On the Foundations of Multinomial Sequence Based Estimation

  • B. John OommenEmail author
  • Sang-Woon Kim
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9875)


This paper deals with the relatively new field of sequence-based estimation which involves utilizing both the information in the observations and in their sequence of appearance. Our intention is to obtain Maximum Likelihood estimates by “extracting” the information contained in the observations when perceived as a sequence rather than as a set. The results of [15] introduced the concepts of Sequence Based Estimation (SBE) for the Binomial distribution. This current paper generalizes these results for the multinomial “two-at-a-time” scenario. We invoke a novel phenomenon called “Occlusion” that can be described as follows: By “concealing” certain observations, we map the estimation problem onto a lower-dimensional binomial space. Once these occluded SBEs have been computed, we demonstrate how the overall Multinomial SBE (MSBE) can be obtained by mapping several lower-dimensional estimates onto the original higher-dimensional space. We formally prove and experimentally demonstrate the convergence of the corresponding estimates.


Estimation using sequential information Sequence Based Estimation Estimation of multinomials Fused estimation methods Sequential information 


  1. 1.
    Bickel, P., Doksum, K.: Mathematical Statistics: Basic Ideas and Selected Topics, vol. 1, 2nd edn. Prentice Hall, Upper Saddle River (2000)zbMATHGoogle Scholar
  2. 2.
    Bunke, H.: Structural and syntactic pattern recognition. In: Chen, C.H., Pau, L.F., Wang, P.S.P. (eds.) Handbook of Pattern Recognition and Computer Vision, pp. 163–209. World Scientific-25, River Edge (1993)CrossRefGoogle Scholar
  3. 3.
    Casella, G., Berger, R.: Statistical Inference, 2nd edn. Brooks/Cole Publisher Company, Pacific Grove (2001)zbMATHGoogle Scholar
  4. 4.
    Duda, R., Hart, P., Stork, D.: Pattern Classification, 2nd edn. John Wiley and Sons, Inc., New York (2000)zbMATHGoogle Scholar
  5. 5.
    El-Gendy, M.A., Bose, A., Shin, K.G.: Evolution of the internet QoS and support for soft real-time applications. Proc. IEEE 91, 1086–1104 (2003)CrossRefGoogle Scholar
  6. 6.
    Friedman, M., Kandel, A.: Introduction to Pattern Recognition - Statistical, Structural, Neural and Fuzzy Logic Approaches. World Scientific, New Jersey (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fukunaga, K.: Introduction to Statistical Pattern Recognition. Academic Press, San Diego (1990)zbMATHGoogle Scholar
  8. 8.
    Goldberg, S.: Probability: An Introduction. Prentice-Hall, Englewood Cliffs (1960)Google Scholar
  9. 9.
    Herbrich, R.: Learning Kernel Classifiers: Theory and Algorithms. MIT Press, Cambridge (2001)Google Scholar
  10. 10.
    Jones, B., Garthwaite, P., Jolliffe, I.: Statistical Inference, 2nd edn. Oxford University Press, New York (2002)zbMATHGoogle Scholar
  11. 11.
    Kittler, J., Hatef, M., Duin, R.P.W., Matas, J.: On combining classifiers. IEEE Trans. Pattern Anal. Mach. Intell. PAMI–20, 226–239 (1998)CrossRefGoogle Scholar
  12. 12.
    Kreyszig, E.: Advanced Engineering Mathematics, 8th edn. John Wiley & Sons, New York (1999)zbMATHGoogle Scholar
  13. 13.
    Kuncheva, L.I., Bezdek, J.C., Duin, R.P.W.: Decision templates for multiple classifier fusion: an experimental comparison. Pattern Recogn. 34, 299–414 (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kuncheva, L.I.: A theoretical study on six classifier fusion strategies. IEEE Trans. Pattern Anal. Mach. Intell. PAMI–24, 281–286 (2002)CrossRefGoogle Scholar
  15. 15.
    Oommen, B.J., Kim, S.-W., Horn, G.: On the estimation of independent binomial random variables using occurrence and sequential information. Pattern Recogn. 40(11), 3263–3276 (2007)CrossRefzbMATHGoogle Scholar
  16. 16.
    Oommen, B.J., Kim, S-W.: Occlusion-based estimation of independent multinomial random variables using occurrence and sequential information. To be submitted for PublicationGoogle Scholar
  17. 17.
    Ross, S.: Introduction to Probability Models, 2nd edn. Academic Press, Orlando (2002)Google Scholar
  18. 18.
    Shao, J.: Mathematical Statistics, 2nd edn. Springer, Heidelberg (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sprinthall, R.: Basic Statistical Analysis. Allyn and Bacon, Boston (2002)Google Scholar
  20. 20.
    van der Heijden, F., Duin, R.P.W., de Ridder, D., Tax, D.M.J.: Classification, Parameter Estimation and State Estimation: An Engineering Approach using MATLAB. John Wiley and Sons Ltd, England (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Webb, A.: Statistical Pattern Recognition, 2nd edn. John Wiley & Sons, New York (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Computer EngineeringMyongji UniversityYonginSouth Korea

Personalised recommendations