Skip to main content
Springer Nature Link
Log in

Additive Fuzzy Systems as Generalized Probability Mixture Models

  • Chapter
  • First Online:
Fifty Years of Fuzzy Logic and its Applications

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 326))

  • 1688 Accesses

Abstract

Additive fuzzy systems generalize the popular mixture-density models of machine learning. Additive fuzzy systems map inputs to outputs by summing fired then-parts sets and then taking the centroid of the sum. This additive structure produces a simple convex structure: Outputs are convex combinations of the centroids of the fired then-part sets. Additive systems are uniform function approximators and admit simple learning laws that grow and tune rules from sample data. They also behave as conditional expectations with conditional variances and other higher moment that describe their uncertainty. But they suffer from exponential rule explosion in high dimensions. Extending finite-rule additive systems to fuzzy systems with continuum-many rules overcomes the problem of rule explosion if a higher-level mixture structure acts as a system of tunable meta-rules. Monte Carlo sampling can then compute fuzzy-system outputs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Kosko, B.: Neural Networks and Fuzzy Systems, Prentice-Hall (1991)

    Google Scholar 

  2. Kosko, B.: Fuzzy systems as universal approximators. IEEE Trans. Comput. 43(11), 1329–1333 (1994)

    Article  MATH  Google Scholar 

  3. Kosko, B.: Optimal fuzzy rules cover extrema. Int. J. Intell. Syst. 10, 249–255 (1995)

    Article  Google Scholar 

  4. Dickerson, J.A., Kosko, B.: “Fuzzy Function Approximation with Ellipsoidal Rules”, with J.A. Dickerson. IEEE Trans. Syst. Man Cybern. 26(4), 542–560 (1996)

    Article  Google Scholar 

  5. Kosko, B., Fuzzy Engineering. Prentice-Hall (1996)

    Google Scholar 

  6. Kosko, B.: Global stability of generalized additive fuzzy systems. IEEE Trans. Syst. Man Cybern. 28(3), 441–452 (1998)

    Article  Google Scholar 

  7. Mitaim, S., Kosko, B.: Neural fuzzy agents for profile learning and adaptive object matching. Presence 7(6), 617–637 (1998)

    Article  Google Scholar 

  8. Mitaim, S., Kosko, B.: The shape of fuzzy sets in adaptive function approximation. IEEE Trans. Fuzzy Syst. 9(4), 637–656 (2001)

    Article  Google Scholar 

  9. Lee, I., Anderson, W.F., Kosko, B.: Modeling of gunshot bruises in soft body armor with an adaptive fuzzy system. IEEE Trans. Syst. Man Cybern. 35(6), 1374–1390 (2005)

    Article  Google Scholar 

  10. Kandel, A.: Fuzzy Mathematical Techniques with Applications. Addison-Wesley (1986)

    Google Scholar 

  11. Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainty, and Information. Prentice-Hall (1988)

    Google Scholar 

  12. Terano, T., Asai, A., Sugeno, M.: Fuzzy Systems Theory and its Applications. Academic Press (1992)

    Google Scholar 

  13. Zimmerman, H.J.: Fuzzy Set Theory and its Application. Kluwer (1985)

    Google Scholar 

  14. Isaka, S., Kosko, B.: Fuzzy Logic. Sci. Am. 269, 76–81 (1993)

    Google Scholar 

  15. Jang, J.-S.R., Sun, C.-T.: Functional equivalence between radial basis function networks and fuzzy inference systems. IEEE Trans. Neural Netw. 4(1), 156–159 (1993)

    Article  Google Scholar 

  16. Moody, J., Darken, C.: Fast learning in networks of locally tuned processing units. Neural Comput. 1, 281–294 (1989)

    Article  Google Scholar 

  17. Specht, D.F.: A general regression neural network. IEEE Trans. Neural Netw. 4, 549–557 (1991)

    Google Scholar 

  18. Wang, L.-X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Netw. 3, 802–814 (1992)

    Google Scholar 

  19. Watkins, F.A.: Fuzzy Engineering. Ph.D. Dissertation, Department of Electrical Engineering, UC Irvine, Irvine, CA (1994)

    Google Scholar 

  20. Watkins, F.A.: The representation problem for additive fuzzy systems. In: Proceedings of the IEEE Int. Conference on Fuzzy Systems (IEEE FUZZ), vol. 1, pp. 117–122, March 1995

    Google Scholar 

  21. Osoba, O., Mitaim, S., Kosko, B.: Bayesian inference with adaptive fuzzy priors and likelihoods. IEEE Trans. Syst. Man Cybern.-B 41(5), 1183–1197 (2011)

    Article  Google Scholar 

  22. Osoba, O., Mitaim, S., Kosko, B.: Triply fuzzy function approximation for hierarchical bayesian inference. Fuzzy Optim. Decis. Making 11(3), 241–268 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hogg, R.V., McKean, J.W., Craig, A.T.: Introduction to Mathematical Statistics, 7th edn. Prentice Hall, New York (2013)

    Google Scholar 

  24. Osoba, O., Mitaim, S., Kosko, B.: The noisy expectation-maximization algorithm. Fluctuation Noise Lett. 12(3), 1350012-1–1350012-30 (2013)

    Google Scholar 

  25. Audhkhasi, K., Osoba, O., Kosko, B.: Noise benefits in backpropagation and deep bidirectional pre-training. In: Proceedings of the 2013 International Joint Conference on Neural Networks, pp. 2254–2261, August (2013)

    Google Scholar 

  26. Audhkhasi, K., Osoba, O., Kosko, B.: Noise benefits in convolutional neural networks. In: Proceedings of the 2014 International Conference on Advances in Big Data Analytics, pp. 73–80, July (2014)

    Google Scholar 

  27. Kong, S.G., Kosko, B.: “Adaptive fuzzy systems for backing up a truck-and- trailer”, with S.G. Kong. IEEE Trans. Neural Netw. 3(2), 211–223 (1992)

    Article  Google Scholar 

  28. Cappe, O., Douc, R., Guillin, A., Marin, J.-M., Robert, C.P.: Adaptive importance sampling in general mixture classes. Stat Comput., 18(4), 447–459 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Signal and Image Processing Institute, University of Southern California, Los Angeles, CA, USA

    Bart Kosko

Authors
  1. Bart Kosko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bart Kosko .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Texas State University, San Marcos, Texas, USA

    Dan E. Tamir

  2. School of Computing and Information Sciences, Florida International University, Miami, Florida, USA

    Naphtali D. Rishe

  3. School of Computing and Information Sciences, The University of South Florida, Tampa, Florida, USA, and Florida International University, Miami, Florida, USA

    Abraham Kandel

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kosko, B. (2015). Additive Fuzzy Systems as Generalized Probability Mixture Models. In: Tamir, D., Rishe, N., Kandel, A. (eds) Fifty Years of Fuzzy Logic and its Applications. Studies in Fuzziness and Soft Computing, vol 326. Springer, Cham. https://doi.org/10.1007/978-3-319-19683-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-19683-1_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19682-4

  • Online ISBN: 978-3-319-19683-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics