Equation for generalization error of Multinomial classifier is derived and tested. Particular attention is paid to imbalanced training sets. It is shown that artificial growth of training vectors of less probable class could be harmful. Use of predictive Bayes approach to estimate cell probabilities of the classifier reduces both the generalization error and effect of unequal training sample sizes.


BKS rule Complexity Generalization error Learning Imbalan-ced training sets Multinomial classifier Sample size 


  1. 1.
    Lachenbruch, P.A., Goldstein, M.: Discriminant analysis. Biometrics 5(3), 9–85 (1979)MathSciNetGoogle Scholar
  2. 2.
    Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley, New York (2000)Google Scholar
  3. 3.
    Hughes, G.F.: On the mean accuracy of statistical pattern recognizers. IEEE Trans. on Information Theory IT-14, 55–63 (1965)Google Scholar
  4. 4.
    Chandrasekaran, B., Jain, A.K.: Quantization complexity and independent measurements. IEEE Trans. Computers 23, 102–106 (1974)zbMATHCrossRefGoogle Scholar
  5. 5.
    Duin, R.P.W.: On the Accuracy of Satistical Pattern Recognizers. Ph.D. dissertation. Delft University of Technology, Delft (1978)Google Scholar
  6. 6.
    Griskevicius, D., Raudys, S.: On the expected probability of the classification error of the classifier for discrete variables. In: Raudys, S. (ed.) Statistical Problems of Control. Institute of Mathematics and Informatics, Vilnius, vol. 38, pp. 95–112 (1979) (in Russian)Google Scholar
  7. 7.
    Serych, A.P.: On use of nonparametric density estimates in pattern recognition. In: Proc. of Siberian Physic technical V.D. Kuznetsov Institute, vol. 63, pp. 13–41 (1973) (in Russian)Google Scholar
  8. 8.
    Raudys, S.: Statistical and Neural Classifiers: An integrated approach to design, p. 312. Springer, New York (2000)Google Scholar
  9. 9.
    Amari, S.: A universal theorem on learning curves. Neural Networks 6, 161–166 (1993)CrossRefGoogle Scholar
  10. 10.
    Vidyasagar, M.: A Theory of Learning and Generalization. Springer, London (1997)zbMATHGoogle Scholar
  11. 11.
    Vapnik, V.: Statistical Learning Theory. John Wiley and Sons, New York (1998)zbMATHGoogle Scholar
  12. 12.
    Skurichina, M., Raudys, S., Duin, R.P.W.: K-NN directed noise injection in multilayer perceptron training. IEEE Trans. on Neural Networks 11(2), 504–511 (2000)CrossRefGoogle Scholar
  13. 13.
    Raudys, S.: Experts’ boasting in trainable fusion rule. IEEE Trans. on Pattern Analysis and Machine Intelligence, PAMI 25(9), 1178–1182 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sarunas Raudys
    • 1
  1. 1.Vilnius Gediminas Technical UniversityVilniusLithuania

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