Skip to main content
SpringerLink
Log in

Generalization of Dempster–Shafer theory: A complex mass function

  • Published:
Applied Intelligence Aims and scope Submit manuscript

Abstract

Dempster–Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster–Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster–Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster’s combination rule is exploited. In contrast to the classical Dempster’s combination rule, the condition in terms of the conflict coefficient between the evidences is released in the generalized Dempster’s combination rule. Hence, it is more general and applicable than the classical Dempster’s combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences is less than 1. In a word, this generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Thanks to this advantage, an algorithm for decision-making is devised based on the generalized Dempster–Shafer evidence theory. Finally, an application in a medical diagnosis illustrates the efficiency and practicability of the proposed algorithm.

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
Fig. 4

Similar content being viewed by others

References

  1. Yager RR (2018) On using the shapley value to approximate the Choquet integral in cases of uncertain arguments. IEEE Trans Fuzzy Syst 26(3):1303–1310

    Google Scholar 

  2. Zavadskas EK, Antucheviciene J, Hajiagha SHR, Hashemi SS (2014) Extension of weighted aggregated sum product assessment with interval-valued intuitionistic fuzzy numbers (WASPAS-IVIF). Appl Soft Comput 24:1013–1021

    Google Scholar 

  3. Fu C, Liu W, Chang W (2018) Data-driven multiple criteria decision making for diagnosis of thyroid cancer. Ann Oper Res pp, 1–30

  4. Fei L (2019) On interval-valued fuzzy decision-making using soft likelihood functions. Int J Intell Syst 34(7):1631–1652

    Google Scholar 

  5. Xiao F (2019) A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems. IEEE Transactions on Systems, Man, and Cybernetics: Systems p. https://doi.org/10.1109/TSMC.2019.2958635

  6. Feng F, Cho J, Pedrycz W, Fujita H, Herawan T (2016) Soft set based association rule mining. Knowl-Based Syst 111:268–282

    Google Scholar 

  7. Wang Y, Zhang K, Deng Y (2019) Base belief function: an efficient method of conflict management. J Ambient Intell Humaniz Comput 10(9):3427–3437

    Google Scholar 

  8. Jiang W (2018) A correlation coefficient for belief functions. Int J Approx Reason 103:94–106

    MathSciNet  MATH  Google Scholar 

  9. Zhao J, Deng Y (2019) Performer selection in human reliability analysis: D numbers approach. Int J Comput Commun Control 14(4):521–536

    MathSciNet  Google Scholar 

  10. Deng X, Jiang W (2019) D number theory based game-theoretic framework in adversarial decision making under a fuzzy environment. Int J Approx Reason 106:194–213

    MathSciNet  MATH  Google Scholar 

  11. Jiang W, Cao Y, Deng X (2019) A novel Z-network model based on Bayesian network and Z-number. IEEE Transactions on Fuzzy Systems p. https://doi.org/10.1109/TFUZZ.2019.2918999

  12. Liu Q, Tian Y, Kang B (2019) Derive knowledge of Z-number from the perspective of Dempster-Shafer evidence theory. Eng Appl Artif Intel 85:754–764

    Google Scholar 

  13. Seiti H, Hafezalkotob A, Martínez L (2019) R-numbers, A new risk modeling associated with fuzzy numbers and its application to decision making. Inform Sci 483:206–231

    MathSciNet  Google Scholar 

  14. Seiti H, Hafezalkotob A, Martinez L (2019) R-sets, comprehensive fuzzy sets risk modeling for risk-based information fusion and decision-making. IEEE Transactions on Fuzzy Systems p. https://doi.org/10.1109/TFUZZ.2019.2955061

  15. Cao Z, Lin CT (2018) Inherent fuzzy entropy for the improvement of EEG complexity evaluation. IEEE Trans Fuzzy Syst 26(2):1032–1035

    MathSciNet  Google Scholar 

  16. Ma J, Yu W, Liang P, Li C, Jiang J (2019) FusionGAN: A generative adversarial network for infrared and visible image fusion. Inf Fusion 48:11–26

    Google Scholar 

  17. Fujita H, Cimr D (2019) Decision support system for arrhythmia prediction using convolutional neural network structure without preprocessing. Appl Intell 49(9):3383–3391

    Google Scholar 

  18. Fujita H, Cimr D (2019) Computer aided detection for fibrillations and flutters using deep convolutional neural network. Inf Sci 486:231–239

    Google Scholar 

  19. Yager RR, Petry FE (2019) Using quality measures in the intelligent fusion of probabilistic information. In: Information quality in information fusion and decision making, pp 51–77. Springer

  20. Seiti H, Hafezalkotob A, Fattahi R (2018) Extending a pessimistic–optimistic fuzzy information axiom based approach considering acceptable risk: Application in the selection of maintenance strategy. Appl Soft Comput 67:895–909

    Google Scholar 

  21. Deng X, Jiang W (2019) Evaluating green supply chain management practices under fuzzy environment: a novel method based on D number theory. Int J Fuzzy Syst 21:1389–1402

    Google Scholar 

  22. Geng J, Ma X, Zhou X, Wang H (2019) Saliency-guided deep neural networks for SAR image change detection. IEEE Trans Geosci Remote Sens 57(10):7365–7377. https://doi.org/10.1109/TGRS.2019.4332913095

    Google Scholar 

  23. Fujita H, Gaeta A, Loia V, Orciuoli F (2019) Improving awareness in early stages of security analysis: A zone partition method based on GrC. Appl Intell 49(3):1063–1077

    Google Scholar 

  24. Xiao F, Zhang Z, Abawajy J (2019) Workflow scheduling in distributed systems under fuzzy environment. Journal of Intelligent & Fuzzy Systems 37(4):5323–5333. https://doi.org/10.3233/JIFS--190483

  25. Zhou D, Al-Durra A, Zhang K, Ravey A, Gao F (2019) A robust prognostic indicator for renewable energy technologies: A novel error correction grey prediction model. IEEE Trans Ind Electron 66:9312–9325

    Google Scholar 

  26. Cao Z, Lin CT, Lai KL, Ko LW, King JT, Liao KK, Fuh JL, Wang SJ (2019) Extraction of SSVEPs-based inherent fuzzy entropy using a wearable headband EEG in migraine patients. IEEE Transactions on Fuzzy Systems p. https://doi.org/10.1109/TFUZZ.2019.2905823

  27. Ma J, Ma Y, Li C (2019) Infrared and visible image fusion methods and applications: a survey. Inf Fusion 45:153–178

    Google Scholar 

  28. Kang B, Zhang P, Gao Z, Chhipi-Shrestha G, Hewage K, Sadiq R (2019) Environmental assessment under uncertainty using Dempster–Shafer theory and Z-numbers. Journal of Ambient Intelligence and Humanized Computing pp. https://doi.org/10.1007/s12652--019--01228--y

  29. Feng F, Fujita H, Ali MI, Yager RR, Liu X (2018) Another view on generalized intuitionistic fuzzy soft sets and related multiattribute decision making methods. IEEE Trans Fuzzy Syst 27(3):474–488

    Google Scholar 

  30. Xiao F (2019) A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion. Information Sciences 514:462–339. https://doi.org/10.1016/j.ins.2019.11.022

    MathSciNet  Google Scholar 

  31. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38 (2):325–339

    MathSciNet  MATH  Google Scholar 

  32. Shafer G, et al. (1976) A mathematical theory of evidence, vol 1, Princeton University Press, Princeton

  33. Deng X, Jiang W, Wang Z (2019) Zero-sum polymatrix games with link uncertainty: A Dempster-Shafer theory solution. Appl Math Comput 340:101–112

    MathSciNet  MATH  Google Scholar 

  34. Su X, Li L, Qian H, Sankaran M, Deng Y (2019) A new rule to combine dependent bodies of evidence. Soft Comput 23(20):9793–9799

    MATH  Google Scholar 

  35. Yager RR, Elmore P, Petry F (2017) Soft likelihood functions in combining evidence. Inf Fusion 36:185–190

    Google Scholar 

  36. Su X, Li L, Shi F, Qian H (2018) Research on the fusion of dependent evidence based on mutual information. IEEE Access 6:71839–71845

    Google Scholar 

  37. Yager RR (2018) Satisfying uncertain targets using measure generalized Dempster-Shafer belief structures. Knowl-Based Syst 142:1–6

    Google Scholar 

  38. Seiti H, Hafezalkotob A, Najafi S, Khalaj M (2018) A risk-based fuzzy evidential framework for FMEA analysis under uncertainty: An interval-valued DS approach. Journal of Intelligent & Fuzzy Systems (Preprint), 1–12

  39. Yager RR (2019) Generalized Dempster–Shafer structures. IEEE Trans Fuzzy Syst 27(3):428–435

    MathSciNet  Google Scholar 

  40. Wang H, Deng X, Zhang Z, Jiang W (2019) A new failure mode and effects analysis method based on Dempster–Shafer theory by integrating evidential network. IEEE Access 7:79579– 79591

    Google Scholar 

  41. Yager RR (2018) Fuzzy rule bases with generalized belief structure inputs. Eng Appl Artif Intel 72:93–98

    Google Scholar 

  42. Zavadskas EK, Bausys R, Juodagalviene B, Garnyte-Sapranaviciene I (2017) Model for residential house element and material selection by neutrosophic MULTIMOORA method. Eng Appl Artif Intel 64:315–324

    Google Scholar 

  43. Zhou M, Liu XB, Chen YW, Yang JB (2018) Evidential reasoning rule for MADM with both weights and reliabilities in group decision making. Knowl-Based Syst 143:142–161

    Google Scholar 

  44. Liao H, Jiang L, Lev B, Fujita H (2019) Novel operations of PLTSs based on the disparity degrees of linguistic terms and their use in designing the probabilistic linguistic ELECTRE III method. Appl Soft Comput 80:450–464

    Google Scholar 

  45. Yager RR (2018) Multi-criteria decision making with interval criteria satisfactions using the golden rule representative value. IEEE Trans Fuzzy Syst 26(2):1023–1031

    Google Scholar 

  46. Feng F, Liang M, Fujita H, Yager RR, Liu X (2019) Lexicographic orders of intuitionistic fuzzy values and their relationships. Mathematics 7(2):1–26

    Google Scholar 

  47. Wang X, Song Y (2018) Uncertainty measure in evidence theory with its applications. Appl Intell 48 (7):1672–1688

    Google Scholar 

  48. Xiao F (2018) A novel multi-criteria decision making method for assessing health-care waste treatment technologies based on d numbers. Eng Appl Artif Intel 71:216–225

    Google Scholar 

  49. Gao X, Liu F, Pan L, Deng Y, Tsai SB (2019) Uncertainty measure based on Tsallis entropy in evidence theory. Int J Intell Syst 34(11):3105–3120

    Google Scholar 

  50. Gong Y, Su X, Qian H, Yang N (2018) Research on fault diagnosis methods for the reactor coolant system of nuclear power plant based on D-S evidence theory. Ann Nucl Energy 112:395–399

    Google Scholar 

  51. Zhang H, Deng Y (2020) Weighted belief function of sensor data fusion in engine fault diagnosis. Soft Computing 24(3):2329–2339. https://doi.org/10.1007/s00500--019--04063--7

    MathSciNet  Google Scholar 

  52. Liu Z, Liu Y, Dezert J, Cuzzolin F (2019) Evidence combination based on credal belief redistribution for pattern classification. IEEE Transactions on Fuzzy Systems p. https://doi.org/10.1109/TFUZZ.2019.2911915

  53. Liu Z, Zhang Z, Liu Y, Dezert J, Pan Q (2019) A new pattern classification improvement method with local quality matrix based on K-NN. Knowl-Based Syst 164:336–347

    Google Scholar 

  54. Xu X, Zheng J, Yang J, Xu D, Chen YW (2017) Data classification using evidence reasoning rule. Knowl-Based Syst 116:144–151

    Google Scholar 

  55. Liu Z, Pan Q, Dezert J, Han JW, He Y (2018) Classifier fusion with contextual reliability evaluation. IEEE Trans Cybern 48(5):1605–1618

    Google Scholar 

  56. Xu X, Li S, Song X, Wen C, Xu D (2016) The optimal design of industrial alarm systems based on evidence theory. Control Eng Pract 46:142–156

    Google Scholar 

  57. Xu X, Xu H, Wen C, Li J, Hou P, Zhang J (2018) A belief rule-based evidence updating method for industrial alarm system design. Control Eng Pract 81:73–84

    Google Scholar 

  58. Song Y, Wang X, Zhu J, Lei L (2018) Sensor dynamic reliability evaluation based on evidence theory and intuitionistic fuzzy sets. Appl Intell 48(11):3950–3962

    Google Scholar 

  59. Fan CL, Song Y, Lei L, Wang X, Bai S (2018) Evidence reasoning for temporal uncertain information based on relative reliability evaluation. Expert Syst Appl 113:264–276

    Google Scholar 

  60. Jiang W, Huang C, Deng X (2019) A new probability transformation method based on a correlation coefficient of belief functions. Int J Intell Syst 34:1337–1347

    Google Scholar 

  61. Yager RR (2019) Entailment for measure based belief structures. Inf Fusion 47:111–116

    Google Scholar 

  62. Xiao F (2019) Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Inf Fusion 46(2019):23–32

    Google Scholar 

  63. Zhang W, Deng Y (2019) Combining conflicting evidence using the DEMATEL method. Soft Comput 23:8207–8216

    Google Scholar 

  64. Xiao F (2019) EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy. IEEE Transactions on Fuzzy Systems p. https://doi.org/10.1109/TFUZZ.2019.2936368

  65. Sun R, Deng Y (2019) A new method to determine generalized basic probability assignment in the open world. IEEE Access 7(1):52827–52835

    Google Scholar 

  66. Gao X, Deng Y (2019) The negation of basic probability assignment. IEEE Access 7(1):107006–107014

    Google Scholar 

  67. Luo Z, Deng Y (2019) A matrix method of basic belief assignment’s negation in Dempster-Shafer theory. IEEE Trans Fuzzy Syst 27:1–1. https://doi.org/10.1109/TFUZZ.2019.2930027

    Google Scholar 

  68. Han D, Dezert J, Yang Y (2016) Belief interval-based distance measures in the theory of belief functions. IEEE Trans Syst, Man, and Cybern Syst 48(6):833–850

    Google Scholar 

  69. Song Y, Wang X, Lei L, Yue S (2016) Uncertainty measure for interval-valued belief structures. Measurement 80:241–250

    Google Scholar 

  70. Song Y, Deng Y (2019) Divergence measure of belief function and its application in data fusion. IEEE Access 7(1):107465–107472

    Google Scholar 

  71. Cui H, Liu Q, Zhang J, Kang B (2019) An improved deng entropy and its application in pattern recognition. IEEE Access 7:18284–18292

    Google Scholar 

  72. Yager RR (2008) Entropy and specificity in a mathematical theory of evidence. In: Classic works of the Dempster-Shafer theory of belief functions, pp 291–310. Springer

  73. Li M, Xu H, Deng Y (2019) Evidential decision tree based on belief entropy. Entropy 21(9):897

    Google Scholar 

  74. Dezert J, Tchamova A, Han D (2018) Total belief theorem and conditional belief functions. Int J Intell Syst 33(12):2314–2340

    Google Scholar 

  75. Jiang W, Zhang Z, Deng X (2019) A novel failure mode and effects analysis method based on fuzzy evidential reasoning rules. IEEE Access 7:113605–113615

    Google Scholar 

  76. Li Y, Deng Y (2019) TDBF: Two dimension belief function. Int J Intell Syst 34(8):1968–1982

    Google Scholar 

  77. Liu ZG, Pan Q, Dezert J, Martin A (2018) Combination of classifiers with optimal weight based on evidential reasoning. IEEE Trans Fuzzy Syst 26(3):1217–1230

    Google Scholar 

  78. liu B, Deng Y (2019) Risk evaluation in failure mode and effects analysis based on D numbers theory. Int J Comput Commun Control 14(5):672–691

    Google Scholar 

  79. Yager RR, Alajlan N (2017) Maxitive belief structures and imprecise possibility distributions. IEEE Trans Fuzzy Syst 25(4):768–774

    Google Scholar 

  80. Pan L, Deng Y (2020) An association coefficient of belief function and its application in target recognition system. Int J Intell Syst 35:85–104

    Google Scholar 

  81. Huang Z, Yang L, Jiang W (2019) Uncertainty measurement with belief entropy on the interference effect in the quantum-like Bayesian Networks. Appl Math Comput 347:417–428

    MathSciNet  MATH  Google Scholar 

  82. Zhou M, Liu XB, Yang JB, Chen YW, Wu J (2019) Evidential reasoning approach with multiple kinds of attributes and entropy-based weight assignment. Knowl-Based Syst 163:358–375

    Google Scholar 

  83. Fu C, Chang W, Xue M, Yang S (2019) Multiple criteria group decision making with belief distributions and distributed preference relations. Eur J Oper Res 273(2):623–633

    MathSciNet  Google Scholar 

  84. Zhou M, Liu X, Yang J (2017) Evidential reasoning approach for MADM based on incomplete interval value. J Intell Fuzzy Syst 33(6):3707–3721

    Google Scholar 

  85. Dezert J, Han D, Tacnet JM, Carladous S, Yang Y (2016) Decision-making with belief interval distance. In: International conference on belief functions, pp 66–74. Springer

  86. Garg H, Rani D (2019) A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making. Appl Intell 49(2):496–512

    Google Scholar 

Download references

Acknowledgements

The author greatly appreciates the reviewers’ suggestions and the editor’s encouragement. This research is supported by the Fundamental Research Funds for the Central Universities (No. XDJK2019C085) and Chongqing Overseas Scholars Innovation Program (No. cx2018077).

Author information

Authors and Affiliations

  1. School of Computer and Information Science, Southwest University, No. 2 Tiansheng Road, BeiBei District, Chongqing, 400715, China

    Fuyuan Xiao

Authors
  1. Fuyuan Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fuyuan Xiao.

Ethics declarations

Conflict of interests

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, F. Generalization of Dempster–Shafer theory: A complex mass function. Appl Intell 50, 3266–3275 (2020). https://doi.org/10.1007/s10489-019-01617-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10489-019-01617-y

Keywords

Search

Navigation