pp. 708–717.

    Article  Google Scholar 

  83. Zadeh, L.A. (1965), “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353.

    Article  MathSciNet  MATH  Google Scholar 

  84. Zadeh, L.A. (1984), “Review of books: A mathematical theory of evidence,” The AI Magazine, pp. 81–83.

    Google Scholar 

  85. Zadeh, L.A. (1996), “Fuzzy logic = Computing with words,” IEEE Trans. Fuzzy Systems, vol. 4, no. 2, pp. 103–111.

    Article  MathSciNet  Google Scholar 

  86. Zeng, X-J. and Singh, M.G. (1997), “Fuzzy bounded least-squares method for the identification of linear systems,” IEEE Trans. Systems, Man, Cybernetics — A, vol. 27, no. 5, pp. 624–635.

    Article  Google Scholar 

  87. Zimmermann, H.J. (1991), Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, Norwell, MA.

    Book  MATH  Google Scholar 

  88. Zouhal, L.M. and Denoeux, T. (1998), “An evidence-theoretic k-NN Rule with parameter optimization,” IEEE Trans. Systems, Man, and Cybernetics–C, vol. 28, no. 2, pp. 263–271.

    Article  Google Scholar 

  89. Zurada, J.M., Marks II, R.J., and Robinson, C.J. (Eds.) (1994), Computational Intelligence — Imitating Life, IEEE Press, New York, NY.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, GR-54006, Thessaloniki, Greece

    V. G. Kaburlasos & V. Petridis

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  1. V. G. Kaburlasos
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  2. V. Petridis
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  1. Knowledge-Based Intelligent Engineering Systems Centre, University of South Australia, 5095, Adelaide, Mawson Lakes, South Australia, Australia

    Lakhmi C. Jain

  2. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

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Kaburlasos, V.G., Petridis, V. (2002). Learning and Decision-Making in the Framework of Fuzzy Lattices. In: Jain, L.C., Kacprzyk, J. (eds) New Learning Paradigms in Soft Computing. Studies in Fuzziness and Soft Computing, vol 84. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1803-1_3

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