Skip to main content
SpringerLink
Log in

A new distance between BPAs based on the power-set-distribution pignistic probability function

  • Published:
Applied Intelligence Aims and scope Submit manuscript

Abstract

In Dempster-Shafer evidence theory, the pignistic probability function is used to transform the basic probability assignment (BPA) into pignistic probabilities. Since the transformation is from the power set of the frame of discernment to the set itself, it may cause some information loss. The distance between betting commitments is constructed on the basis of the pignistic probability function and is used to measure the dissimilarity between two BPAs. However, it is a pseudo-metric and it may bring unreasonable results in some cases. To solve such problem, we propose a power-set-distribution (PSD) pignistic probability function based on the new explanation of the non-singleton focal elements in the BPA. The new function is directly operated on the power set, so it takes more information contained in the BPA than the pignistic probability function does. Based on the new function, the distance between PSD betting commitments which can better measure the dissimilarity between two BPAs is also proposed, and the proof that it is a metric is provided. In order to demonstrate the performance of the new distance, numerical examples are given to compare it with three existing dissimilarity measures. Moreover, its applications in combining the conflicting BPAs are also presented through two examples.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38 (1):325–339

    Article  MathSciNet  MATH  Google Scholar 

  2. Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton

    MATH  Google Scholar 

  3. Zadeh L (1984) Review of a mathematical theory of evidence. AI Mag 5(2):235–247

    Google Scholar 

  4. Zadeh L (1986) A simple view of the Dempster–Shafer theory of evidence and its implication for the rule of combination. AI Mag 7(1):85–90

    Google Scholar 

  5. Liu W (2006) Analyzing the degree of conflict among belief functions. Artif Intell 170(9):909–924

    Article  MathSciNet  MATH  Google Scholar 

  6. Jousselme AL, Maupin P (2012) Distances in evidence theory: Comprehensive survey and generalizations. Int J Approx Reason 53(1):118–145

    Article  MathSciNet  MATH  Google Scholar 

  7. Jousselme AL, Maupin P On some properties of distances in evidence theory Workshop on the theory of belief functions, Brest, France, p 2010

  8. Perry WL, Stephanou HE (1991) Belief function divergence as a classifier Proceedings of the 1991 IEEE international symposium on intelligent control, Arlington, VA, USA

  9. Blackman S, Popoli R (1999) Design and analysis of modern tracking systems. Artech House, Norwood

    MATH  Google Scholar 

  10. Tessem B (1993) Approximations for efficient computation in the theory of evidence. Artif Intell 61(1):315–329

    Article  MathSciNet  Google Scholar 

  11. Jousselme AL, Grenier D, Bossé É. (2001) A new distance between two bodies of evidence. Information Fusion 2(1):91–101

    Article  Google Scholar 

  12. Cuzzolin F Consistent approximations of belief functions 6Th international symposium on imprecise probability: Theories and applications, Durham, UK, p 2009

  13. Zouhal L, Denoeux T (1998) An evidence-theoretic k–NN rule with parameter optimization. IEEE Trans Syst Man Cybern Part C Appl Rev 28(1):263–271

    Article  Google Scholar 

  14. Ristic B, Smets P (2006) The TBM global distance measure for the association of uncertain combat ID declarations. Information Fusion 7:276–284

    Article  Google Scholar 

  15. Wen C, Wang Y, Xu X (2008) Fuzzy information fusion algorithm of fault diagnosis based on similarity measure of evidence Advances in neural networks, lecture notes in computer science, vol 5264. Springer, Berlin, pp 506–515

  16. Florea MC, Bossé E. Crisis management using Dempster Shafer theory: using dissimilarity measures to characterize sources’ reliability C3i in crisis, emergency and consequence management, RTO–MP–IST–086, Bucharest, Romania, p 2009

  17. Hellinger E (1909) Neaue begründung der theorie der quadratischen formen von unendlichen vielen veränderlichen. Journal fü,r die Reine und Aangewandte Mathematik 136:210–271

    MathSciNet  MATH  Google Scholar 

  18. Denoeux T (2001) Inner and outer approximation of belief structures using a hierarchical clustering approach. Int J Uncertainty Fuzziness Knowledge Based Syst 9(3):437–460

    Article  MathSciNet  MATH  Google Scholar 

  19. Denoeux T (1997) A neural network classifier based on Dempster–Shafer theory. IEEE Trans Syst Man Cybern Syst Hum 30(1):131–150

    Google Scholar 

  20. Han DQ, Deng Y, Han CZ, et al. (2012) Some notes on betting commitment distance in evidence theory. Science China–Information Sciences 55(2):558–565

    Article  MathSciNet  Google Scholar 

  21. Smets Ph, Kennes R (1994) The transferable belief model. Artif Intell 66(1):191–234

    Article  MathSciNet  MATH  Google Scholar 

  22. Smets Ph (1990) The combination of evidence in the transferable belief model. IEEE Trans Pattern Anal Mach Intell 12(4):447–458

    Article  Google Scholar 

  23. Chen S, Deng Y, Wu J (2013) Fuzzy sensor fusion based on evidence theory and its application. Appl Artif Intell 27(2):235–248

    Article  Google Scholar 

  24. Du HL, Lv F, Du N (2012) A method of multi–classifier combination based on Dempster–Shafer evidence theory and the application in the fault diagnosis. Adv Mater Res 490–495:1402– 1406

    Google Scholar 

  25. Huang SY, Su XY, Hu Y, et al. (2014) A new decision–making method by incomplete preferences based on evidence distance. Knowl-Based Syst 56(2):264–272

    Article  Google Scholar 

  26. Cuzzolin F (2004) Geometry of Dempster’s rule of combination. IEEE Trans Syst Man Cybern B Cybern 34 (1):961–977

    Article  Google Scholar 

  27. Cuzzolin F (2008) A geometric approach to the theory of evidence. IEEE Trans Syst Man Cybern Part C Appl Rev 38(3):522– 534

    Article  Google Scholar 

  28. Bouchard M, Jousselme AL, Dore PE (2013) A proof for the positive definiteness of the Jaccard index matrix. Int J Approx Reason 54(4):615–626

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu ZG, Dezert J, Pan Q, Mercier G (2011) Combination of sources of evidence with different discounting factors based on a new dissimilarity measure. Decis Support Syst 52(1):133–141

    Article  Google Scholar 

  30. Deng Y, Shi WK, Zhu ZF, Liu Q (2004) Combining belief functions based on distance of evidence. Decis Support Syst 38(2):489–493

    Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China under grants 61273275, 60975026 and 61573375. The authors also want to thank the anonymous reviewers for their detailed comments that have helped greatly in improving the quality of this paper.

Author information

Authors and Affiliations

  1. Air Force Engineering University, Xi’an, 710051, People’s Republic of China

    Jingwei Zhu, Xiaodan Wang & Yafei Song

Authors
  1. Jingwei Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaodan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yafei Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaodan Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, J., Wang, X. & Song, Y. A new distance between BPAs based on the power-set-distribution pignistic probability function. Appl Intell 48, 1506–1518 (2018). https://doi.org/10.1007/s10489-017-1018-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10489-017-1018-9

Keywords

Search

Navigation