Advertisement

Data-Driven and Knowledge-Based Modeling

  • Erich Peter Klement
  • Edwin Lughofer
  • Johannes Himmelbauer
  • Bernhard Moser
Chapter
  • 522 Downloads

Abstract

This chapter describes some highlights of successful research focusing on knowledge-based and data-driven models for industrial and decision processes. This research has been carried out during the last ten years in a close cooperation of two research institutions in Hagenberg:

- the Fuzzy Logic Laboratorium Linz-Hagenberg (FLLL), a part of the Department of Knowledge-Based Mathematical Systems of the Johannes Kepler University Linz which is located in the Softwarepark Hagenberg since 1993,

- the Software Competence Center Hagenberg (SCCH), initiated by several departments of the Johannes Kepler University Linz as a non-academic research institution under the Kplus Program of the Austrian Government in 1999 and transformed into a K1 Center within the COMET Program (also of the Austrian Government) in 2008.

Keywords

Fuzzy System Fuzzy Model Fuzzy Controller Fuzzy Regression Fuzzy Decision Tree 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Abo03]
    J. Abonyi. Fuzzy Model Identification for Control. Birkhäuser, Boston, 2003.zbMATHGoogle Scholar
  2. [ABS02]
    J. Abonyi, R. Babuska, and F. Szeifert. Modified Gath-Geva fuzzy clustering for identification of Takagi-Sugeno fuzzy models. IEEE Trans. Syst. Man Cybern. B, 32:612–621, 2002.CrossRefGoogle Scholar
  3. [AF04]
    P. Angelov and D. Filev. An approach to online identification of Takagi-Sugeno fuzzy models. IEEE Trans. Syst. Man Cybern. B, 34:484–498, 2004.CrossRefGoogle Scholar
  4. [AFS06]
    C. Alsina, M. J. Frank, and B. Schweizer. Associative Functions: Triangular Norms and Copulas. World Scientific, Singapore, 2006.zbMATHGoogle Scholar
  5. [AGG+06]
    P. Angelov, V. Giglio, C. Guardiola, E. Lughofer, and J. M. Luján. An approach to model-based fault detection in industrial measurement systems with application to engine test benches. Measurement Science and Technology, 17:1809–1818, 2006.CrossRefGoogle Scholar
  6. [AKM95]
    T. Aach, A. Kaup, and R. Mester. On texture analysis: local energy transforms versus quadrature filters. Signal Process., 45:173–181, 1995.zbMATHCrossRefGoogle Scholar
  7. [ALZ08]
    P. Angelov, E. Lughofer, and X. Zhou. Evolving fuzzy classifiers using different model architectures. Fuzzy Sets and Systems, 159:3160–3182, 2008.CrossRefMathSciNetGoogle Scholar
  8. [AZ06]
    P. Angelov and X. Zhou. Evolving fuzzy systems from data streams in realtime. In Proceedings International Symposium on Evolving Fuzzy Systems 2006, pages 29–35, 2006.Google Scholar
  9. [Bab98]
    R. Babuska. Fuzzy Modeling for Control. Kluwer, Boston, 1998.Google Scholar
  10. [Bez81]
    J. C. Bezdek. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York, 1981.zbMATHGoogle Scholar
  11. [BFSO93]
    L. Breiman, J. Friedman, C. J. Stone, and R. A. Olshen. Classification and Regression Trees. Chapman and Hall, Boca Raton, 1993.Google Scholar
  12. [BHBE02]
    M. Burger, J. Haslinger, U. Bodenhofer, and H. W. Engl. Regularized data-driven construction of fuzzy controllers. J. Inverse Ill-Posed Probl., 10:319–344, 2002.zbMATHMathSciNetGoogle Scholar
  13. [BKLM95]
    P. Bauer, E. P. Klement, A. Leikermoser, and B. Moser. Modeling of control functions by fuzzy controllers. In Nguyen et al. [NSTY95], chapter 5, pages 91–116.Google Scholar
  14. [BLK+06]
    J. Botzheim, E. Lughofer, E. P. Klement, L. T. Kóczy, and T. D. Gedeon. Separated antecedent and consequent learning for Takagi-Sugeno fuzzy systems. In Proceedings FUZZ-IEEE 2006, pages 2263–2269, Vancouver, 2006.Google Scholar
  15. [BM98]
    C. L. Blake and C. J. Merz. UCI repository of machine learning databases. Univ. of California, Irvine, Dept. of Information and Computer Sciences, 1998. http://www.ics.uci.edu/ mlearn/MLRepository.html.Google Scholar
  16. [Bod03a]
    U. Bodenhofer. A note on approximate equality versus the Poincaré paradox. Fuzzy Sets and Systems, 133:155–160, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Bod03b]
    U. Bodenhofer. Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets and Systems, 137:113–136, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [Buc93]
    J. J. Buckley. Sugeno type controllers are universal controllers. Fuzzy Sets and Systems, 53:299–303, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [Cas95]
    J. L. Castro. Fuzzy logic controllers are universal approximators. IEEE Trans. Syst. Man Cybernet., 25:629–635, 1995.CrossRefGoogle Scholar
  20. [CCH92]
    J. Coulon, J.-L. Coulon, and U. Höhle. Classification of extremal subobjects over SM-SET. In Rodabaugh et al. [RKH92], pages 9–31.Google Scholar
  21. [CCHM03]
    J. Casillas, O. Cordon, F. Herrera, and L. Magdalena. Interpretability Issues in Fuzzy Modeling. Springer, Berlin, 2003.zbMATHGoogle Scholar
  22. [CD96]
    J. L. Castro and M. Delgado. Fuzzy systems with defuzzification are universal approximators. IEEE Trans. Syst. Man Cybern. B, 26:149–152, 1996.CrossRefGoogle Scholar
  23. [CDM00]
    R. Cignoli, I. M. L. D’Ottaviano, and D. Mundici. Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht, 2000.zbMATHGoogle Scholar
  24. [CFM02]
    G. Castellano, A. M. Fanelli, and C. Mencar. A double-clustering approach for interpretable granulation of data. In Proceedings IEEE Int. Conf. on Syst. Man Cybern. 2002, Hammamet, 2002.Google Scholar
  25. [CGH+04]
    O. Cordon, F. Gomide, F. Herrera, F. Hoffmann, and L. Magdalena. Ten years of genetic fuzzy systems: current framework and new trends. Fuzzy Sets and Systems, 141:5–31, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [CH99]
    O. Cordon and F. Herrera. A two-stage evolutionary process for designing TSK fuzzy rule-based systems. IEEE Trans. Syst. Man Cybern. B, 29:703– 715, 1999.CrossRefGoogle Scholar
  27. [CHHM01]
    O. Cordon, F. Herrera, F. Hoffmann, and L. Magdalena. Genetic Fuzzy Systems—Evolutionary Tuning and Learning of Fuzzy Knowledge Bases. World Scientific, Singapore, 2001.zbMATHGoogle Scholar
  28. [Chi94]
    S. Chiu. Fuzzy model identification based on cluster estimation. Journal of Intelligent and Fuzzy Systems, 2:267–278, 1994.CrossRefGoogle Scholar
  29. [CST01]
    N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press, Cambridge, 2001.zbMATHGoogle Scholar
  30. [DGP92]
    D. Dubois, M. Grabisch, and H. Prade. Gradual rules and the approximation of functions. In Proceedings 2nd International Conference on Fuzzy Logic and Neural Networks, Iizuka, pages 629–632, 1992.Google Scholar
  31. [DH06]
    M. Drobics and J. Himmelbauer. Creating comprehensible regression models: Inductive learning and optimization of fuzzy regression trees using comprehensible fuzzy predicates. Soft Comput., 11:421–438, 2006.CrossRefGoogle Scholar
  32. [DHR93]
    D. Driankov, H. Hellendoorn, and M. Reinfrank. An Introduction to Fuzzy Control. Springer, Berlin, 1993.zbMATHGoogle Scholar
  33. [DHS00]
    R. O. Duda, P. E. Hart, and D. G. Stork. Pattern Classification. Wiley, Chichester, 2000.Google Scholar
  34. [Dro04]
    M. Drobics. Choosing the best predicates for data-driven fuzzy modeling. In Proceedings 13th IEEE Int. Conf. on Fuzzy Systems, pages 245–249, Budapest, 2004.Google Scholar
  35. [EGHL08]
    C. Eitzinger, M. Gmainer, W. Heidl, and E. Lughofer. Increasing classification performance with adaptive features. In A. Gasteratos, M. Vincze, and J. K. Tsotsos, editors, Proceedings ICVS 2008, Santorini Island, volume 5008 of LNCS, pages 445–453. Springer, Berlin, 2008.Google Scholar
  36. [EHL+09]
    C. Eitzinger, W. Heidl, E. Lughofer, S. Raiser, J. E. Smith, M. A. Tahir, D. Sannen, and H. van Brussel. Assessment of the influence of adaptive components in trainable surface inspection systems. Machine Vision and Applications, 2009. To appear.Google Scholar
  37. [FT06]
    D. P. Filev and F. Tseng. Novelty detection based machine health prognostics. In Proceedings 2006 International Symposium on Evolving Fuzzy Systems, pages 193–199, Lake District, 2006.Google Scholar
  38. [GK79]
    D. Gustafson and W. Kessel. Fuzzy clustering with a fuzzy covariance matrix. In Proceedings IEEE CDC, pages 761–766, San Diego, 1979.Google Scholar
  39. [Got01]
    S. Gottwald. A Treatise on Many-Valued Logic. Studies in Logic and Computation. Research Studies Press, Baldock, 2001.Google Scholar
  40. [Gra84]
    R. M. Gray. Vector quantization. IEEE ASSP Magazine, 1:4–29, 1984.CrossRefGoogle Scholar
  41. [Háj98]
    P. Hájek. Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998.zbMATHGoogle Scholar
  42. [Hal50]
    P. R. Halmos. Measure Theory. Van Nostrand Reinhold, New York, 1950.zbMATHGoogle Scholar
  43. [HGC01]
    R. Herbrich, T. Graepel, and C. Campbell. Bayes point machines. Journal of Machine Learning Research, 1:245–279, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  44. [HKKR99]
    F. Höppner, F. Klawonn, R. Kruse, and T. A. Runkler. Fuzzy Cluster Analysis—Methods for Image Recognition, Classification, and Data Analysis. John Wiley & Sons, Chichester, 1999.Google Scholar
  45. [HØ82]
    L. P. Holmblad and J. J. Østergaard. Control of a cement kiln by fuzzy logic. In M. M. Gupta and E. Sanchez, editors, Fuzzy Information and Decision Processes, pages 389–399. North-Holland, Amsterdam, 1982.Google Scholar
  46. [Höh92]
    U. Höhle. M-valued sets and sheaves over integral commutative CL-monoids. In Rodabaugh et al. [RKH92], pages 33–72.Google Scholar
  47. [Höh98]
    U. Höhle. Many-valued equalities, singletons and fuzzy partitions. Soft Computing, 2:134–140, 1998.Google Scholar
  48. [HTF01]
    T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining, Inference and Prediction. Springer, New York, 2001.zbMATHGoogle Scholar
  49. [Jan93]
    J.-S. R. Jang. ANFIS: Adaptive-network-based fuzzy inference systems. IEEE Trans. Syst. Man Cybern., 23:665–685, 1993.CrossRefGoogle Scholar
  50. [Jan98]
    C. Z. Janikow. Fuzzy decision trees: Issues and methods. IEEE Trans. Syst. Man Cybern. B, 28:1–14, 1998.CrossRefGoogle Scholar
  51. [Kas01]
    N. Kasabov. Evolving fuzzy neural networks for supervised/unsupervised online knowledge-based learning. IEEE Trans. Syst. Man Cybern. B, 31:902–918, 2001.CrossRefGoogle Scholar
  52. [Kas02]
    N. Kasabov. Evolving Connectionist Systems—Methods and Applications in Bioinformatics, Brain Study and Intelligent Machines. Springer, London, 2002.zbMATHGoogle Scholar
  53. [KK97]
    F. Klawonn and R. Kruse. Constructing a fuzzy controller from data. Fuzzy Sets and Systems, 85:177–193, 1997.CrossRefMathSciNetGoogle Scholar
  54. [KKM99]
    E. P. Klement, L. T. Kóczy, and B. Moser. Are fuzzy systems universal approximators? Internat. J. Gen. Systems, 28:259–282, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  55. [KM05]
    E. P. Klement and R. Mesiar, editors. Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier, Amsterdam, 2005.zbMATHGoogle Scholar
  56. [KMP00]
    E. P. Klement, R. Mesiar, and E. Pap. Triangular Norms. Kluwer, Dordrecht, 2000.zbMATHGoogle Scholar
  57. [Kos92]
    B. Kosko. Fuzzy systems as universal approximators. In Proceedings IEEE International Conference on Fuzzy Systems 1992, San Diego, pages 1153–1162. IEEE Press, Piscataway, 1992.Google Scholar
  58. [Kun00]
    L. Kuncheva. Fuzzy Classifier Design. Physica-Verlag, Heidelberg, 2000.zbMATHGoogle Scholar
  59. [Kun04]
    L. Kuncheva. Combining Pattern Classifiers: Methods and Algorithms. Wiley, Chichester, 2004.Google Scholar
  60. [LA09]
    E. Lughofer and P. Angelov. Detecting and reacting on drifts and shifts in on-line data streams with evolving fuzzy systems. In Proceedings IF-SA/EUSFLAT 2009 Conference, Lisbon, 2009. To appear.Google Scholar
  61. [LAZ07]
    E. Lughofer, P. Angelov, and X. Zhou. Evolving single- and multi-model fuzzy classifiers with FLEXFIS-Class. In Proceedings FUZZ-IEEE 2007, pages 363–368, London, 2007.Google Scholar
  62. [LG08]
    E. Lughofer and C. Guardiola. On-line fault detection with data-driven evolving fuzzy models. Journal of Control and Intelligent Systems, 36:307– 317, 2008.Google Scholar
  63. [LHBG09]
    E. Lima, M. Hell, R. Ballini, and F. Gomide. Evolving fuzzy modeling using participatory learning. In P. Angelov, D. Filev, and N. Kasabov, editors, Evolving Intelligent Systems: Methodology and Applications. John Wiley & Sons, New York, 2009. To appear.Google Scholar
  64. [LHK05]
    E. Lughofer, E. Hüllermeier, and E. P. Klement. Improving the interpretability of data-driven evolving fuzzy systems. In Proceedings EUSFLAT 2005, pages 28–33, Barcelona, Spain, 2005.Google Scholar
  65. [Lju99]
    L. Ljung. System Identification: Theory for the User. Prentice Hall, Upper Saddle River, 1999.Google Scholar
  66. [LK08]
    E. Lughofer and S. Kindermann. Improving the robustness of data-driven fuzzy systems with regularization. In Proceedings IEEE World Congress on Computational Intelligence (WCCI) 2008, pages 703–709, Hongkong, 2008.Google Scholar
  67. [LK09]
    E. Lughofer and S. Kindermann. Rule weight optimization and feature selection in fuzzy systems with sparsity constraints. In Proceedings IF-SA/EUSFLAT 2009 Conference, Lisbon, Portugal, 2009. To appear.Google Scholar
  68. [LMP05]
    G. Leng, T. M McGinnity, and G. Prasad. An approach for on-line extraction of fuzzy rules using a self-organising fuzzy neural network. Fuzzy Sets and Systems, 150:211–243, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  69. [Lug08a]
    E. Lughofer. Evolving Fuzzy Models—Incremental Learning, Interpretability and Stability Issues, Applications. VDM Verlag Dr. Müller, Saarbrücken, 2008.Google Scholar
  70. [Lug08b]
    E. Lughofer. Evolving vector quantization for classification of on-line data streams. In Proceedings Conference on Computational Intelligence for Modelling, Control and Automation (CIMCA 2008), pages 780–786, Vienna, 2008.Google Scholar
  71. [Lug08c]
    E. Lughofer. Extensions of vector quantization for incremental clustering. Pattern Recognition, 41:995–1011, 2008.zbMATHCrossRefGoogle Scholar
  72. [Lug08d]
    E. Lughofer. FLEXFIS: A robust incremental learning approach for evolving TS fuzzy models. IEEE Trans. Fuzzy Syst., 16:1393–1410, 2008.CrossRefGoogle Scholar
  73. [Lug09]
    E. Lughofer. Towards robust evolving fuzzy systems. In P. Angelov, D. Filev, and N. Kasabov, editors, Evolving Intelligent Systems: Methodology and Applications. John Wiley & Sons, New York, 2009. To appear.Google Scholar
  74. [MA75]
    E. H. Mamdani and S. Assilian. An experiment in linguistic synthesis with a fuzzy logic controller. Intern. J. Man-Machine Stud., 7:1–13, 1975.zbMATHCrossRefGoogle Scholar
  75. [Men42]
    K. Menger. Statistical metrics. Proc. Nat. Acad. Sci. U.S.A., 8:535–537, 1942.CrossRefMathSciNetGoogle Scholar
  76. [Mer09]
    J. Mercer. Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. Roy. Soc. London, 209:415–446, 1909.CrossRefGoogle Scholar
  77. [MH08a]
    B. Moser and P. Haslinger. Texture classification with SVM based on Hermann Weyl’s discrepancy norm. In Proceedings of QCAV09, Wels, 2008. To appear.Google Scholar
  78. [MH08b]
    B. Moser and T. Hoch. Misalignment measure based on Hermann Weyl’s discrepancy. In A. Kuijper, B. Heise, and L. Muresan, editors, Proceedings 32nd Workshop of the Austrian Association for Pattern Recognition (AAPR/OAGM), volume 232, pages 187–197. Austrian Computer Society, 2008.Google Scholar
  79. [Mil02]
    A. Miller. Subset Selection in Regression Second Edition. Chapman and Hall/CRC, Boca Raton, 2002.Google Scholar
  80. [MKH08]
    B. Moser, T. Kazmar, and P. Haslinger. On the potential of Hermann Weyl’s discrepancy norm for texture analysis. In Proceedings Intern. Conf. on Computational Intelligence for Modelling, Control and Automation, 2008. To appear.Google Scholar
  81. [MLMRJRT00]
    H. Maturino-Lozoya, D. Munoz-Rodriguez, F. Jaimes-Romera, and H. Tawfik. Handoff algorithms based on fuzzy classifiers. IEEE Transactions on Vehicular Technology, 49:2286–2294, 2000.CrossRefGoogle Scholar
  82. [Mos99]
    B. Moser. Sugeno controllers with a bounded number of rules are nowhere dense. Fuzzy Sets and Systems, 104:269–277, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  83. [Mos06a]
    B. Moser. On representing and generating kernels by fuzzy equivalence relations. J. Machine Learning Research, 7:2603–2620, 2006.Google Scholar
  84. [Mos06b]
    B. Moser. On the T-transitivity of kernels. Fuzzy Sets and Systems, 157:1787–1796, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  85. [Mos09]
    B. Moser. A similarity measure for images and volumetric data based on Hermann Weyl’s discrepancy. IEEE Trans. on Pattern Analysis and Machine Intelligence, 2009. To appear.Google Scholar
  86. [NFI00]
    O. Nelles, A. Fink, and R. Isermann. Local linear model trees (LOLIMOT) toolbox for nonlinear system identification. In Proceedings 12th IFAC Symposium on System Identification, Santa Barbara, 2000.Google Scholar
  87. [NK92]
    H. T. Nguyen and V. Kreinovich. On approximations of controls by fuzzy systems. Technical Report 92–93/302, LIFE Chair of Fuzzy Theory, Tokyo Institute of Technology, Nagatsuta, Yokohama, 1992.Google Scholar
  88. [NK98]
    D. Nauck and R. Kruse. NEFCLASS-X—a soft computing tool to build readable fuzzy classifiers. BT Technology Journal, 16:180–190, 1998.CrossRefGoogle Scholar
  89. [NSTY95]
    H. T. Nguyen, M. Sugeno, R. Tong, and R. R. Yager, editors. Theoretical Aspects of Fuzzy Control. Wiley, New York, 1995.zbMATHGoogle Scholar
  90. [OW03]
    C. Olaru and L. Wehenkel. A complete fuzzy decision tree technique. Fuzzy Sets and Systems, 138:221–254, 2003.CrossRefMathSciNetGoogle Scholar
  91. [Par62]
    E. Parzen. Extraction and detection problems and reproducing kernel hilbert spaces. Journal of the Society for Industrial and Applied Mathematics. Series A, On control, 1:35–62, 1962.zbMATHCrossRefMathSciNetGoogle Scholar
  92. [PF01]
    Y. Peng and P. A. Flach. Soft discretization to enhance the continuous decision tree induction. In Proceedings ECML/PKDD01 Workshop Integrating Aspects of Data Mining, Decision Support and Meta-Learning, pages 109–118, 2001.Google Scholar
  93. [RKH92]
    S. E. Rodabaugh, E. P. Klement, and U. Höhle, editors. Applications of Category Theory to Fuzzy Subsets. Kluwer, Dordrecht, 1992.zbMATHGoogle Scholar
  94. [RS01]
    H. Roubos and M. Setnes. Compact and transparent fuzzy models and classifiers through iterative complexity reduction. IEEE Trans. on Fuzzy Syst., 9:516–524, 2001.CrossRefGoogle Scholar
  95. [RSA03]
    J. A. Roubos, M. Setnes, and J. Abonyi. Learning fuzzy classification rules from data. Inform. Sci., 150:77–93, 2003.CrossRefMathSciNetGoogle Scholar
  96. [RSHS06]
    H.-J. Rong, N. Sundararajan, G.-B. Huang, and P. Saratchandran. Sequential adaptive fuzzy inference system (SAFIS) for nonlinear system identification and prediction. Fuzzy Sets and Systems, 157:1260–1275, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  97. [Rud76]
    W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, Düsseldorf, 1976.zbMATHGoogle Scholar
  98. [SDJ99]
    R. Santos, E. R. Dougherty, and J. T. Astola Jaakko. Creating fuzzy rules for image classification using biased data clustering. In SPIE Proceedings Series, volume 3646, pages 151–159. Society of Photo-Optical Instrumentation Engineers, Bellingham, 1999.Google Scholar
  99. [SNS+08]
    D. Sannen, M. Nuttin, J. E. Smith, M. A. Tahir, E. Lughofer, and C. Eitzinger. An interactive self-adaptive on-line image classification framework. In A. Gasteratos, M. Vincze, and J.K. Tsotsos, editors, Proceedings ICVS 2008, Santorini Island, volume 5008 of LNCS, pages 173–180. Springer, Berlin, 2008.Google Scholar
  100. [SS83]
    B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, New York, 1983.zbMATHGoogle Scholar
  101. [SS95]
    H. Schwetlick and T. Schuetze. Least squares approximation by splines with free knots. BIT, 35:361–384, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  102. [SS01]
    B. Schölkopf and A. J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (Adaptive Computation and Machine Learning). The MIT Press, 2001.Google Scholar
  103. [Sto74]
    M. Stone. Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society, 36:111–147, 1974.zbMATHGoogle Scholar
  104. [TA77]
    A. N. Tikhonov and V. Y. Arsenin. Solutions of Ill-Posed Problems. Winston & Sons, Washington, 1977.zbMATHGoogle Scholar
  105. [TJ93]
    M. Tuceryan and A. K. Jain. Texture analysis. In C. H. Chen, L. F. Pau, and P. S. P. Wang, editors, The Handbook of pattern recognition & computer vision, pages 235–276. World Scientific, River Edge, 1993.Google Scholar
  106. [TS85]
    T. Takagi and M. Sugeno. Fuzzy identification of systems and its application to modelling and control. IEEE Trans. Syst. Man Cybernet., 15:116–132, 1985.zbMATHGoogle Scholar
  107. [Tsy04]
    A. Tsymbal. The problem of concept drift: definitions and related work. Technical Report TCD-CS-2004-15, Department of Computer Science, Trinity College Dublin, Ireland, 2004.Google Scholar
  108. [Vap95]
    V. Vapnik. The Nature of Statistical Learning Theory. Springer, New York, 1995.zbMATHGoogle Scholar
  109. [Vap98]
    V. Vapnik. Statistical Learning Theory. Wiley, New York, 1998.zbMATHGoogle Scholar
  110. [Wan92]
    L. X. Wang. Fuzzy systems are universal approximators. In Proceedings IEEE International Conference on Fuzzy Systems 1992, San Diego, pages 1163–1169. IEEE, Piscataway, 1992.CrossRefGoogle Scholar
  111. [Was93]
    P. D. Wasserman. Advanced Methods in Neural Computing. Van Nostrand Reinhold, New York, 1993.zbMATHGoogle Scholar
  112. [WCQY00]
    X. Wang, B. Chen, G. Qian, and F. Ye. On the optimization of fuzzy decision trees. Fuzzy Sets and Systems, 112:117–125, 2000.CrossRefMathSciNetGoogle Scholar
  113. [Wey16]
    H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 77:313–352, 1916.zbMATHCrossRefMathSciNetGoogle Scholar
  114. [WKQ+06]
    X. Wu, V. Kumar, J. R. Quinlan, J. Gosh, Q. Yang, H. Motoda, G. J. MacLachlan, A. Ng, B. Liu, P. S. Yu, Z.-H. Zhou, M. Steinbach, D. J. Hand, and D. Steinberg. Top 10 algorithms in data mining. Knowledge and Information Systems, 14:1–37, 2006.zbMATHCrossRefGoogle Scholar
  115. [Yag90]
    R. R. Yager. A model of participatory learning. IEEE Trans. Syst. Man Cybern., 20:1229–1234, 1990.CrossRefMathSciNetGoogle Scholar
  116. [YF93]
    R. R. Yager and D. P. Filev. Learning of fuzzy rules by mountain clustering. In Proceedings SPIE Conf. on Application of Fuzzy Logic Technology, volume 2061, pages 246–254. International Society for Optical Engineering, Boston, 1993.Google Scholar
  117. [Yin98]
    H. Ying. Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno fuzzy systems with linear rule consequents. IEEE Trans. Syst. Man and Cybern. A, 28:515–520, 1998.CrossRefGoogle Scholar
  118. [Zad65]
    L. A. Zadeh. Fuzzy sets. Inform. and Control, 8:338–353, 1965.zbMATHCrossRefMathSciNetGoogle Scholar
  119. [ZS96]
    J. Zeidler and M. Schlosser. Continuous valued attributes in fuzzy decision trees. In Proceedings 8th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pages 395–400, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Erich Peter Klement
    • 1
  • Edwin Lughofer
    • 1
  • Johannes Himmelbauer
    • 2
  • Bernhard Moser
    • 2
  1. 1.Dept. of Knowledge-Based Mathematical Systems, Fuzzy Logic Laboratorium Linz-Hagenberg (FLLL)Johannes Kepler University Linz (JKU)LinzAustria
  2. 2.Software Competence Center Hagenberg (SCCH)HagenbergAustria

Personalised recommendations