Skip to main content
SpringerLink
Log in

A belief classification rule for imprecise data

  • Published:
Applied Intelligence Aims and scope Submit manuscript

Abstract

The classification of imprecise data is a difficult task in general because the different classes can partially overlap. Moreover, the available attributes used for the classification are often insufficient to make a precise discrimination of the objects in the overlapping zones. A credal partition (classification) based on belief functions has already been proposed in the literature for data clustering. It allows the objects to belong (with different masses of belief) not only to the specific classes, but also to the sets of classes called meta-classes which correspond to the disjunction of several specific classes. In this paper, we propose a new belief classification rule (BCR) for the credal classification of uncertain and imprecise data. This new BCR approach reduces the misclassification errors of the objects difficult to classify by the conventional methods thanks to the introduction of the meta-classes. The objects too far from the others are considered as outliers. The basic belief assignment (bba) of an object is computed from the Mahalanobis distance between the object and the center of each specific class. The credal classification of the object is finally obtained by the combination of these bba’s associated with the different classes. This approach offers a relatively low computational burden. Several experiments using both artificial and real data sets are presented at the end of this paper to evaluate and compare the performances of this BCR method with respect to other classification methods.

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
Fig. 3

Similar content being viewed by others

Notes

  1. In this work, our classification method is a supervised learning technique because it classifies the objects using a given training data set. The unsupervised learning technique used for the cluster analysis without training data is called a clustering method.

  2. I.e. the set of the specific classes that the objects are close to.

  3. The center of a specific class is calculated by the simple arithmetic average value of the training data belonging to this class.

  4. This rule is mathematically defined only if the denominator is strictly positive, i.e. the sources are not totally conflicting.

  5. The subsets A of Θ such that m(A)>0 are called the focal elements of m(⋅).

  6. There exist other methods of construction of bba’s [29, 30], but they need more parameters to tune, and have a higher computation complexity which makes them not easy to use.

  7. The focal elements of each bba are nested.

References

  1. Silveman BW, Jones MC, Fix E, Hodges JL (1951) An important contribution to nonparametric discriminant analysis and density estimation. Int Stat Rev 57(3):233–247. Dec 1989

    Article  Google Scholar 

  2. Bishop CM (1995) Neural networks for pattern recognition. Oxford University Press, Oxford

    Google Scholar 

  3. Lee LH, Wan CH, Rajkumar R, Isa D (2012) An enhanced support vector machine classification framework by using Euclidean distance function for text document categorization. Appl Intell 37(1):80–99

    Article  Google Scholar 

  4. Verma B, Hassan SZ (2011) Hybrid ensemble approach for classification. Appl Intell 34(2):258–278

    Article  Google Scholar 

  5. Dudani SA (1976) The distance-weighted k-nearest-neighbor rule. IEEE Trans Syst Man Cybern SMC 6:325–327

    Article  Google Scholar 

  6. Kim KJ, Ahn H (2012) Simultaneous optimization of artificial neural networks for financial forecasting. Appl Intell 36(4):887–898

    Article  MathSciNet  Google Scholar 

  7. Jousselme A-L, Maupin P, Bossé E (2003) Uncertainty in a situation analysis perspective. In: Proc of fusion 2003 int conf, July, pp 1207–1213

    Google Scholar 

  8. Jousselme A-L, Liu C, Grenier D, Bossé E (2006) Measuring ambiguity in the evidence theory. IEEE Trans Syst Man Cybern, Part A 36(5):890–903. (See also comments in [9])

    Article  Google Scholar 

  9. Klir G, Lewis H (2008) Remarks on “Measuring ambiguity in the evidence theory”. IEEE Trans Syst Man Cybern, Part A 38(4):895–999

    Article  Google Scholar 

  10. Shafer G (1976) A mathematical theory of evidence. Princeton Univ. Press, Princeton

    MATH  Google Scholar 

  11. Smarandache F, Dezert J (eds) (2004–2009) Advances and applications of DSmT for information fusion, vols 1–3. American Research Press, Rehoboth. http://fs.gallup.unm.edu/DSmT.htm

    Google Scholar 

  12. Smets P (1990) The combination of evidence in the transferable belief model. IEEE Trans Pattern Anal Mach Intell 12(5):447–458

    Article  Google Scholar 

  13. Smets P, Kennes R (1994) The transferable belief model. Artif Intell 66(2):191–243

    Article  MATH  MathSciNet  Google Scholar 

  14. Denœux T (1995) A k-nearest neighbor classification rule based on Dempster-Shafer theory. IEEE Trans Syst Man Cybern 25(5):804–813

    Article  Google Scholar 

  15. Denœux T, Smets P (2006) Classification using belief functions: relationship between case-based and model-based approaches. IEEE Trans Syst Man Cybern, Part B 36(6):1395–1406

    Article  Google Scholar 

  16. Sutton-Charani N, Destercke S, Denœux T (2012) Classification trees based on belief functions. In: Denoeux T, Masson M-H (eds) Belief functions: theory and appl. AISC, vol 164, pp 77–84

    Chapter  Google Scholar 

  17. Karem F, Dhibi M, Martin A (2012) Combination of supervised and unsupervised classification using the theory of belief functions. In: Denoeux T, Masson M-H (eds) Belief functions: theory and appl. AISC, vol 164, pp 85–92

    Chapter  Google Scholar 

  18. Kanj S, Abdallah F, Denœux T (2012) Evidential multi-label classification using the random k-label sets approach. In: Denoeux T, Masson M-H (eds) Belief functions: theory and appl. AISC, vol 164, pp 21–28

    Chapter  Google Scholar 

  19. Lu Y (1996) Knowledge integration in a multiple classifier system. Appl Intell 6(2):75–86

    Article  Google Scholar 

  20. Denœux T, Masson M-H (2004) EVCLUS: EVidential CLUStering of proximity data. IEEE Trans Syst Man Cybern, Part B 34(1):95–109

    Article  Google Scholar 

  21. Masson M-H, Denœux T (2008) ECM: an evidential version of the fuzzy c-means algorithm. Pattern Recognit 41(4):1384–1397

    Article  MATH  Google Scholar 

  22. Masson M-H, Denœux T (2004) Clustering interval-valued data using belief functions. Pattern Recognit Lett 25(2):163–171

    Article  Google Scholar 

  23. Liu Z-g, Dezert J, Mercier G, Pan Q (2012) Belief C-means: an extension of fuzzy C-means algorithm in belief functions framework. Pattern Recognit Lett 33(3):291–300

    Article  Google Scholar 

  24. Liu Z-g, 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 52:133–141

    Article  Google Scholar 

  25. Li XD, Dai XZ, Dezert J (2010) Fusion of imprecise qualitative information. Appl Intell 33(3):340–351

    Article  Google Scholar 

  26. Denoeux T (2000) A neural network classifier based on Dempster-Shafer theory. IEEE Trans Syst Man Cybern A 30(2):131–150

    Article  MathSciNet  Google Scholar 

  27. Dubois D, Prade H (1988) Representation and combination of uncertainty with belief functions and possibility measures. Comput Intell 4(4):244–264

    Article  Google Scholar 

  28. Altincay H (2006) On the independence requirement in Dempster-Shafer theory for combining classifiers providing statistical evidence. Appl Intell 25(1):73–90

    Article  Google Scholar 

  29. Dezert J, Liu Z-g, Mercier G (2011) Edge detection in color images based on DSmT. In: Proceedings of fusion 2011 int conf, Chicago, USA, July 2011

    Google Scholar 

  30. Klein J, Colot O (2012) A belief function model for pixel data. In: Denoeux T, Masson M-H (eds) Belief functions: theory and appl AISC, vol 164, pp 189–196

    Chapter  Google Scholar 

  31. Zouhal LM, Denœux T (1998) An evidence-theoretic k-NN rule with parameter optimization. IEEE Trans Syst Man Cybern, Part C 28(2):263–271

    Article  Google Scholar 

  32. Frank A, Asuncion A (2010) UCI Machine Learning Repository [http://archive.ics.uci.edu/ml]. University of California, School of Information and Computer Science, Irvine, CA, USA

  33. Geisser S (1993) Predictive inference: an introduction. Chapman and Hall, New York

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work has been partially supported by National Natural Science Foundation of China (Nos. 61075029, 61135001) and Ph.D. Thesis Innovation Fund from Northwestern Polytechnical University (No. cx201015).

Author information

Authors and Affiliations

  1. School of Automation, Northwestern Polytechnical University, Xi’an, China

    Zhun-ga Liu & Quan Pan

  2. ONERA—The French Aerospace Lab, 91761, Palaiseau, France

    Jean Dezert

Authors
  1. Zhun-ga Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Quan Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean Dezert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhun-ga Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Zg., Pan, Q. & Dezert, J. A belief classification rule for imprecise data. Appl Intell 40, 214–228 (2014). https://doi.org/10.1007/s10489-013-0453-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10489-013-0453-5

Keywords

Search

Navigation